Join irreducible semigroups

TL;DR

This paper systematically studies finite semigroups that generate join irreducible pseudovarieties, characterizes them, introduces operators preserving this property, and describes all such pseudovarieties generated by small semigroups, expanding existing theory.

Contribution

It introduces a new operator preserving join irreducibility, characterizes finite semigroups generating join irreducible pseudovarieties, and classifies all such pseudovarieties from semigroups of order up to five.

Findings

Characterization of semigroups generating join irreducible pseudovarieties.

Introduction of an operator that preserves join irreducibility.

Complete classification of join irreducible pseudovarieties from small semigroups.

Abstract

We begin a systematic study of finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups $S$ that generate join irreducible pseudovarieties are characterized as follows: whenever $S$ divides a direct product $A \times B$ of finite semigroups, then $S$ divides either $A^n$ or $B^n$ for some $n \geq 1$. We present a new operator ${ \mathbf{V} \mapsto \mathbf{V}^\mathsf{bar} }$ that preserves the property of join irreducibility, as does the dual operator, and show that iteration of these operators on any nontrivial join irreducible pseudovariety leads to an infinite hierarchy of join irreducible pseudovarieties. We also describe all join irreducible pseudovarieties generated by a semigroup of order up to five. It turns out that there are $30$ such pseudovarieties, and there is a relatively easy way to remember them. In addition, we survey most results known about join irreducible pseudovarieties to date and generalize a number of results in Sec. 7.3 of The $q$-theory of Finite Semigroups, Springer Monographs in Mathematics (Springer, Berlin, 2009).

Tables

Table 1. Table 1. Some generators of primitive 𝗃𝗂 𝗃𝗂 \mathsf{ji} pseudovarieties

$n$ Semigroups of order $n$ that generate primitive $\mathrm{𝗃𝗂}$ pseudovarieties 2 ${\mathbb{Z}}_2$, $N_2$, $L_2$ 3 ${\mathbb{Z}}_3$, $N_3$ 4 ${\mathbb{Z}}_4$, $N_4$, $A_0$ 5 ${\mathbb{Z}}_5$, $N_5$, $A_2$, $B_2$, ${\ell }_3^{\mathrm{𝖻𝖺𝗋}}$

Table 2. Table 2. Multiplication tables of  ℤ 2 𝖻𝖺𝗋 superscript subscript ℤ 2 𝖻𝖺𝗋 \mathbb{Z}_{2}^{\mathsf{bar}} and  N 2 𝖻𝖺𝗋 superscript subscript 𝑁 2 𝖻𝖺𝗋 N_{2}^{\mathsf{bar}}

Table 3. Table 3. Multiplication tables of S ​ l 2 𝑆 subscript 𝑙 2 Sl_{2} , L 2 subscript 𝐿 2 L_{2} , and  R 2 subscript 𝑅 2 R_{2}

Table 4. Table 4. Multiplication table of L 2 𝖻𝖺𝗋 superscript subscript 𝐿 2 𝖻𝖺𝗋 L_{2}^{\mathsf{bar}}

Table 5. Table 5. Multiplication tables of A 2 subscript 𝐴 2 A_{2} and  B 2 subscript 𝐵 2 B_{2}

Table 6. Table 6. Multiplication tables of A 0 subscript 𝐴 0 A_{0} and B 0 subscript 𝐵 0 B_{0}

Table 7. Table 7. Multiplication tables of  ℓ 3 subscript ℓ 3 \ell_{3} and ℓ 3 𝖻𝖺𝗋 superscript subscript ℓ 3 𝖻𝖺𝗋 \ell_{3}^{\mathsf{bar}}

Table 8. Table 8. Multiplication table of ( ℓ 3 𝖻𝖺𝗋 ) 𝗈𝗉 superscript superscript subscript ℓ 3 𝖻𝖺𝗋 𝗈𝗉 (\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}

Table 9. Table 9. Multiplication table of T 6 subscript 𝑇 6 T_{6}

Table 10. Table 10. Multiplication table of W 𝑊 W

Table 11. Table 11. Multiplication table of T 11 subscript 𝑇 11 T_{11}

Table 12. Table 12. Results for join irreducibility of pseudovarieties in Theorem  7.1

Pseudovarieties Join irreducible by $⟨⟨{\mathbb{Z}}_2⟩⟩$, $⟨⟨{\mathbb{Z}}_3⟩⟩$, $⟨⟨{\mathbb{Z}}_4⟩⟩$, $⟨⟨{\mathbb{Z}}_5⟩⟩$ Theorem 5.3 $⟨⟨{\mathbb{Z}}_2^{\mathrm{𝖻𝖺𝗋}}⟩⟩$, $⟨⟨{\left({\mathbb{Z}}_2^{\mathrm{𝖻𝖺𝗋}}\right)}^{\mathrm{𝗈𝗉}}⟩⟩$ Theorem 5.5 $⟨⟨N_2⟩⟩$, $⟨⟨N_3⟩⟩$, $⟨⟨N_4⟩⟩$, $⟨⟨N_5⟩⟩$ Theorem 5.7 $⟨⟨N_1^I⟩⟩$, $⟨⟨N_2^I⟩⟩$, $⟨⟨N_3^I⟩⟩$, $⟨⟨N_4^I⟩⟩$ Theorem 5.9 $⟨⟨N_2^{\mathrm{𝖻𝖺𝗋}}⟩⟩$, $⟨⟨{\left(N_2^{\mathrm{𝖻𝖺𝗋}}\right)}^{\mathrm{𝗈𝗉}}⟩⟩$ Theorem 5.11 $⟨⟨{\left(N_2^{\mathrm{𝖻𝖺𝗋}}\right)}^I⟩⟩$, $⟨⟨{\left({\left(N_2^{\mathrm{𝖻𝖺𝗋}}\right)}^I\right)}^{\mathrm{𝗈𝗉}}⟩⟩$ Theorem 5.13 $⟨⟨L_2⟩⟩$, $⟨⟨L_2^{\mathrm{𝗈𝗉}}⟩⟩$ Theorem 5.15 $⟨⟨L_2^I⟩⟩$, $⟨⟨{\left(L_2^I\right)}^{\mathrm{𝗈𝗉}}⟩⟩$ Theorem 5.17 $⟨⟨L_2^{\mathrm{𝖻𝖺𝗋}}⟩⟩$, $⟨⟨{\left(L_2^{\mathrm{𝖻𝖺𝗋}}\right)}^{\mathrm{𝗈𝗉}}⟩⟩$ Theorem 5.19 $⟨⟨A_0⟩⟩$ Theorem 5.21 $⟨⟨A_0^I⟩⟩$ Theorem 5.23 $⟨⟨A_2⟩⟩$ Theorem 5.25 $⟨⟨B_2⟩⟩$ Theorem 5.27 $⟨⟨{\ell }_3^{\mathrm{𝖻𝖺𝗋}}⟩⟩$, $⟨⟨{\left({\ell }_3^{\mathrm{𝖻𝖺𝗋}}\right)}^{\mathrm{𝗈𝗉}}⟩⟩$ Theorem 5.29

Table 13. Table 13. Number of 𝗃𝗂 𝗃𝗂 \mathsf{ji} semigroups of order up to five

$n=2$ $n=3$ $n=4$ $n=5$ $2\le n\le 5$ Number of $\mathrm{𝗃𝗂}$ 4 8 33 196 241 semigroups of order $n$ Number of non-$\mathrm{𝗃𝗂}$ 0 10 93 964 1067 semigroups of order $n$ Number of 4 18 126 1160 1308 semigroups of order $n$

Equations

$$\ell \le \bigvee \mathcal{X}⟹\ell \le \bigvee \mathcal{F}\text{ for some finite }\mathcal{F}\subseteq \mathcal{X};$$

$$\ell \le \bigvee \mathcal{X}⟹\ell \le x\text{ for some }x\in \mathcal{X};$$

$$\ell \le \bigvee \mathcal{F}⟹\ell \le x\text{ for some }x\in \mathcal{F};$$

$$\ell \ge \bigwedge \mathcal{X}⟹\ell \ge x\text{ for some }x\in \mathcal{X};$$

$$\ell \ge \bigwedge \mathcal{F}⟹\ell \ge x\text{ for some }x\in \mathcal{F};$$

$$\ell =\bigvee \mathcal{X}⟹\ell \in \mathcal{X};$$

$$\ell =\bigvee \mathcal{F}⟹\ell \in \mathcal{F};$$

$$\ell =\bigwedge \mathcal{X}⟹\ell \in \mathcal{X};$$

$$\ell =\bigwedge \mathcal{F}⟹\ell \in \mathcal{F}.$$

$$\mathbf{V}=\bigvee \{⟨⟨S⟩⟩\mid S\in \mathbf{V}\}.$$

$$S\prec T_1\times T_2⟹S\prec T_1^n\text{ or }S\prec T_2^n\text{ for some }n\ge 1,$$

$$S\prec T_1\times T_2⟹S\prec T_1\text{ or }S\prec T_2;$$

$$\{\text{KN semigroups}\}⫋\{\text{×-prime semigroups}\}⫋\{\text{ji semigroups}\}$$

$$S\mapsto \{\begin{array}{ll}1& \text{if ⟨⟨S⟩⟩ is sji}\\ 0& \text{otherwise}\end{array}$$

$$S\mapsto \{\begin{array}{ll}1& \text{if ⟨⟨S⟩⟩ is ji}\\ 0& \text{otherwise}\end{array}$$

$$\mathbf{DS}=\big{[}\big{[}((xy)^{\omega}(yx)^{\omega}(xy)^{\omega})^{\omega}\approx(xy)^{\omega}\big{]}\big{]}$$

$$\mathbf{V}\mapsto {\mathbf{V}}^{\mathsf{op}}=\{S^{\mathsf{op}}\mid S\in \mathbf{V}\}$$

$$S^{\bullet }=\{\begin{array}{ll}S& \text{if S is a monoid},\\ S^I& \text{otherwise}.\end{array}$$

$$⟨⟨S^{\mathsf{bar}}⟩⟩,⟨⟨(S^{\mathsf{bar}}{)}^{♭}⟩⟩,⟨⟨((S^{\mathsf{bar}}{)}^{♭}{)}^{\mathsf{bar}}⟩⟩,⟨⟨(((S^{\mathsf{bar}}{)}^{♭}{)}^{\mathsf{bar}}{)}^{♭}⟩⟩,\dots ,$$

$$(x^{\omega }{y}^{\omega }{)}^{n+\omega }\approx (x^{\omega }{y}^{\omega }{)}^n\text{and}(x^{\omega }{y}^{\omega }{)}^{n+\omega }{x}^{\omega }\approx (x^{\omega }{y}^{\omega }{)}^n{x}^{\omega },$$

$$\mathbf{Com}=(\mathbf{Com}\cap \mathbf{G})\vee (\mathbf{Com}\cap \mathbf{A}),$$

$$S=\mathscr{M}^{0}\big{(}\mathbb{Z}_{2},\{1,2\},\{1,2\};\big{[}\begin{smallmatrix}1&0\\ 0&1\end{smallmatrix}\big{]}\big{)}$$

$$(X,S)\times (Y,T)=(X\times Y,S\times T)$$

$$\overline{(X,S)}\prec \overline{(S^{\bullet },S)}\prec {\overline{(X,S)}}^{|X|}.$$

$$(S^{\bullet },S^{\mathsf{bar}})=\overline{(S^{\bullet },S)}\prec \overline{(T^{\bullet },T)}=(T^{\bullet },T^{\mathsf{bar}}).$$

$$((S$$

$$=\overline{((S\times T{)}^{\bullet },S\times T)}\prec \overline{(S^{\bullet }\times T^{\bullet },S\times T)}$$

$$=\overline{(S^{\bullet },S)\times (T^{\bullet },T)}\prec \overline{(S^{\bullet },S)}\times \overline{(T^{\bullet },T)}$$

$$=(S^{\bullet },S^{\mathsf{bar}})\times (T^{\bullet },T^{\mathsf{bar}})=(S^{\bullet }\times T^{\bullet },S^{\mathsf{bar}}\times T^{\mathsf{bar}}).$$

$${\mathbf{V}}^{\mathsf{bar}}=⟨⟨S^{\mathsf{bar}}\mid S\in \mathbf{V}⟩⟩.$$

$${\mathbb{Z}}_n=⟨\mathsf{g}\mid {\mathsf{g}}^n=1⟩=\{1,\mathsf{g},{\mathsf{g}}^2,\dots ,{\mathsf{g}}^{n-1}\}.$$

$$N_n=⟨\mathsf{a}\mid {\mathsf{a}}^n=0⟩=\{0,\mathsf{a},{\mathsf{a}}^2,\dots ,{\mathsf{a}}^{n-1}\}.$$

Full text

The second author was supported by Simons Foundation Collaboration Grants for Mathematicians #313548. The third author was supported by Simons Foundation #245268, United States–Israel Binational Science Foundation #2012080, and NSA MSP #H98230-16-1-0047.

1. 1 Introduction

2. 1.1 Groups

3. 1.2 $\mathscr{J}$-trivial semigroups

4. 1.3 Commutative semigroups

5. 1.4 Bands

6. 1.5 Kovács–Newman semigroups

7. 1.6 The subdirectly indecomposable viewpoint

8. 1.7 Organization

9. 2 Augmented semigroups

10. 3 Some important semigroups

11. 3.1 Cyclic groups

12. 3.2 Nilpotent semigroups

13. 3.3 Bands

14. 3.4 Completely 0-simple semigroups

15. 4 Some general results on join irreducibility

16. 4.1 Non-definability by simple pseudoidentities

17. 4.2 Large exclusion pseudovarieties

18. 4.3 Augmentation preserves join irreducibility

19. 4.4 Iterating augmentation and its dual to bands

20. 4.5 From non-$\mathsf{ji}$ pseudovarieties to $\mathsf{ji}$ pseudovarieties

21. 4.6 A sufficient condition for the join irreducibility of groups

22. 4.7 Join irreducibility of the pseudovariety $\langle\langle D_{4}\rangle\rangle=\langle\langle Q_{8}\rangle\rangle$

23. 5 Join irreducible pseudovarieties

24. 5.1 Preliminaries

25. 5.2 The pseudovariety $\langle\langle\mathbb{Z}_{n}\rangle\rangle$

26. 5.3 The pseudovariety $\langle\langle\mathbb{Z}_{2}^{\mathsf{bar}}\rangle\rangle$

27. 5.4 The pseudovariety $\langle\langle N_{n}\rangle\rangle$

28. 5.5 The pseudovariety $\langle\langle N_{n}^{I}\rangle\rangle$

29. 5.6 The pseudovariety $\langle\langle N_{2}^{\mathsf{bar}}\rangle\rangle$

30. 5.7 The pseudovariety $\langle\langle(N_{2}^{\mathsf{bar}})^{I}\rangle\rangle$

31. 5.8 The pseudovariety $\langle\langle L_{2}\rangle\rangle$

32. 5.9 The pseudovariety $\langle\langle L_{2}^{I}\rangle\rangle$

33. 5.10 The pseudovariety $\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle$

34. 5.11 The pseudovariety $\langle\langle A_{0}\rangle\rangle$

35. 5.12 The pseudovariety $\langle\langle A_{0}^{I}\rangle\rangle$

36. 5.13 The pseudovariety $\langle\langle A_{2}\rangle\rangle$

37. 5.14 The pseudovariety $\langle\langle B_{2}\rangle\rangle$

38. 5.15 The pseudovariety $\langle\langle\ell_{3}^{\mathsf{bar}}\rangle\rangle$

39. 6 Non-$\mathsf{ji}$ pseudovarieties

40. 6.1 The pseudovariety $\langle\langle\mathbb{Z}_{3},\mathbb{Z}_{4},\mathbb{Z}_{2}^{\mathsf{bar}},(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}},N_{3}^{I}\rangle\rangle$

41. 6.2 The pseudovariety $\langle\langle\mathbb{Z}_{m},N_{n}^{I},L_{2}^{I},R_{2}^{I},A_{0}^{I}\rangle\rangle$

42. 6.3 Pseudovarieties of noncommutative nilpotent semigroups

43. 6.4 The pseudovariety $\langle\langle N_{n+r},N_{n}^{I}\rangle\rangle$

44. 6.5 The pseudovariety $\langle\langle N_{n}^{I},N_{2}^{\mathsf{bar}}\rangle\rangle$

45. 6.6 The pseudovariety $\langle\langle N_{2}^{I},(L_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$

46. 6.7 The pseudovariety $\langle\langle L_{2}^{I},\ell_{3},\ell_{3}^{\mathsf{op}}\rangle\rangle$

47. 6.8 The pseudovariety $\langle\langle A_{0},\ell_{3}^{I},(\ell_{3}^{\mathsf{op}})^{I}\rangle\rangle$

48. 6.9 The pseudovariety $\langle\langle(N_{2}^{\mathsf{bar}})^{I},L_{2}^{\mathsf{bar}}\rangle\rangle$

49. 7 Pseudovarieties generated by a semigroup of order up to five

50. 7.1 Conditions sufficient for join irreducibility

51. 7.2 Conditions sufficient for non-join irreducibility

1. Introduction

In the 1970s, Eilenberg [4] highlighted the importance of $\mathbf{PV}$, the algebraic lattice of all pseudovarieties of finite semigroups, via his research with Schützenberger, by providing a correspondence between $\mathbf{PV}$ and varieties of regular languages. Specifically, they proved that the lattice $\mathbf{PV}$ is isomorphic to the algebraic lattice of varieties of regular languages; see the monograph by the second and third authors [23, Introduction] and the references therein.

The $\mathfrak{q}$-theory of finite semigroups focuses on $\mathbf{PV}$, but in a different manner, and can be viewed in analogy with the classical real analysis theory of continuous and differentiable functions from $[0,1]$ into $[0,1]$. The analogy is given by replacing $[0,1]$ with $\mathbf{PV}$, continuous functions with $\mathbf{Cnt}(\mathbf{PV})$, and differentiable functions with $\mathbf{GMC}(\mathbf{PV})$; see [23, Chapter 2].

From a number of points of view, $\mathbf{PV}$ is an important algebraic lattice with many interesting properties, and several theories have been developed for its investigation. For instance, the theorem of Reiterman [21] characterized pseudovarieties as exactly the classes defined by pseudoidentities. This led to the syntactic approach—employed by Almeida in his work and monograph [2]—that has became a fundamental tool in finite semigroup theory. Some of these results and techniques will be employed in this paper. Another important approach is the abstract spectral theory of $\mathbf{PV}$ going back to Stone with lattice theoretic foundations going back to Birkhoff; see [23, Chapter 7].

Since $\mathbf{PV}$ is a lattice, it is natural to investigate its elements that satisfy important lattice properties. For any element $\ell$ in a lattice $\mathscr{L}$,

1. (1)

$\ell$ is compact if, for any $\mathscr{X}\subseteq\mathscr{L}$,

[TABLE] 2. (2)

$\ell$ is join irreducible ($\mathsf{ji}$) if, for any $\mathscr{X}\subseteq\mathscr{L}$,

[TABLE] 3. (3)

$\ell$ is finite join irreducible ($\mathsf{fji}$) if, for any finite $\mathscr{F}\subseteq\mathscr{L}$,

[TABLE] 4. (4)

$\ell$ is meet irreducible ($\mathsf{mi}$) if, for any set $\mathscr{X}\subseteq\mathscr{L}$,

[TABLE] 5. (5)

$\ell$ is finite meet irreducible ($\mathsf{fmi}$) if, for any finite $\mathscr{F}\subseteq\mathscr{L}$,

[TABLE] 6. (6)

$\ell$ is strictly join irreducible ($\mathsf{sji}$) if, for any set $\mathscr{X}\subseteq\mathscr{L}$,

[TABLE] 7. (7)

$\ell$ is strictly finite join irreducible ($\mathsf{sfji}$) if, for any finite $\mathscr{F}\subseteq\mathscr{L}$,

[TABLE] 8. (8)

$\ell$ is strictly meet irreducible ($\mathsf{smi}$) if, for any $\mathscr{X}\subseteq\mathscr{L}$,

[TABLE] 9. (9)

$\ell$ is strictly finite meet irreducible ($\mathsf{sfmi}$) if, for any finite $\mathscr{F}\subseteq\mathscr{L}$,

[TABLE]

An algebraic lattice is a complete lattice that is join generated by its compact elements. The compact elements of $\mathbf{PV}$ are the finitely generated pseudovarieties. The pseudovariety generated by a finite semigroup $S$ is denoted by $\langle\langle S\rangle\rangle$. It is clear that for any $\mathbf{V}\in\mathbf{PV}$,

[TABLE]

The abstract spectral theory of a lattice is closely connected to the computation of its maximal distributive image, which is determined by the lattice’s $\mathsf{fji}$ and $\mathsf{fmi}$ elements; see [23, Chapter 7] and the references therein. The $\mathsf{fji}$ and $\mathsf{fmi}$ elements of $\mathbf{PV}$ are thus very important. The $\mathsf{ji}$ pseudovarieties are just the compact $\mathsf{fji}$ pseudovarieties, as is easy to see, so we are interested in finite semigroups that generate pseudovarieties that are $\mathsf{fji}$ or equivalently $\mathsf{ji}$.

By abuse of terminology, we say that a finite semigroup $S$ is join irreducible ($\mathsf{ji}$) if the pseudovariety $\langle\langle S\rangle\rangle$ is $\mathsf{ji}$; finite semigroups that satisfy the properties in (3)–(9) are similarly defined. A finite semigroup $S$ is $\mathsf{ji}$ if and only if for all finite semigroups $T_{1}$ and $T_{2}$,

[TABLE]

where $A\prec B$ means that $A$ is a homomorphic image of a subsemigroup of $B$, and $A^{n}=A\times A\times\cdots\times A$ is the direct product of $n$ copies of $A$. For finite semigroups, there are several properties stronger than being $\mathsf{ji}$: a finite semigroup $S$ is $\times$-prime [2, Section 9.3] if for all finite semigroups $T_{1}$ and $T_{2}$,

[TABLE]

a semigroup $S$ is Kovács–Newman ($\mathsf{KN}$) if whenever $f:T\twoheadrightarrow S$ is a surjective homomorphism where $T$ is a subsemigroup of $T_{1}\times T_{2}$ for some finite semigroups $T_{1}$ and $T_{2}$, subdirectly embedded, then $f$ factors through one of the projections. Semigroups that are $\mathsf{KN}$ have been completely classified [23, Section 7.4].

The proper inclusions

[TABLE]

are known to hold. For example, while any simple non-abelian group is $\mathsf{KN}$, any cyclic group $\mathbb{Z}_{p}$ of prime order $p$ is $\times$-prime but not $\mathsf{KN}$. The well-known Brandt semigroup $B_{2}$ of order five is $\mathsf{ji}$ but not $\times$-prime [23, Example 7.4.3].

Since the lattice $\mathbf{PV}$ is algebraic, it follows from a well-known theorem of Birkhoff that its $\mathsf{smi}$ elements constitute the unique minimal set of meet generators [23, Section 7.1]. It easily follows from Reiterman’s theorem [23, Section 3.2] that each $\mathsf{smi}$ pseudovariety is defined by a single pseudoidentity but not conversely. Now the reverse of the lattice $\mathbf{PV}$ is not algebraic but is locally dually algebraic, so the $\mathsf{sji}$ elements of $\mathbf{PV}$ constitute the unique minimal set of join generators for $\mathbf{PV}$ [23, Section 7.2]. The $\mathsf{sji}$ pseudovarieties are precisely those having a unique proper maximal subpseudovariety.

Every $\mathsf{ji}$ pseudovariety is $\mathsf{sji}$, but the converse does not hold, as demonstrated by several known examples [23, Proposition 7.3.22] and additional examples in Propositions 3.1 and 6.30. Hence $\mathsf{ji}$ pseudovarieties do not join generate the lattice $\mathbf{PV}$. This prompts the following tantalizing question.

Question 1.1**.**

What do the $\mathsf{ji}$ elements in $\mathbf{PV}$ join generate?

It is well known and not difficult to prove that the function

[TABLE]

on the class of finite semigroups is computable; see, for example, Proposition 4.1 and its proof. On the other hand, it is unknown if the function

[TABLE]

on the class of finite semigroups is decidable.

Question 1.2**.**

Is $\mathsf{ji}$ decidable, that is, is the above function computable?

If $\mathsf{ji}$ is not decidable, then a systematic study of $\mathsf{ji}$ semigroups seems doomed in general. But even if $\mathsf{ji}$ is decidable, then it is probably hopeless, in practice, to find all $\mathsf{ji}$ semigroups. In any case, an important step is to find methods to produce new $\mathsf{ji}$ semigroups and methods to identify and eliminate finite semigroups that are not $\mathsf{ji}$. This paper develops several new methods. For semigroups of small order, in particular, the (Birkhoff) equational theory is crucial and is often used.

A pleasant feature of a finite semigroup $S$ being $\mathsf{ji}$ is the “five for one phenomenon” related to the exclusion class $\operatorname{\mathsf{Excl}}(S)$ of $S$, the class of all finite semigroups $T$ for which $S\notin\langle\langle T\rangle\rangle$. Indeed, a finite semigroup $S$ is $\mathsf{ji}$ if and only if $\operatorname{\mathsf{Excl}}(S)$ is a pseudovariety [23, Theorem 7.1.2]. In this case, $\operatorname{\mathsf{Excl}}(S)$ is $\mathsf{mi}$ and so is defined by a single pseudoidentity, and since $\operatorname{\mathsf{Excl}}(S)$ is also $\mathsf{smi}$, it has $\operatorname{\mathsf{Excl}}(S)\vee\langle\langle S\rangle\rangle$ as a unique cover. Further, $\langle\langle S\rangle\rangle\cap\operatorname{\mathsf{Excl}}(S)$ is the unique maximal subpseudovariety of $\langle\langle S\rangle\rangle$, and so $\operatorname{\mathsf{Excl}}(S)$ determines $\langle\langle S\rangle\rangle$; see [23, Section 7.1]. For example, the Brandt semigroup $B_{2}$ is $\mathsf{ji}$, the exclusion class $\operatorname{\mathsf{Excl}}(B_{2})$ coincides with the pseudovariety

[TABLE]

of finite semigroups whose $\mathscr{J}$-classes are subsemigroups [23, Example 7.3.4], and $\langle\langle B_{2}\rangle\rangle\cap\mathbf{DS}=\langle\langle B_{0}\rangle\rangle$ is the unique maximal subpseudovariety of $\langle\langle B_{2}\rangle\rangle$, where $B_{0}$ is a subsemigroup of $B_{2}$ of order four [6]; see Subsection 3.4. More examples of maximal subpseudovarieties can be found in Section 5.

As mentioned earlier, a goal of this paper is to find new $\mathsf{ji}$ semigroups. One approach—and a very important problem in its own right—is to find new operators on $\mathbf{PV}$ that preserve the property of being $\mathsf{ji}$. The following are some known examples.

Example 1.3**.**

For any semigroup $S$, the opposite semigroup $S^{\mathsf{op}}$ of $S$ is obtained by reversing the multiplication on $S$. Then the dual operator

[TABLE]

on $\mathbf{PV}$ preserves the property of being $\mathsf{ji}$.

Example 1.4** (See Lemma 5.2).**

For any semigroup $S$, let $S^{I}$ denote the monoid obtained by adjoining an external identity element $I$ to $S$, and define

[TABLE]

Then the operator $\mathbf{V}\mapsto\mathbf{V}^{\bullet}=\{S^{\bullet}\mid S\in\mathbf{V}\}$ on $\mathbf{PV}$ preserves the property of being $\mathsf{ji}$.

Example 1.3 is not surprising; in fact, in many investigations, such as the finite basis problem for small semigroups [14, 30], it is common to identify $S^{\mathsf{op}}$ with $S$. The situation for the operator $\mathbf{V}\mapsto\mathbf{V}^{\bullet}$, however, can be different because it is possible that no new $\mathsf{ji}$ pseudovariety is produced. Indeed, if a pseudovariety $\mathbf{V}$ is generated by some monoid, then $\mathbf{V}^{\bullet}=\mathbf{V}$ cannot be a new example of $\mathsf{ji}$ pseudovariety. But if $\mathbf{V}=\langle\langle S\rangle\rangle$ is a $\mathsf{ji}$ pseudovariety that is not generated by any monoid, then $\mathbf{V}^{\bullet}=\langle\langle S^{I}\rangle\rangle$ is a $\mathsf{ji}$ pseudovariety properly containing $\mathbf{V}$. Note that the operator $\mathbf{V}\mapsto\mathbf{V}^{I}=\{S^{I}\mid S\in\mathbf{V}\}$ does not preserve the property of being $\mathsf{ji}$. For example, the cyclic group $\mathbb{Z}_{p}$ of any prime order $p$ generates a $\mathsf{ji}$ pseudovariety, but the pseudovariety $\langle\langle\mathbb{Z}_{p}\rangle\rangle^{I}=\langle\langle\mathbb{Z}_{p}^{I}\rangle\rangle$ is not $\mathsf{ji}$ because $\langle\langle\mathbb{Z}_{p}^{I}\rangle\rangle=\langle\langle\mathbb{Z}_{p}\rangle\rangle\vee\langle\langle Sl_{2}\rangle\rangle$, where $Sl_{2}$ is the semilattice of order two.

On the other hand, it is possible for $\mathbf{V}^{I}$ to be $\mathsf{ji}$ even though $\mathbf{V}$ is not $\mathsf{ji}$. For example, if $S=Sl_{2}\times R_{2}$, where $R_{2}$ is the right zero semigroup of order two, then the pseudovariety $\langle\langle S\rangle\rangle=\langle\langle Sl_{2}\rangle\rangle\vee\langle\langle R_{2}\rangle\rangle$ is not $\mathsf{ji}$ but $\langle\langle S\rangle\rangle^{I}=\langle\langle Sl_{2}^{I}\times R_{2}^{I}\rangle\rangle=\langle\langle R_{2}^{I}\rangle\rangle$ is $\mathsf{ji}$ [23, Example 7.3.1].

Remark 1.5**.**

It is clear that the operator $\mathbf{V}\mapsto\mathbf{V}^{\mathsf{op}}$ also preserves the property of being $\mathsf{sji}$, but the operator $\mathbf{V}\mapsto\mathbf{V}^{\bullet}$ does not preserve this property. For instance, the pseudovariety $\langle\langle B_{0}\rangle\rangle$ is $\mathsf{sji}$ while $\langle\langle B_{0}\rangle\rangle^{\bullet}=\langle\langle B_{0}^{I}\rangle\rangle$ is not $\mathsf{sji}$; see Proposition 3.1.

Given a finite semigroup $S$, consider the right regular representation $(S^{\bullet},S)$ of $S$ acting on $S^{\bullet}$ by right multiplication. Then $S^{\mathsf{bar}}$ is defined by adding all constant maps on $S^{\bullet}$ to $S$, where multiplication is composition with the variable written on the left. Note that if $(S,S)$ is a faithful transformation semigroup, then we shall see later that the semigroup obtained from $S$ by adjoining the constant mappings on $S$ generates the same pseudovariety as $S^{\mathsf{bar}}$ and hence we sometimes (abusively) denote this latter semigroup by $S^{\mathsf{bar}}$ as well. Some small examples of $S^{\mathsf{bar}}$ can be found in Section 3.

It turns out that the operator $\mathbf{V}\mapsto\mathbf{V}^{\mathsf{bar}}=\langle\langle S^{\mathsf{bar}}\mid S\in\mathbf{V}\rangle\rangle$ on $\mathbf{PV}$ preserves the property of being $\mathsf{ji}$. This result, the details of which are given in Subsection 4.3, is important: for any finite nontrivial $\mathsf{ji}$ semigroup $S$, the pseudovarieties

[TABLE]

where $X^{\flat}=((X^{\mathsf{op}})^{\mathsf{bar}})^{\mathsf{op}}$, constitute an infinite increasing chain of $\mathsf{ji}$ pseudovarieties (Corollary 4.11) whose complete union is an $\mathsf{fji}$ pseudovariety that is not compact [23, Chapter 7].

Unsurprisingly, irregularities do show up when the operator $\mathbf{V}\mapsto\mathbf{V}^{\mathsf{bar}}$ is applied. For instance, it is sometimes possible for $\langle\langle S^{\mathsf{bar}}\rangle\rangle=\langle\langle S\rangle\rangle$, so that no new $\mathsf{ji}$ pseudovariety is obtained. Further, it is possible for $\langle\langle S^{\mathsf{bar}}\rangle\rangle$ to be $\mathsf{ji}$ even though $\langle\langle S\rangle\rangle$ is not $\mathsf{ji}$. An important class of examples will be given in Subsection 4.5.

A main result of this paper is the complete classification of all $\mathsf{ji}$ pseudovarieties generated by a semigroup of order up to five. We want to give the reader an easy way to remember their generators. First, we have the three operators $S\mapsto S^{\mathsf{op}}$, $S\mapsto S^{\bullet}$, and $S\mapsto S^{\mathsf{bar}}$, and their iterations such as $(((S^{\mathsf{op}})^{\mathsf{bar}})^{\mathsf{op}}$ and $(((((S^{\mathsf{bar}})^{\mathsf{op}})^{\bullet})^{\mathsf{bar}})^{\mathsf{op}})^{\mathsf{bar}}$. If we have a list of $\mathsf{ji}$ semigroups, applying the three operators and their iterations give $\mathsf{ji}$ semigroups that may or may not generate new $\mathsf{ji}$ pseudovarieties.

A $\mathsf{ji}$ pseudovariety $\mathbf{V}$ is primitive if $\mathbf{V}\neq\langle\langle S^{\bullet}\rangle\rangle$ and $\mathbf{V}\neq\langle\langle S^{\mathsf{bar}}\rangle\rangle$ for any finite semigroup $S$ that generates a $\mathsf{ji}$ proper subpseudovariety of $\mathbf{V}$. Now we are only interested in knowing the primitive $\mathsf{ji}$ pseudovarieties up to isomorphism and anti-isomorphism of members since the others can be obtained by applying the operators. Therefore when describing $\mathsf{ji}$ pseudovarieties generated by a semigroup of order up to five, it suffices to list, up to isomorphism and anti-isomorphism, only generators of those that are primitive; see Table 1. Presentations and multiplication tables of these semigroups can be found in Section 3. The only new example of a semigroup of order five that generates a primitive $\mathsf{ji}$ pseudovariety is $\ell_{3}^{\mathsf{bar}}$; all the other semigroups were previously known to be $\mathsf{ji}$. Note that $\ell_{3}^{\mathsf{bar}}$ is $\mathsf{ji}$ but $\ell_{3}$ is not; see Subsection 4.5, where this example is extended to an infinite family of examples.

The statement of the above result regarding semigroups of order up to five is straightforward, but its proof is not so; it requires knowledge of subpseudovarieties of pseudovarieties generated by small semigroups [6, 9, 12, 13, 15, 16, 27, 31, 32] and of bases of pseudoidentities for many pseudovarieties of the form $\bigvee_{i=1}^{k}\langle\langle S_{i}\rangle\rangle$, and advanced algebraic theory of finite semigroups [23].

The following are all other $\mathsf{ji}$ semigroups known to us, except for some well-known results on completely simple semigroups.

1.1. Groups

It is an easy observation that a finite group generates a $\mathsf{ji}$ pseudovariety of semigroups if and only if it generates a $\mathsf{ji}$ pseudovariety of groups; see [23, Chapter 7]. A pseudovariety $\mathbf{V}$ of groups is called saturated if whenever $\varphi:G\to H$ is a homomorphism of finite groups with $H\in\mathbf{V}$, there exists a subgroup $K\leq G$ such that $K\in\mathbf{V}$ and $K\varphi=H$. It is observed in [24, Example 7.6.5] that any pseudovariety of groups closed under extension is saturated. In particular, for any prime $p$, the pseudovariety of $p$-groups is saturated. It is almost immediate from the definition that if $\mathbf{V}$ is a saturated pseudovariety of groups, then a group $G\in\mathbf{V}$ generates a $\mathsf{ji}$ pseudovariety in the lattice of all semigroup pseudovarieties if and only if it generates a $\mathsf{ji}$ member of the lattice of subpseudovarieties of $\mathbf{V}$. In particular, a $p$-group $G$ is $\mathsf{ji}$ if and only if whenever $G$ divides a direct product $A\times B$ of $p$-groups, then $G$ divides either $A^{n}$ or $B^{n}$ for some $n\geq 1$.

Abelian groups

The following statements on any directly indecomposable finite abelian group $A$ are equivalent: $A$ is $\mathsf{ji}$, $A$ is $\times$-prime, and $A\cong\mathbb{Z}_{p^{n}}$ for some prime $p$ and $n\geq 1$. This result follows from the Fundamental Theorem of Finite Abelian Groups and that $\mathbb{Z}_{p^{n}}$ lifts in the sense that whenever $\mathbb{Z}_{p^{n}}$ is a homomorphic image of some semigroup $S$, then $\mathbb{Z}_{p^{n+r}}$ embeds into $S$ for some $r\geq 0$.

Monolithic groups

A finite group $G$ is monolithic if it contains a unique minimal nontrivial normal subgroup $N$; in this case, $N$ is called the monolith of $G$, and it is well known that $N\cong H^{n}$ for some simple group $H$ and $n\geq 1$.

A finite group is monolithic if and only if it is subdirectly indecomposable; recall that a semigroup $S$ is a subdirect product of $S_{1}$ and $S_{2}$, written $S\ll S_{1}\times S_{2}$, if $S$ is a subsemigroup of $S_{1}\times S_{2}$ mapping onto both $S_{1}$ and $S_{2}$ via the projections $\pi_{i}$. A semigroup $S$ is subdirectly indecomposable ($\mathsf{sdi}$) if $S\ll S_{1}\times S_{2}$ implies that at least one of the projections $\pi_{i}:S\twoheadrightarrow S_{i}$ is an isomorphism. Therefore when locating $\mathsf{ji}$ groups from among finite groups, it suffices to concentrate on those that are monolithic.

Groups with non-abelian monolith

Kovács and Newman proved that any monolithic group with non-abelian monolith is $\mathsf{KN}$ [23, Section 7.4] and so also $\times$-prime and $\mathsf{ji}$. Therefore, all simple non-abelian groups are $\mathsf{ji}$.

Groups with abelian monolith

An abelian monolith $N$ of a finite group $G$ splits if there exists a subgroup $K$ of $G$ so that $N\cap K=\{1\}$ and $NK=G$. A finite subdirectly indecomposable group with an abelian monolith that splits is $\mathsf{ji}$; this result is due to G. M. Bergman and its proof is given in Subsection 4.6. Therefore, the symmetric group $\mathsf{Sym}_{3}$ over three symbols is $\mathsf{ji}$.

Groups of small order

The $\mathsf{ji}$ pseudovarieties generated by a group of order seven or less are $\langle\langle\mathbb{Z}_{2}\rangle\rangle$, $\langle\langle\mathbb{Z}_{3}\rangle\rangle$, $\langle\langle\mathbb{Z}_{4}\rangle\rangle$, $\langle\langle\mathbb{Z}_{5}\rangle\rangle$, $\langle\langle\mathsf{Sym}_{3}\rangle\rangle$, and $\langle\langle\mathbb{Z}_{7}\rangle\rangle$. Regarding groups of order eight that generate other $\mathsf{ji}$ pseudovarieties, besides $\mathbb{Z}_{8}$, there are two nontrivial cases: the dihedral group $D_{4}$ of the square and the quaternion group $Q_{8}$. Let $G\in\{D_{4},Q_{8}\}$. Then forming $G\times G$ and dividing out the two centers identified, $(G\times G)/\{(1,1),(a,a)\}$ gives isomorphic groups, denoted by $G\circ G$. Since $G\leq G\circ G$, it follows that $\langle\langle D_{4}\rangle\rangle=\langle\langle Q_{8}\rangle\rangle$. Therefore, the groups $D_{4}$ and $Q_{8}$ are not $\times$-prime and so also not $\mathsf{KN}$. However, the pseudovariety $\langle\langle D_{4}\rangle\rangle=\langle\langle Q_{8}\rangle\rangle$ is $\mathsf{ji}$; see Subsection 4.7. This result is due independently to Kearnes [5] and the anonymous reviewer.

Other $\mathsf{ji}$ pseudovarieties generated by a group of order up to 11 are $\langle\langle\mathbb{Z}_{9}\rangle\rangle$, $\langle\langle D_{5}\rangle\rangle$, and $\langle\langle\mathbb{Z}_{11}\rangle\rangle$.

1.2. $\mathscr{J}$-trivial semigroups

Presently, the only $\mathsf{ji}$ pseudovarieties of $\mathscr{J}$-trivial semigroups known in the literature are generated by the following:

1. $\bullet$

$N_{n}=\langle\mathsf{a}\mid\mathsf{a}^{n}=0\rangle$, $n\geq 1$; 2. $\bullet$

$H_{n}=\langle\mathsf{e},\mathsf{f}\mid\mathsf{e}^{2}=\mathsf{e},\,\mathsf{f}^{2}=\mathsf{f},\,(\mathsf{e}\mathsf{f})^{n}=0\rangle$, $n\geq 1$; 3. $\bullet$

$K_{n}=\langle\mathsf{e},\mathsf{f}\mid\mathsf{e}^{2}=\mathsf{e},\,\mathsf{f}^{2}=\mathsf{f},\,(\mathsf{e}\mathsf{f})^{n}\mathsf{e}=0\rangle$, $n\geq 1$; 4. $\bullet$

$N_{n}^{I}$, $H_{n}^{I}$, $K_{n}^{I}$, $n\geq 1$.

The pseudoidentity defining the pseudovariety $\operatorname{\mathsf{Excl}}(N_{n})$ is given in Subsection 5.4, while the pseudovarieties $\operatorname{\mathsf{Excl}}(H_{n})$ and $\operatorname{\mathsf{Excl}}(K_{n})$ are defined by the pseudoidentities

[TABLE]

respectively [11, Propositions 2.3 and 3.3].

1.3. Commutative semigroups

The pseudovariety $\mathbf{Com}$ of finite commutative semigroups can be decomposed as

[TABLE]

where $\mathbf{G}$ is the pseudovariety of finite groups and $\mathbf{A}$ is the pseudovariety of finite aperiodic semigroups [2, Figure 9.1]. Therefore any $\mathsf{ji}$ pseudovariety of commutative semigroups is contained in either $\mathbf{Com}\cap\mathbf{G}$ or $\mathbf{Com}\cap\mathbf{A}$. As noted in Subsection 1.1, the $\mathsf{ji}$ subpseudovarieties of $\mathbf{Com}\cap\mathbf{G}$ are each generated by a cyclic group $\mathbb{Z}_{p^{n}}$ of prime power order. As for $\mathbf{Com}\cap\mathbf{A}$, each of its finite semigroups satisfies the identity $x^{n+1}\approx x^{n}$ for all sufficiently large $n\geq 1$ and so belongs to $\langle\langle N_{n}^{I}\rangle\rangle$; see Proposition 5.10(i). A complete description of $\mathsf{ji}$ subpseudovarieties of $\mathbf{Com}$ is thus dependent on the answer to the following question.

Question 1.6**.**

For each $n\geq 1$, what are the $\mathsf{ji}$ subpseudovarieties of $\langle\langle N_{n}^{I}\rangle\rangle$?

Presently, the only known examples of $\mathsf{ji}$ subpseudovarieties of $\langle\langle N_{n}^{I}\rangle\rangle$ are $\langle\langle N_{k}\rangle\rangle$ and $\langle\langle N_{k}^{I}\rangle\rangle$, where $1\leq k\leq n$.

1.4. Bands

The pseudovariety $\mathbf{B}$ of finite bands is $\mathsf{fji}$ (Corollary 4.12). Each proper subpseudovariety of $\mathbf{B}$ is compact and a complete description of the lattice of subpseudovarieties of $\mathbf{B}$ is well known; see, for example, Almeida [2, Section 5.5]. The atoms of this lattice are $\mathbf{Sl}=\langle\langle Sl_{2}\rangle\rangle$, $\mathbf{LZ}=\langle\langle L_{2}\rangle\rangle$, and $\mathbf{RZ}=\langle\langle R_{2}\rangle\rangle$; see Subsection 3.3.

Let $\mathbf{LNB}=\mathbf{Sl}\vee\mathbf{LZ}$. For any pseudovariety $\mathbf{V}$, define the Mal’cev products \widetilde{\alpha}\mathbf{V}=\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V} and \widetilde{\beta}\mathbf{V}=\mathbf{LZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V}; see Subsection 4.2. Then by [19], the proper, nontrivial $\mathsf{sji}$ pseudovarieties of bands are as follows:

1. $\bullet$

$\mathbf{LZ}$, $\mathbf{RZ}$, and $\mathbf{Sl}$; 2. $\bullet$

$(\widetilde{\alpha}\widetilde{\beta})^{n}\mathbf{Sl}$ and $\widetilde{\beta}(\widetilde{\alpha}\widetilde{\beta})^{n}\mathbf{Sl}$, $n\geq 1$, and their duals; 3. $\bullet$

$(\widetilde{\beta}\widetilde{\alpha})^{n+1}\mathbf{LNB}$ and $\widetilde{\alpha}(\widetilde{\beta}\widetilde{\alpha})^{n}\mathbf{LNB}$, $n\geq 0$, and their duals.

However, we observe that $\widetilde{\alpha}\mathbf{LNB}=\widetilde{\alpha}\mathbf{LZ}$. Since $S\mapsto S^{\mathsf{bar}}$ preserves join irreducibility, it follows that the pseudovariety generated by a finite band is $\mathsf{ji}$ if and only if it is $\mathsf{sji}$; see Theorem 4.14. As observed after Question 1.1, it is decidable if a finite semigroup generates a $\mathsf{sji}$ pseudovariety. Therefore, Question 1.2 is affirmatively answered for bands.

1.5. Kovács–Newman semigroups

All $\mathsf{KN}$ semigroups are known [23, Section 7]. These are semigroups with kernel (minimal two-sided ideal) a Rees matrix semigroup over a monolithic group with non-abelian monolith that acts faithfully on the right and left of the kernel.

1.6. The subdirectly indecomposable viewpoint

Since every finite semigroup divides (in fact, is a subdirect product of) its $\mathsf{sdi}$ homomorphic images, we can restrict our search for new $\mathsf{ji}$ semigroups to $\mathsf{sdi}$ semigroups, just as in the case of groups, when we can restrict our search to monolithic finite groups.

In more detail, to find the $\mathsf{ji}$ pseudovarieties, we clearly need only to find finite semigroups $S$ such that $\langle\langle S\rangle\rangle$ is $\mathsf{ji}$ and there exist no semigroups $T$ with $|T|<|S|$ and $\langle\langle T\rangle\rangle=\langle\langle S\rangle\rangle$. Such a semigroup $S$ is called a minimal order generator for the compact pseudovariety $\langle\langle S\rangle\rangle$.

Now the minimal order generators of $\mathsf{ji}$ pseudovarieties, in fact of $\mathsf{sji}$ pseudovarieties, must be $\mathsf{sdi}$. To see this, suppose that $S$ is any finite semigroup that is not $\mathsf{sdi}$. Then $S\ll S_{1}\times S_{2}\times\cdots\times S_{k}$ for some homomorphic images $S_{j}$ of $S$ such that $|S_{j}|<|S|$. But since $\langle\langle S\rangle\rangle=\langle\langle S_{1}\rangle\rangle\vee\langle\langle S_{2}\rangle\rangle\vee\cdots\vee\langle\langle S_{k}\rangle\rangle$ and $\langle\langle S\rangle\rangle$ is $\mathsf{sji}$, it follows that $\langle\langle S\rangle\rangle=\langle\langle S_{j}\rangle\rangle$ for some $j$, whence $S$ is not a minimal order generator

If a finite semigroup $S$ is $\times$-prime (e.g. $\mathsf{KN}$), then $S$ is a minimal order generator and any minimal order generator for $\langle\langle S\rangle\rangle$ is isomorphic to $S$. The proof is clear. However, minimal order generators for the same $\mathsf{ji}$ pseudovariety need not be isomorphic; for example, $\langle\langle Q_{8}\rangle\rangle=\langle\langle D_{4}\rangle\rangle$ is $\mathsf{ji}$ and $Q_{8}\ncong D_{4}$.

It should be pointed out that a finite semigroup $S$ being $\mathsf{sdi}$ does not imply that the pseudovariety $\langle\langle S\rangle\rangle$ is $\mathsf{ji}$ or even $\mathsf{sji}$. For example, the Rees matrix semigroup

[TABLE]

is $\mathsf{sdi}$, but $\langle\langle S\rangle\rangle=\langle\langle B_{2}\rangle\rangle\vee\langle\langle\mathbb{Z}_{2}\rangle\rangle$ is not $\mathsf{sji}$; see [23, Section 4.7].

1.7. Organization

The article is organized as follows. In Section 2, the operator $\mathbf{V}\mapsto\mathbf{V}^{\mathsf{bar}}$ is introduced in detail and some related results are established. In Section 3, some important small semigroups that are required for this paper are defined. In Section 4, some general results regarding $\mathsf{ji}$ pseudovarieties are established. In Section 5, some explicit pseudovarieties are shown to be $\mathsf{ji}$, and conditions sufficient for a finite semigroup to generate one of them are established. In Section 6, some conditions sufficient for a finite semigroup to generate a non-$\mathsf{ji}$ pseudovariety are established. Results in Sections 4–6 are then employed in Section 7 to prove that among all pseudovarieties generated by a semigroup of order up to five, only 30 are $\mathsf{ji}$.

2. Augmented semigroups

All semigroups and transformation semigroups, with the exception of free semigroups and free profinite semigroups, are assumed finite. Notation in the monograph [23] will often be followed closely.

Let $(X,S)$ be a transformation semigroup where $S$ is a semigroup that acts faithfully on the right of a set $X$. Then $\overline{(X,S)}=(X,S\cup\overline{X})$ where $\overline{X}$ is the set of constant maps on $X$. The constant map to a fixed element $x\in X$ is denoted by $\overline{x}$. If $(X,S)$ and $(Y,T)$ are transformation semigroups, then

[TABLE]

with the action $(x,y)(s,t)=(xs,yt)$.

Refer to Eilenberg [4] for the definition of division $\prec$ of transformation semigroups.

Lemma 2.1** (Eilenberg [4, Exercise I.4.1, Propositions I.5.4, and page 20]).**

Let $(X,S)$ and $(Y,T)$ be any transformation semigroups.* Then*

1. (i)

$(X,S)\prec(Y,T)$* implies that $\overline{(X,S)}\prec\overline{(Y,T)}$;* 2. (ii)

$(X,S)\prec(Y,T)$* implies that $S\prec T$;* 3. (iii)

$\overline{(X,S)\times(Y,T)}\prec\overline{(X,S)}\times\overline{(Y,T)}$.**

Lemma 2.1(ii) holds because the mappings involved are total.

Lemma 2.2** (D. Allen; see Eilenberg [4, Proposition I.9.8]).**

If $(X,S)$ is any transformation semigroup,* then $(S^{\bullet},S)\prec(X,S)^{|X|}$*.**

Following [23, Chapter 4], write $\overline{(S^{\bullet},S)}=(S^{\bullet},S^{\mathsf{bar}})$ and call $S^{\mathsf{bar}}$ the augmentation of $S$. Note that if $(X,S)\prec(S^{\bullet},S)$, then $(X,S)\prec(S^{\bullet},S)\prec(X,S)^{|X|}$ by Lemma 2.2 and hence

[TABLE]

Thus if $S^{\prime}=S\cup\overline{X}$, then $S^{\prime}\prec S^{\mathsf{bar}}\prec(S^{\prime})^{|X|}$, yielding the following result.

Corollary 2.3**.**

If $(X,S)$ is a transformation semigroup such that $(X,S)\prec(S^{\bullet},S)$,* then $\langle\langle S\cup\overline{X}\rangle\rangle=\langle\langle S^{\mathsf{bar}}\rangle\rangle$. In particular*,* if $S$ is any semigroup and $J$ is any right ideal of $S$ on which it acts faithfully*,* then $\langle\langle S^{\mathsf{bar}}\rangle\rangle=\langle\langle S\cup\overline{J}\rangle\rangle$*.**

The following are some elementary properties enjoyed by augmentation.

Proposition 2.4**.**

Let $S$ and $T$ be any finite semigroups.* Then*

1. (i)

$S\prec T$* implies that $S^{\mathsf{bar}}\prec T^{\mathsf{bar}}$;* 2. (ii)

$(S\times T)^{\mathsf{bar}}\prec S^{\mathsf{bar}}\times T^{\mathsf{bar}}$.**

Proof.

(i) Suppose that $S\prec T$, so that $(S^{\bullet},S)\prec(T^{\bullet},T)$ by Eilenberg [4, Proposition I.5.8]. Then by Lemma 2.1(i),

[TABLE]

Therefore, $S^{\mathsf{bar}}\prec T^{\mathsf{bar}}$ by Lemma 2.1(ii).

(ii) First note that $((S\times T)^{\bullet},S\times T)\prec(S^{\bullet}\times T^{\bullet},S\times T)$. Then

[TABLE]

Therefore, $(S\times T)^{\mathsf{bar}}\prec S^{\mathsf{bar}}\times T^{\mathsf{bar}}$ by Lemma 2.1(ii). ∎

In the following, augmentation is viewed as a continuous operator on the lattice $\mathbf{PV}$ of pseudovarieties. An operator is continuous if it preserves order and directed joins [23]. For any pseudovariety $\mathbf{V}$, define

[TABLE]

Recall that $\mathbf{RZ}=\langle\langle R_{2}\rangle\rangle$ is the pseudovariety of right zero semigroups.

Proposition 2.5**.**

The operator on $\mathbf{PV}$ defined by $\mathbf{V}\mapsto\mathbf{V}^{\mathsf{bar}}$ is continuous,* non-decreasing*,* and idempotent*.* Further*,**

1. (i)

$\langle\langle S\rangle\rangle^{\mathsf{bar}}=\langle\langle S^{\mathsf{bar}}\rangle\rangle$* for any finite semigroup $S$;* 2. (ii)

$\mathbf{RZ}\subseteq\mathbf{V}^{\mathsf{bar}}$* for any nontrivial pseudovariety $\mathbf{V}$.*

Consequently,* if $\langle\langle S\rangle\rangle=\langle\langle T\rangle\rangle$, then $\langle\langle S^{\mathsf{bar}}\rangle\rangle=\langle\langle T^{\mathsf{bar}}\rangle\rangle$*.**

Proof.

Clearly augmentation is order preserving. Let $\{\mathbf{V}_{\delta}\mid\delta\in D\}$ be any directed set of pseudovarieties, so that the complete join $\mathbf{V}=\bigvee_{\delta\in D}\mathbf{V}_{\delta}$ is a union. The inclusion $\mathbf{V}_{\delta}^{\mathsf{bar}}\subseteq\mathbf{V}^{\mathsf{bar}}$ clearly holds for all $\delta\in D$, so that $\bigvee_{\delta\in D}\mathbf{V}_{\delta}^{\mathsf{bar}}\subseteq\mathbf{V}^{\mathsf{bar}}$. Conversely, if $S\in\mathbf{V}^{\mathsf{bar}}$, say $S\prec T_{1}^{\mathsf{bar}}\times T_{2}^{\mathsf{bar}}\times\cdots\times T_{k}^{\mathsf{bar}}$ for some $T_{1},T_{2},\ldots,T_{k}\in\mathbf{V}$, then due to directedness, there exists $\delta\in D$ with $T_{1},T_{2},\ldots,T_{k}\in\mathbf{V}_{\delta}$, whence $S\in\mathbf{V}_{\delta}^{\mathsf{bar}}$. Therefore, augmentation is continuous.

Since $S\prec S^{\mathsf{bar}}$, it is obvious that augmentation is non-decreasing and the inclusion $\mathbf{V}^{\mathsf{bar}}\subseteq(\mathbf{V}^{\mathsf{bar}})^{\mathsf{bar}}$ holds. To establish the reverse inclusion, it suffices to prove that $(S^{\mathsf{bar}})^{\mathsf{bar}}\in\mathbf{V}^{\mathsf{bar}}$ for all $S\in\mathbf{V}$. But $S^{\mathsf{bar}}$ acts faithfully on the right of its minimal ideal $\overline{S^{\bullet}}$ and it contains all the constant mappings. Thus $(\overline{S^{\bullet}},S^{\mathsf{bar}})=\overline{(\overline{S^{\bullet}},S^{\mathsf{bar}})}$ and $(\overline{S^{\bullet}},S^{\mathsf{bar}})\prec((S^{\mathsf{bar}})^{\bullet},S^{\mathsf{bar}})$. It follows from Corollary 2.3 that $S^{\mathsf{bar}}=S^{\mathsf{bar}}\cup\overline{S^{\bullet}}$ generates the same pseudovariety as $(S^{\mathsf{bar}})^{\mathsf{bar}}$. This shows that $(S^{\mathsf{bar}})^{\mathsf{bar}}\in\mathbf{V}^{\mathsf{bar}}$, so that augmentation is idempotent.

It remains to establish parts (i) and (ii).

(i) The inclusion $\langle\langle S^{\mathsf{bar}}\rangle\rangle\subseteq\langle\langle S\rangle\rangle^{\mathsf{bar}}$ holds trivially. To establish the reverse inclusion, suppose that $T\in\langle\langle S\rangle\rangle^{\mathsf{bar}}$, so that $T\prec U^{\mathsf{bar}}$ for some $U\in\langle\langle S\rangle\rangle$. Then $U\prec S^{n}$ for some $n\geq 0$ and so $T\prec U^{\mathsf{bar}}\prec(S^{n})^{\mathsf{bar}}\prec(S^{\mathsf{bar}})^{n}$ by Proposition 2.4(ii). Therefore, $T\in\langle\langle S^{\mathsf{bar}}\rangle\rangle$. Consequently, $\langle\langle S^{\mathsf{bar}}\rangle\rangle=\langle\langle S\rangle\rangle^{\mathsf{bar}}$.

(ii) If $S$ is a nontrivial semigroup in $\mathbf{V}$, then the right zero semigroup $R_{2}$ is a subsemigroup of $S^{\mathsf{bar}}$, whence $\mathbf{RZ}\subseteq\mathbf{V}$. ∎

Corollary 2.6**.**

Let $S$ be any finite semigroup whose minimal ideal $J$ consists of right zeroes.* Suppose that $S$ acts faithfully on the right of $J$. Then $\langle\langle S\rangle\rangle^{\mathsf{bar}}=\langle\langle S\rangle\rangle$*.**

Proof.

By Proposition 2.5, it suffices to prove that $\langle\langle S^{\mathsf{bar}}\rangle\rangle=\langle\langle S\rangle\rangle$. But since $\overline{(J,S)}=(J,S)$, it follows that $S=S\cup\overline{J}$. The desired conclusion then follows from Corollary 2.3. ∎

3. Some important semigroups

In this section, semigroups that are required throughout the paper are introduced. Semigroups are given by their presentations, and whenever feasible, multiplication tables. In presentations, the symbols $\mathsf{e}$ and $\mathsf{f}$ are exclusively reserved for idempotent elements.

3.1. Cyclic groups

The cyclic group of order $n\geq 1$ is

[TABLE]

The augmentation of $\mathbb{Z}_{2}=\{1,\mathsf{g}\}$ is the semigroup $\mathbb{Z}_{2}^{\mathsf{bar}}=\{1,\mathsf{g},\overline{1},\overline{\mathsf{g}}\}$ given in Table 2. The semigroup $\mathbb{Z}_{2}^{\mathsf{bar}}$ is isomorphic to the semigroup of transformations of the set $\{1,2\}$. Information on identities satisfied by the semigroups $\mathbb{Z}_{n}$ and $\mathbb{Z}_{2}^{\mathsf{bar}}$ is given in Subsections 5.2 and 5.3, respectively.

3.2. Nilpotent semigroups

The monogenic nilpotent semigroup of order $n\geq 1$ is

[TABLE]

The augmentation of $N_{2}=\{0,\mathsf{a}\}$ is the semigroup $N_{2}^{\mathsf{bar}}=\{\overline{0},\mathsf{a},\overline{\mathsf{a}},\overline{I}\}$ given in Table 2. Information on identities satisfied by the semigroups $N_{n}$, $N_{n}^{I}$, $N_{2}^{\mathsf{bar}}$, and $(N_{2}^{\mathsf{bar}})^{I}$ is given in Subsections 5.4, 5.5, 5.6, and 5.7, respectively.

3.3. Bands

The smallest nontrivial bands are the semilattice $Sl_{2}=\{0,1\}$ and the left zero and right zero semigroups of order two:

[TABLE]

see Table 3. Note that $Sl_{2}\cong N_{1}^{I}$ and $L_{2}^{\mathsf{op}}\cong R_{2}$. It is well known that $Sl_{2}$ generates the pseudovariety $\mathbf{Sl}$ of semilattices, $L_{2}$ generates the pseudovariety $\mathbf{LZ}$ of left zero semigroups, and $R_{2}$ generates the pseudovariety $\mathbf{RZ}$ of right zero semigroups.

The augmentation of $L_{2}$ is the semigroup $L_{2}^{\mathsf{bar}}=\{\mathsf{e},\mathsf{f},\overline{\mathsf{e}},\overline{\mathsf{f}},\overline{I}\}$ given in Table 4. Information on identities satisfied by the semigroups $L_{2}$, $L_{2}^{I}$, and $L_{2}^{\mathsf{bar}}$ is given in Subsections 5.8, 5.9, and 5.10, respectively.

3.4. Completely 0-simple semigroups

The smallest completely 0-simple semigroups with zero divisors are the idempotent-generated semigroup

[TABLE]

and the Brandt semigroup

[TABLE]

see Table 5. The Rees matrix representations of these semigroups are

[TABLE]

The semigroups $A_{2}$ and $B_{2}$ contain subsemigroups isomorphic to

[TABLE]

respectively; see Table 6. The semigroup

[TABLE]

and its augmentation $\ell_{3}^{\mathsf{bar}}=\{\overline{0},\mathsf{a},\mathsf{e},\overline{\mathsf{a}},\overline{\mathsf{e}}\}$ are given in Table 7. Note that

[TABLE]

Information on identities satisfied by the semigroups $A_{0}$, $A_{0}^{I}$, $A_{2}$, $B_{2}$, and $\ell_{3}^{\mathsf{bar}}$ is given in Subsections 5.11, 5.12, 5.13, 5.14, and 5.15, respectively.

It is shown in Subsection 4.5 that the semigroup $\ell_{3}$ belongs to an infinite class of semigroups $S$ with the property that $\langle\langle S\rangle\rangle$ is not $\mathsf{ji}$ but $\langle\langle S\rangle\rangle^{\mathsf{bar}}$ is $\mathsf{ji}$. The semigroup $B_{0}$ serves as a counterexample to the implications

[TABLE]

mentioned in the introduction.

Proposition 3.1**.**

1. (i)

The pseudovariety $\langle\langle B_{0}\rangle\rangle$ is **$\mathsf{sji}$.** 2. (ii)

The pseudovariety $\langle\langle B_{0}\rangle\rangle$ is not **$\mathsf{ji}$.** 3. (iii)

The pseudovariety $\langle\langle B_{0}^{I}\rangle\rangle$ is not **$\mathsf{sji}$.**

Proof.

The pseudovariety $\langle\langle B_{0}\rangle\rangle$ is $\mathsf{sji}$ since it has a unique maximal proper subpseudovariety [8, Lemma 5(b)]. The pseudovariety $\langle\langle B_{0}^{I}\rangle\rangle$ is not $\mathsf{sji}$ since it has two maximal proper subpseudovarieties [8, Lemma 6(b)]. In particular, the pseudovariety $\langle\langle B_{0}^{I}\rangle\rangle$ is not $\mathsf{ji}$, whence the pseudovariety $\langle\langle B_{0}\rangle\rangle$ is also not $\mathsf{ji}$; see Lemma 5.2. ∎

4. Some general results on join irreducibility

The pseudovariety defined by a class $\Sigma$ of pseudoidentities is denoted by $[[\Sigma]]$, while the pseudovariety generated by a class $\mathscr{K}$ of finite semigroups is denoted by $\langle\langle\mathscr{K}\rangle\rangle$. A pseudovariety is compact if it is generated by a single finite semigroup.

Proposition 4.1**.**

Every compact pseudovariety contains positively and only finitely many maximal subpseudovarieties.**

Proof.

Let $S=\{s_{1},s_{2},\ldots,s_{n}\}$ be any finite semigroup and let $\mathcal{V}$ denote the variety generated by $S$. Since the lattice of subvarieties of $\mathcal{V}$ is isomorphic to the lattice of subpseudovarieties of $\langle\langle S\rangle\rangle$, it suffices to show that $\mathcal{V}$ contains positively and only finitely many maximal subvarieties. Given any identity $\mathbf{u}\approx\mathbf{v}$ such that $S\not\models\mathbf{u}\approx\mathbf{v}$, there exists some substitution $\varphi$ into $S$ such that $\mathbf{u}\varphi\neq\mathbf{v}\varphi$. Then $\varphi$ induces a substitution $\varphi^{\prime}$ into the set $\mathscr{X}_{n}=\{x_{1},x_{2},\ldots,x_{n}\}$ given by $x\varphi^{\prime}=x_{i}$ if $x\varphi=s_{i}$. Therefore $\mathbf{u}\varphi^{\prime}\approx\mathbf{v}\varphi^{\prime}$ is an identity over $\mathscr{X}_{n}$ such that $\mathbf{u}\approx\mathbf{v}\vdash\mathbf{u}\varphi^{\prime}\approx\mathbf{v}\varphi^{\prime}$ and $S\not\models\mathbf{u}\varphi^{\prime}\approx\mathbf{v}\varphi^{\prime}$. It follows that every proper subvariety of $\mathcal{V}$ satisfies some identity over $\mathscr{X}_{n}$. Modulo the equational theory of the semigroup $S$, there can only be finitely many identities over $\mathscr{X}_{n}$ that are violated by $S$; these identities form a finite preordered set $\mathcal{P}$ under equational deduction $\vdash$. Each greatest element of $(\mathcal{P},\vdash)$ defines within $\mathcal{V}$ a maximal subvariety. ∎

The exclusion class $\operatorname{\mathsf{Excl}}(S)$ of a finite semigroup $S$ is the class of all finite semigroups $T$ for which $S\notin\langle\langle T\rangle\rangle$. Recall that a finite semigroup $S$ is $\mathsf{ji}$ if and only if $\operatorname{\mathsf{Excl}}(S)$ is a pseudovariety [23, Theorem 7.1.2].

In this section, some results on the property of being $\mathsf{ji}$ are established. There are seven subsections. The main result of Subsection 4.1 demonstrates that many exclusion classes of $\mathsf{ji}$ semigroups in this paper are not definable by a certain type of pseudoidentities. In Subsection 4.2, the notion of a “large” pseudovariety is introduced. It turns out that the exclusion class of a $\mathsf{ji}$ semigroup that is right letter mapping, left letter mapping, or group mapping satisfies this largeness condition. In Subsection 4.3, it is shown that the operator $\mathbf{V}\mapsto\mathbf{V}^{\mathsf{bar}}$ on $\mathbf{PV}$ preserves the property of being $\mathsf{ji}$. More specifically, if $\mathbf{u}\approx\mathbf{v}$ is a pseudoidentity that defines the exclusion class $\operatorname{\mathsf{Excl}}(S)$ of a $\mathsf{ji}$ semigroup $S$, then it is shown how a pseudoidentity that defines $\operatorname{\mathsf{Excl}}(S^{\mathsf{bar}})$ can be obtained from $\mathbf{u}\approx\mathbf{v}$.

In Subsection 4.4, it is shown that alternately performing the operators $\mathbf{V}\mapsto\mathbf{V}^{\mathsf{bar}}$ and $\mathbf{V}\mapsto\mathbf{V}^{\flat}=\langle\langle((S^{\mathsf{op}})^{\mathsf{bar}})^{\mathsf{op}}\mid S\in\mathbf{V}\rangle\rangle$ on a nontrivial pseudovariety $\langle\langle S\rangle\rangle$ results in an infinite increasing chain of pseudovarieties; if the semigroup $S$ is $\mathsf{ji}$ to begin with, then the pseudovarieties are all $\mathsf{ji}$. In Subsection 4.5, an infinite class $\{O_{k}\mid k\geq 2\}$ of finite semigroups is introduced and shown to satisfy the following property: for each $k\geq 2$, the pseudovariety $\langle\langle O_{k}\rangle\rangle$ is not $\mathsf{ji}$, while the pseudovariety $\langle\langle O_{k}\rangle\rangle^{\mathsf{bar}}$ is $\mathsf{ji}$.

In Subsection 4.6, a sufficient condition, due to G. M. Bergman, is presented under which a finite $\mathsf{sdi}$ group is $\mathsf{ji}$. In Subsection 4.7, the pseudovariety $\langle\langle Q_{8}\rangle\rangle=\langle\langle D_{4}\rangle\rangle$ is shown to be $\mathsf{ji}$; this result is due independently to Kearnes [5] and the anonymous reviewer.

4.1. Non-definability by simple pseudoidentities

For this subsection, the assumption that all semigroups are finite is temporarily abandoned. The free profinite semigroup on a set $\mathscr{A}$ is denoted by $\widehat{\mathscr{A}^{+}}$. A pseudoidentity $\mathbf{u}\approx\mathbf{v}$ is simple if $\mathbf{u}$ and $\mathbf{v}$ belong to the smallest subsemigroup $F(\mathscr{A})$ of $\widehat{\mathscr{A}^{+}}$ containing $\mathscr{A}$ that is closed under product and unary implicit operations; the latter condition means that ${\overline{\{\mathbf{w}\}^{+}}\subseteq F(\mathscr{A})}$ for all $\mathbf{w}\in F(\mathscr{A})$.

The following theorem was essentially proved by Almeida and Volkov [3], based on an earlier variant of Rhodes [22].

Theorem 4.2**.**

Suppose that $\mathbf{V}$ is any proper pseudovariety of semigroups containing all semigroups with abelian maximal subgroups.* Then $\mathbf{V}$ cannot be defined by simple pseudoidentities*.**

Proof.

Let $\mathscr{A}$ be a fixed countably infinite set and for any $m,n\geq 1$, let $\mathcal{B}_{m,n}$ be the variety of semigroups defined by the identity $x^{m}\approx x^{m+n}$. Then the free semigroup $B(1,m,n)$ on one-generator in $\mathcal{B}_{m,n}$ is finite and if $x^{\eta}\in\widehat{\{x\}^{+}}$, then there exists an integer $n_{\eta}\leq m+n-1$ such that $x^{\eta}=x^{n_{\eta}}$ in $B(1,m,n)$. Thus each implicit operation in $F(\mathscr{A})$ has a natural interpretation on any semigroup in $\mathcal{B}_{m,n}$ which agrees with its usual interpretation in finite semigroups (namely interpret $w^{\eta}$ as $w^{n_{\eta}}$ for every element $w$ of a semigroup $S\in\mathcal{B}_{m,n}$).

Suppose that $\mathbf{V}$ is defined by a set $\Sigma$ of simple pseudoidentities. Let $\mathcal{W}$ be the variety of universal algebras defined by $\Sigma$ in the signature $\tau$ consisting of multiplication and all unary implicit operations and let $T$ be a finite semigroup. Then there exist $m\geq 6$ and $n\geq 1$ such that $T$ belongs to $\mathcal{B}_{m,n}$. As discussed above, $\mathcal{B}_{m,n}$ can be viewed as a variety in the signature $\tau$ such that the unary implicit operations have their usual interpretations in all finite semigroups in $\mathcal{B}_{m,n}$.

Now McCammond [17] has shown that for each integer $k\geq 1$, the semigroup $B(k,m,n)$ has cyclic maximal subgroups and that there is a system of cofinite ideals for $B(k,m,n)$ with empty intersection. Therefore, $B(k,m,n)$ is an infinite subdirect product of finite semigroups with abelian maximal subgroups. Since $\mathcal{W}$ contains all finite semigroups with abelian maximal subgroups, it follows that $B(k,m,n)\in\mathcal{W}$, whence $\mathcal{B}_{m,n}\subseteq\mathcal{W}$. Therefore, $T$ belongs to $\mathcal{W}$ and so satisfies the pseudoidentities $\Sigma$. Consequently, $T\in\mathbf{V}$ and hence $\mathbf{V}$ is the pseudovariety of all finite semigroups. ∎

In this paper, pseudoidentities involving idempotents from the minimal ideal of a free profinite semigroup are often used to define the exclusion pseudovarieties of $\mathsf{ji}$ semigroups. Since many of these exclusion pseudovarieties contain all semigroups with abelian maximal subgroups, Theorem 4.2 implies that, in general, simple pseudoidentities cannot be used in their definition. It is presently unknown if one must use idempotents from the minimal ideal.

4.2. Large exclusion pseudovarieties

If $\mathbf{V}$ and $\mathbf{W}$ are pseudovarieties of semigroups, then their Mal’cev product \mathbf{V}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{W} is the pseudovariety generated by all semigroups $S$ with a homomorphism $\varphi\colon S\to T$ such that $T\in\mathbf{W}$ and $e\varphi^{-1}\in\mathbf{V}$ for all idempotents $e\in T$. A remarkable property of the Mal’cev product is that

[TABLE]

see [23] for details.

Let $\mathbf{1}$ denote the pseudovariety of trivial semigroups. For any $\mathsf{ji}$ semigroup $S$, we say that $\operatorname{\mathsf{Excl}}(S)$ is large if

[TABLE]

If $\operatorname{\mathsf{Excl}}(S)$ is large and $\{\mathbf{V}_{\alpha}\mid\alpha\in A\}$ is a collection of pseudovarieties such that $\bigcap_{\alpha\in A}\mathbf{V}_{\alpha}=\mathbf{1}$, then it follows from (4.1) and the fact that $\operatorname{\mathsf{Excl}}(S)$ is $\mathsf{mi}$ that \mathbf{V}_{\alpha}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\operatorname{\mathsf{Excl}}(S)=\operatorname{\mathsf{Excl}}(S) for some $\alpha\in A$. In particular, either

[TABLE]

where $\mathbf{A}$ is the pseudovariety of finite aperiodic semigroups and $\mathbf{G}$ is the pseudovariety of finite groups. For more examples of pseudovarieties with trivial intersection, see [23].

If $S$ is a finite subdirectly indecomposable semigroup, then $S$ has a unique [math]-minimal ideal $I$ (where if $S$ has no zero, then we consider the minimal ideal as [math]-minimal). Moreover, one of the following cases holds:

1. $\bullet$

$I^{2}=0$ (the null case); 2. $\bullet$

$S$ acts faithfully on the right of the set of $\mathscr{L}$-classes of $I$ (the left letter mapping case); 3. $\bullet$

$S$ acts faithfully on the left of the set of $\mathscr{L}$-classes of $I$ (the right letter mapping case); 4. $\bullet$

$I$ contains a nontrivial maximal subgroup and $S$ acts faithfully on both the left and right of $I$ (the group mapping case).

In the last three cases we say that $S$ is of semisimple type; see [23, Sec. 4.7].

Theorem 4.3**.**

Let $S$ be any subdirectly indecomposable $\mathsf{ji}$ semigroup of semisimple type (left letter mapping, right letter mapping,* or group mapping*).* Then $\operatorname{\mathsf{Excl}}(S)$ is large*.**

Proof.

Obviously, \operatorname{\mathsf{Excl}}(S)\subseteq\mathbf{1}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\operatorname{\mathsf{Excl}}(S). As $\operatorname{\mathsf{Excl}}(S)$ is the largest pseudovariety that fails to contain $S$, it suffices to show that S\notin\mathbf{1}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\operatorname{\mathsf{Excl}}(S). But [23, Theorem 4.6.50] immediately implies that in any of the three cases, S\in\mathbf{1}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V} if and only if $S\in\mathbf{V}$ for any pseudovariety $\mathbf{V}$. Thus S\notin\mathbf{1}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\operatorname{\mathsf{Excl}}(S) and so \mathbf{1}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\operatorname{\mathsf{Excl}}(S)=\operatorname{\mathsf{Excl}}(S). ∎

The proof of Theorem 4.3 is in fact valid if $S$ is left letter mapping, right letter mapping, or group mapping even if it is not $\mathsf{sdi}$.

4.3. Augmentation preserves join irreducibility

In this subsection, augmentation is shown to preserve join irreducibility. Some special cases were previously considered in [23, Section 7.3].

Theorem 4.4**.**

The operator $\mathbf{V}\mapsto\mathbf{V}^{\mathsf{bar}}$ preserves the property of being $\mathsf{ji}$.* In particular*,* if a pseudovariety $\langle\langle S\rangle\rangle$ is $\mathsf{ji}$, then the pseudovariety $\langle\langle S^{\mathsf{bar}}\rangle\rangle$ is also $\mathsf{ji}$. Further*,* if $\operatorname{\mathsf{Excl}}(S)=[[\mathbf{u}\approx\mathbf{v}]]$ where $\mathbf{u},\mathbf{v}\in\widehat{\mathscr{A}^{+}}$, then*

[TABLE]

where $z\notin\mathscr{A}$ and $\mathbf{e}$ is an idempotent in the minimal ideal of $\widehat{(\mathscr{A}\cup\{z\})^{+}}$.**

Proof.

First note that since $S\notin\operatorname{\mathsf{Excl}}(S)$, there exists some homomorphism $\varphi\colon\widehat{\mathscr{A}^{+}}\to S$ such that $\mathbf{u}\varphi\neq\mathbf{v}\varphi$. Let $1$ denote the identity element of $S^{\bullet}$, and extend $\varphi$ to a homomorphism $\widehat{(\mathscr{A}\cup\{z\})^{+}}\to S^{\mathsf{bar}}$ by sending $z$ to $\overline{1}$. Then $(\mathbf{e}z\mathbf{u})^{\omega}\varphi=\overline{\mathbf{u}\varphi}\neq\overline{\mathbf{v}\varphi}=(\mathbf{e}z\mathbf{v})^{\omega}\varphi$ and so $S^{\mathsf{bar}}\notin[[(\mathbf{e}z\mathbf{u})^{\omega}\approx(\mathbf{e}z\mathbf{v})^{\omega}]]$.

To complete the proof, it suffices to assume that $T\notin[[(\mathbf{e}z\mathbf{u})^{\omega}\approx(\mathbf{e}z\mathbf{v})^{\omega}]]$, and show that $S^{\mathsf{bar}}\in\langle\langle T\rangle\rangle$. Replacing $T$ by a subsemigroup if necessary, generality is not lost by assuming the existence of a surjective homomorphism $\psi\colon\widehat{(\mathscr{A}\cup\{z\})^{+}}\to T$ such that $(\mathbf{e}z\mathbf{u})^{\omega}\psi\neq(\mathbf{e}z\mathbf{v})^{\omega}\psi$. Now the semigroup $T$ acts on the right of the set $B$ of $\mathscr{L}$-classes of its minimal ideal $J$; let $(B,\mathsf{RLM}(T))$ denote the resulting faithful transformation semigroup. Note that $(B,\mathsf{RLM}(T))=\overline{(B,\mathsf{RLM}(T))}$ because if $b\in B$, then any element of $T$ in the $\mathscr{L}$-class of $b$ acts on $B$ as a constant map to $b$ by the structure of completely simple semigroups. It follows from Corollary 2.6 that $\langle\langle\mathsf{RLM}(T)\rangle\rangle^{\mathsf{bar}}=\langle\langle\mathsf{RLM}(T)\rangle\rangle$, since the constant mappings form the minimal ideal of $\mathsf{RLM}(T)$.

Since $(\mathbf{e}z)\psi$ is in the minimal ideal $J$ of $T$, the elements $((\mathbf{e}z)\psi)(\mathbf{u}\psi)$ and $((\mathbf{e}z)\psi)(\mathbf{v}\psi)$ are $\mathscr{R}$-equivalent. However, they are not $\mathscr{L}$-equivalent because otherwise they would be $\mathscr{H}$-equivalent and hence have the same idempotent power, as $J$ is completely simple. Thus $\mathbf{u}\psi$ and $\mathbf{v}\psi$ have distinct images under the quotient map $T\to\mathsf{RLM}(T)$. Consequently, there is a homomorphism $\varphi\colon\widehat{\mathscr{A}}^{+}\to\mathsf{RLM}(T)$ such that $\mathbf{u}\varphi\neq\mathbf{v}\varphi$, that is, $\mathsf{RLM}(T)\notin\operatorname{\mathsf{Excl}}(S)$. Therefore, $S\in\langle\langle\mathsf{RLM}(T)\rangle\rangle$, whence $S^{\mathsf{bar}}\in\langle\langle\mathsf{RLM}(T)\rangle\rangle^{\mathsf{bar}}=\langle\langle\mathsf{RLM}(T)\rangle\rangle\subseteq\langle\langle T\rangle\rangle$ as required. ∎

Corollary 4.5**.**

Let $(X,S)$ be any transformation semigroup with $(X,S)\prec(S^{\bullet},S)$.* Suppose that the pseudovariety $\langle\langle S\rangle\rangle$ is $\mathsf{ji}$. Then the pseudovariety $\langle\langle S\cup\overline{X}\rangle\rangle$ is also $\mathsf{ji}$*.**

Proof.

This follows from Corollary 2.3 and Theorem 4.4. ∎

Note that if $S$ is $\mathsf{ji}$, then $\operatorname{\mathsf{Excl}}(S^{\mathsf{bar}})$ will be large by Theorem 4.3 (and the remark following it).

4.4. Iterating augmentation and its dual to bands

For any semigroup $S$, define

[TABLE]

In other words, $S^{\flat}$ is obtained by considering the left action of $S$ on $S^{\bullet}$ and adjoining constant maps. For any pseudovariety $\mathbf{V}$, define

[TABLE]

By symmetry, $\mathbf{V}\mapsto\mathbf{V}^{\flat}$ is a continuous idempotent operator that preserves join irreducibility; see [23, Chapter 2]. Define the operators $\alpha,\beta\colon\mathbf{PV}\to\mathbf{PV}$ by $\alpha\mathbf{V}=\mathbf{V}^{\mathsf{bar}}$ and $\beta\mathbf{V}=\mathbf{V}^{\flat}$. The aim of this subsection is to show that for any nontrivial finite semigroup $S$, the hierarchy

[TABLE]

is strict, as is the dual hierarchy obtained by interchanging the roles of $\alpha$ and $\beta$. An important observation is that $\beta\alpha\langle\langle S\rangle\rangle$ is a compact pseudovariety containing $Sl_{2}$ that is generated by $(S^{\mathsf{bar}})^{\flat}$, which is left mapping with respect to its minimal ideal. Thus it suffices to handle the case that $\mathbf{Sl}\subseteq\langle\langle S\rangle\rangle$ and $S$ is left mapping with respect to its minimal ideal.

Proposition 4.6**.**

For any finite semigroup $S$,**

[TABLE]

Proof.

Clearly, $S^{\mathsf{bar}}/\overline{S^{\bullet}}$ divides the semigroup $S^{0}$ obtained by adjoining an external zero element [math] to $S$. Since $\overline{S^{\bullet}}$ is a right zero semigroup and $\langle\langle S^{0}\rangle\rangle\subseteq\langle\langle S\rangle\rangle\vee\mathbf{Sl}$, the inclusion S^{\mathsf{bar}}\in\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,(\langle\langle S\rangle\rangle\vee\mathbf{Sl}) holds. The second inclusion is dual. ∎

Define the operators $\widetilde{\alpha},\widetilde{\beta}\colon\mathbf{PV}\to\mathbf{PV}$ by \widetilde{\alpha}\mathbf{V}=\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V} and \widetilde{\beta}\mathbf{V}=\mathbf{LZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V}. These operators are idempotent. For any finite semigroup $S$ that contains $Sl_{2}$ as a subsemigroup, define the hierarchy

[TABLE]

Observe that $\mathbf{V}_{n}\subseteq\mathbf{U}_{n}$ for all $n\geq 0$ as a consequence of Proposition 4.6.

Proposition 4.7**.**

Suppose that $S$ is any nontrivial band that is left mapping with respect to its minimal ideal and that $\mathbf{V}$ is any pseudovariety such that $\mathbf{Sl}\subseteq\mathbf{V}$.* Then S^{\mathsf{bar}}\in\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V} if and only if $S\in\mathbf{V}$*.**

Proof.

If $S\in\mathbf{V}$, then S^{\mathsf{bar}}\in\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V} by Proposition 4.6. Conversely, since $S^{\mathsf{bar}}$ is a band, S^{\mathsf{bar}}\in\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V} if and only if S^{\mathsf{bar}}\in\mathbf{D}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V}, where $\mathbf{D}$ is the pseudovariety of semigroups whose idempotents are right zeroes; this occurs if and only if the quotient of $S^{\mathsf{bar}}$ by the intersection $\mathsf{LM}$ of all its left mapping congruences belongs to $\mathbf{V}$ [23, Theorem 4.6.50]. Note that since $S$ is a left mapping band with respect to its minimal ideal, its minimal ideal consists of at least two left zeroes. Therefore, the minimal ideal of $S^{\mathsf{bar}}$ contains no elements of $S$. Then $S^{\mathsf{bar}}/\mathsf{LM}\cong S^{0}$ because $S^{\mathsf{bar}}$ acts trivially on the left of its minimal ideal and acts as $S$ does on the left of its other $\mathscr{J}$-classes. Since $Sl_{2}\in\mathbf{V}$ by assumption, it follows that $S^{\mathsf{bar}}/\mathsf{LM}\in\mathbf{V}$ if and only if $S\in\mathbf{V}$. ∎

Corollary 4.8**.**

Suppose that $S$ is any nontrivial band that is left mapping with respect to its minimal ideal and that $\mathbf{V}$ is any pseudovariety such that $\mathbf{Sl}\subseteq\mathbf{V}$.* Then $(S^{\mathsf{bar}})^{\flat}\in\widetilde{\beta}\widetilde{\alpha}\mathbf{V}$ if and only if $S\in\mathbf{V}$*.**

Proof.

Since $S^{\mathsf{bar}}$ is a nontrivial band that is right mapping with respect to its minimal ideal, the dual of Proposition 4.7 implies that (S^{\mathsf{bar}})^{\flat}\in\mathbf{LZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,(\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V}) if and only if S^{\mathsf{bar}}\in\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V}. An application of Proposition 4.7 then yields that (S^{\mathsf{bar}})^{\flat}\in\mathbf{LZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,(\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{V}) if and only if $S\in\mathbf{V}$. ∎

The hierarchies (4.2) and (4.3) for the case $S=Sl_{2}$ are now analyzed. Recall that $\mathbf{B}$ denotes the pseudovariety of finite bands.

Lemma 4.9**.**

Consider the hierarchies (4.2) and (4.3) with $S=Sl_{2}$.* Then*

1. (i)

$\mathbf{V}_{n}\nsubseteq\mathbf{U}_{n-1}$* for all $n\geq 1$;* 2. (ii)

the hierarchies (4.2) and (4.3) are strict;** 3. (iii)

$\bigcup_{n\geq 0}\mathbf{U}_{n}=\bigcup_{n\geq 0}\mathbf{V}_{n}=\mathbf{B}$.**

Proof.

(i) This is established by induction on $n$. The exclusion $\mathbf{V}_{1}\nsubseteq\mathbf{U}_{0}$ holds since $S^{\mathsf{bar}}\in\mathbf{V}_{1}$ while $S^{\mathsf{bar}}\notin\mathbf{Sl}=\mathbf{U}_{0}$ due to $R_{2}\subseteq S^{\mathsf{bar}}$. Suppose that $\mathbf{V}_{n}\nsubseteq\mathbf{U}_{n-1}$ for some $n\geq 2$. Note that $\mathbf{V}_{n}$ is generated by a band of the form $T=R^{\flat}$ and so $T$ is left mapping with respect to its minimal ideal. Since $T\notin\mathbf{U}_{n-1}$, it follows from Corollary 4.8 that $(T^{\mathsf{bar}})^{\flat}\notin\mathbf{U}_{n}$. Therefore, $(T^{\mathsf{bar}})^{\flat}\in\mathbf{V}_{n+1}\setminus\mathbf{U}_{n}$, whence $\mathbf{V}_{n+1}\nsubseteq\mathbf{U}_{n}$.

(ii) Since $\mathbf{V}_{n}\nsubseteq\mathbf{U}_{n-1}$ by part (i) and $\mathbf{V}_{n-1}\subseteq\mathbf{U}_{n-1}$, the hierarchy (4.2) is strict. Similarly, $\mathbf{V}_{n}\subseteq\mathbf{U}_{n}$ and $\mathbf{V}_{n}\nsubseteq\mathbf{U}_{n-1}$ imply that the hierarchy (4.3) is strict.

(iii) This result holds because the lattice of band pseudovarieties is well known not to contain any strictly increasing infinite chain of subpseudovarieties whose union is not $\mathbf{B}$. ∎

Theorem 4.10**.**

The hierarchy (4.2) is strict for any nontrivial finite semigroup $S$.**

Proof.

Since the hierarchy stabilizes as soon as two consecutive pseudovarieties are identical, replacing $S$ by $(S^{\mathsf{bar}})^{\flat}$ if necessary, $S$ can be assumed to contain $Sl_{2}$ as a subsemigroup. It then follows from Lemma 4.9 that $\bigcup_{n\geq 0}\mathbf{V}_{n}$ contains the pseudovariety $\mathbf{B}$. But since $\mathbf{B}$ is not contained in any compact pseudovariety [25], the union $\bigcup_{n\geq 0}\mathbf{V}_{n}$ is not compact. Since each pseudovariety $\mathbf{V}_{n}$ is compact, the hierarchy is strict. ∎

Corollary 4.11**.**

If $\langle\langle S\rangle\rangle$ is $\mathsf{ji}$,* then the pseudovarieties*

[TABLE]

are $\mathsf{ji}$;* these pseudovarieties are all distinct except possibly for $\langle\langle S\rangle\rangle=\langle\langle S^{\mathsf{bar}}\rangle\rangle$. A dual result holds when ♭ is first applied before bar*.**

Corollary 4.12**.**

The pseudovariety $\mathbf{B}$ is $\mathsf{fji}$.**

Proof.

Since the pseudovariety $\mathbf{Sl}$ is $\mathsf{ji}$, each step in the hierarchy (4.2) is $\mathsf{ji}$ with $S=Sl_{2}$. As the union of a chain of $\mathsf{ji}$ pseudovarieties is $\mathsf{fji}$ [23, Lemma 6.1.13], it follows from Lemma 4.9 that $\mathbf{B}$ is $\mathsf{fji}$. ∎

Using the known structure of the lattice of band pseudovarieties [19] (which coincides with the lattice of all band varieties), we can say more. Namely, we will show that any $\mathsf{sji}$ band is $\mathsf{ji}$. Recall that $\mathbf{LNB}=\mathbf{Sl}\vee\mathbf{LZ}$.

Proposition 4.13**.**

The pseudovariety \mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{LNB} is generated by $L_{2}^{\mathsf{bar}}$.**

Proof.

It follows from Pastijn [19, Figure 3] and the description of the lattice of pseudovarieties of bands (see, for example, Almeida [2, Figure 5.1]) that the pseudovariety \langle\langle L_{2},R_{2}^{I}\rangle\rangle=\big{[}\big{[}x^{2}\approx x,\,xyz\approx xzyz\big{]}\big{]} is the unique maximal subpseudovariety of \mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{LNB}=\big{[}\big{[}x^{2}\approx x,\,xyz\approx xzxyz\big{]}\big{]}. It is then routinely checked that L_{2}^{\mathsf{bar}}\in\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{LNB}\setminus\langle\langle L_{2},R_{2}^{I}\rangle\rangle. Consequently, \langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle=\mathbf{RZ}\,{\mathbin{\hbox{\bigcirc\hbox to0.0pt{\kern-8.5pt\raise 0.5pt\hbox{{\tt m}}\hss}}}}\,\mathbf{LNB}. ∎

By Proposition 4.13 and results from Pastijn [19], a description of proper $\mathsf{sji}$ pseudovarieties of bands can be given as follows. Let $S=L_{2}^{\mathsf{bar}}$ and $T=R_{2}^{\flat}$. Then the proper nontrivial $\mathsf{sji}$ pseudovarieties of bands are $\mathbf{LZ}$, $\mathbf{RZ}$, and those pseudovarieties that can be obtained by applying an alternating word $\mathbf{w}(\widetilde{\alpha},\widetilde{\beta})$ over $\{\widetilde{\alpha},\widetilde{\beta}\}$ to the pseudovarieties generated by $S$, $T$, or $Sl_{2}$ (where the last letter of $\mathbf{w}$ should be $\widetilde{\beta}$ when starting from $\langle\langle S\rangle\rangle$ and should be $\widetilde{\alpha}$ when starting from $\langle\langle T\rangle\rangle$). Further, there are no $\mathsf{sji}$ pseudovarieties strictly in between any successive iterations of these operators. Since $\alpha\mathbf{V}\leq\widetilde{\alpha}\mathbf{V}$ and $\beta\mathbf{V}\leq\widetilde{\beta}\mathbf{V}$ for any pseudovariety $\mathbf{V}$ containing $\mathbf{Sl}$, and each successive iteration of $\alpha$ and $\beta$ starting from the pseudovariety generated by one of $S$, $T$, or $Sl_{2}$ (where the rightmost operator applied must be $\beta$ for $S$ and $\alpha$ for $T$) results in a new $\mathsf{ji}$ pseudovariety, it follows that if $\mathbf{w}(x,y)$ is any alternating word over $\{x,y\}$, then $\mathbf{w}(\alpha,\beta)\mathbf{V}=\mathbf{w}(\widetilde{\alpha},\widetilde{\beta})\mathbf{V}$ whenever $\mathbf{V}$ is one of the pseudovarieties generated by $S$, $T$, or $Sl_{2}$. Consequently, each $\mathsf{sji}$ proper pseudovariety of bands is, in fact, $\mathsf{ji}$ by Corollary 4.11. The following result is thus established.

Theorem 4.14**.**

Any $\mathsf{sji}$ band is $\mathsf{ji}$,* that is*,* a proper pseudovariety of bands is $\mathsf{sji}$ if and only if it is $\mathsf{ji}$*.**

In particular, since $\mathsf{sji}$ is a decidable property, $\mathsf{ji}$ is also decidable for finite bands. The answer to Question 1.2 is thus affirmative for bands.

4.5. From non-$\mathsf{ji}$ pseudovarieties to $\mathsf{ji}$ pseudovarieties

For each $k\geq 2$, define the semigroup

[TABLE]

The main goal of this subsection is to show that the pseudovariety $\langle\langle O_{k}\rangle\rangle$ is not $\mathsf{ji}$ whereas the pseudovariety $\langle\langle O_{k}\rangle\rangle^{\mathsf{bar}}$ is $\mathsf{ji}$. It is also shown that the pseudovarieties $\langle\langle O_{2}\rangle\rangle^{\mathsf{bar}},\langle\langle O_{3}\rangle\rangle^{\mathsf{bar}},\langle\langle O_{4}\rangle\rangle^{\mathsf{bar}},\ldots$ are all distinct.

It is easily seen that the semigroups $O_{2}$ and $\ell_{3}$ are isomorphic by referring to their presentations. Since the semigroup $\ell_{3}^{\mathsf{bar}}$ is of order five (Subsection 3.4), the $\mathsf{ji}$ pseudovariety $\langle\langle\ell_{3}^{\mathsf{bar}}\rangle\rangle=\langle\langle O_{2}\rangle\rangle^{\mathsf{bar}}$ is required later in the paper (Theorem 5.29).

Lemma 4.15**.**

For each $k\geq 2$,* the semigroup $O_{k}$ consists precisely of the following $2k-1$ distinct elements*:**

[TABLE]

Proof.

It is routinely checked that (4.4) are all the elements of $O_{k}$. Therefore, it remains to verify that the elements in (4.4) are distinct. Recall that the right zero semigroup of order two is $R_{2}=\{\mathsf{e},\mathsf{f}\}$ and that the monogenic nilpotent semigroup of order $k$ is

[TABLE]

Consider the subsemigroup $T=(N_{k}^{I}\times R_{2})\setminus\{(I,\mathsf{e})\}$ of $N_{k}^{I}\times R_{2}$ and the ideal $J=\{(0,\mathsf{e}),(0,\mathsf{f}),(\mathsf{a}^{k-1},\mathsf{f})\}$ of $T$. Define $\varphi\colon\{x,e\}^{+}\to T/J$ by $x\varphi=(\mathsf{a},\mathsf{e})$ and $e\varphi=(I,\mathsf{f})$. Then

[TABLE]

It follows that $\varphi$ induces a homomorphism $O_{k}\mapsto T/J$ that separates the elements in (4.4). ∎

Proposition 4.16**.**

The pseudovariety $\langle\langle O_{k}\rangle\rangle$ is not $\mathsf{ji}$.**

Proof.

Since $O_{k}\prec T/J\prec N_{k}^{I}\times R_{2}$ by the proof of Lemma 4.15 (where we retain the notation of that proof), the inclusion $\langle\langle O_{k}\rangle\rangle\subseteq\langle\langle N_{k}^{I}\rangle\rangle\vee\mathbf{RZ}$ holds. But $\langle\langle N_{k}^{I}\rangle\rangle$ consists of commutative semigroups while $\mathbf{RZ}$ consists of bands. Therefore, $\langle\langle O_{k}\rangle\rangle\nsubseteq\langle\langle N_{k}^{I}\rangle\rangle$ and $\langle\langle O_{k}\rangle\rangle\nsubseteq\mathbf{RZ}$. ∎

It remains to prove that the pseudovariety $\langle\langle O_{k}\rangle\rangle^{\mathsf{bar}}$ is $\mathsf{ji}$.

Lemma 4.17**.**

Suppose that $U$ is any semigroup generated by two elements $f$ and $y$ such that $f^{2}=f$,* $fy=y$, and $y^{k-1}\notin\{y^{n}\mid n\geq k\}$. Then*

1. (i)

$y,y^{2},\ldots,y^{k-1}$* are distinct and not in $\{y^{n}\mid n\geq k\}$;* 2. (ii)

$f,yf,y^{2}f,\ldots,y^{k-2}f$* are distinct and not in $\{y^{m}f\mid m\geq k-1\}$;* 3. (iii)

$y^{i}=y^{j}f$* implies that either $i=j$ or $i,j\geq k-1$.*

Proof.

(i) This follows from the structure of monogenic semigroups.

(ii) Suppose that $y^{i}f=y^{j}f$ for some $i,j\geq 0$. Then $y^{i+1}=y^{i}fy=y^{j}fy=y^{j+1}$. Therefore, by part (i), either $i=j$ or $i,j\geq k-1$.

(iii) Suppose that $y^{i}=y^{j}f$ for some $i\geq 1$ and $j\geq 0$. Then $y^{i+1}=y^{j}fy=y^{j+1}$. Therefore, by part (i), either $i=j$ or $i,j\geq k-1$. ∎

Recall that the inclusion $\langle\langle O_{k}\rangle\rangle\subseteq\langle\langle N_{k}^{I}\rangle\rangle\vee\mathbf{RZ}$ was established in the proof of Proposition 4.16; this result is generalized in the following.

Lemma 4.18**.**

Suppose that $T$ is any finite semigroup generated by two elements $d$ and $z$ such that $d^{2}=d$,* $dz=z$, and $z^{k-1}\notin\{z^{n}\mid n\geq k\}$. Then $\langle\langle O_{k}\rangle\rangle\subseteq\langle\langle T\rangle\rangle\vee\mathbf{RZ}$*.**

Proof.

Consider the semigroup $T\times R_{2}$ and its subsemigroup $U=\langle y,f\rangle$ generated by $y=(z,\mathsf{e})$ and $f=(d,\mathsf{f})$. Then it is routinely checked that

1. (a)

$f^{2}=f$, $fy=y$, 2. (b)

$y^{n}=(z^{n},\mathsf{e})$ for all $n\geq 1$, 3. (c)

$y^{n}f=(z^{n}d,\mathsf{f})$ for all $n\geq 1$.

It follows from (b) and the assumption $z^{k-1}\notin\{z^{n}\mid n\geq k\}$ that

1. (d)

$y^{k-1}\notin\{y^{n}\mid n\geq k\}$.

It is clear from (a) that $U=\{y^{i},y^{j}f\mid i\geq 1,j\geq 0\}$. In fact, it follows from (a)–(d) and Lemma 4.17 that

1. (e)

the elements $y,y^{2},y^{3},\ldots,y^{k},f,yf,y^{2}f,\ldots,y^{k-1}f$ of $U$ are distinct.

Now it is routinely checked that the set

[TABLE]

is an ideal of $U$. By (e), the set $U\setminus J$ consists of the elements

[TABLE]

Therefore, $O_{k}\cong U/J$ by Lemma 4.15, whence $\langle\langle O_{k}\rangle\rangle\subseteq\langle\langle T\rangle\rangle\vee\mathbf{RZ}$. ∎

Theorem 4.19**.**

1. (i)

For each $k\geq 2$, the pseudovariety $\langle\langle O_{k}\rangle\rangle^{\mathsf{bar}}$ is $\mathsf{ji}$ and

[TABLE]

where $\mathbf{e}$ is an idempotent in the minimal ideal of $\widehat{\{a,b,c\}^{+}}$. 2. (ii)

The pseudovarieties $\langle\langle O_{2}\rangle\rangle^{\mathsf{bar}},\langle\langle O_{3}\rangle\rangle^{\mathsf{bar}},\langle\langle O_{4}\rangle\rangle^{\mathsf{bar}},\ldots$ are all distinct.**

Proof.

(i) Let $\varphi$ denote the substitution into $O_{k}^{\mathsf{bar}}$ given by $a\mapsto e$, $b\mapsto x$, and $c\mapsto\overline{1}$. Then $(\mathbf{e}c(a^{\omega}b)^{k-1})^{\omega}\varphi=\overline{x^{k-1}}$ and $(\mathbf{e}c((a^{\omega}b)^{k-1})^{\omega+1})^{\omega}\varphi=\overline{x^{k}}$, and these are different elements of $O_{k}^{\mathsf{bar}}$. Therefore, the semigroup $O_{k}^{\mathsf{bar}}$ violates the pseudoidentity in (4.5).

It remains to assume that a semigroup $T$ violates the pseudoidentity in (4.5), and then show that $O_{k}^{\mathsf{bar}}\in\langle\langle T\rangle\rangle$. Replacing $T$ by a subsemigroup if necessary, generality is not lost by assuming the existence of a surjective homomorphism $\psi\colon\widehat{\{a,b,c\}^{+}}\to T$ such that

[TABLE]

Put $f=a^{\omega}\psi$ and $y=(a^{\omega}b)\psi$ and note that $f^{2}=f$ and $fy=y$.

The semigroup $T$ acts on the right of the set $B$ of $\mathscr{L}$-classes of its minimal ideal $J$; denote the corresponding faithful transformation semigroup by $(B,\mathsf{RLM}(T))$. Note that $(B,\mathsf{RLM}(T))=\overline{(B,\mathsf{RLM}(T))}$ because if $b\in B$, then any element of $T$ in the $\mathscr{L}$-class of $b$ acts on $B$ as a constant map to $b$ by the structure of completely simple semigroups. Consequently, $\langle\langle\mathsf{RLM}(T)\rangle\rangle^{\mathsf{bar}}=\langle\langle\mathsf{RLM}(T)\rangle\rangle$ by Corollary 2.6 since the constant mappings form the minimal ideal of $\mathsf{RLM}(T)$.

Since $(\mathbf{e}c)\psi$ is in the minimal ideal $J$ of $T$, it follows that the elements $((\mathbf{e}c)\psi)y^{k-1}$ and $((\mathbf{e}c)\psi)(y^{k-1})^{\omega+1}$ are $\mathscr{R}$-equivalent. However, they are not $\mathscr{L}$-equivalent because otherwise they would be $\mathscr{H}$-equivalent and hence have the same idempotent power, as $J$ is completely simple. Consequently, $\mathsf{RLM}(T)$ is nontrivial and so it follows from Proposition 2.5 that $\langle\langle\mathsf{RLM}(T)\rangle\rangle=\langle\langle\mathsf{RLM}(T)\rangle\rangle\vee\mathbf{RZ}$. Also, if $z$ denotes the image of $y$ under the quotient map $T\to\mathsf{RLM}(T)$ and $d$ denotes the image of $f$ under this map, then $d^{2}=d$, $dz=z$, and $z^{k-1}$ is not a group element (as $z^{k-1}$ and $(z^{k-1})^{\omega+1}$ act differently on the $\mathscr{L}$-class of $(\mathbf{e}c)\psi$). Thus Lemma 4.18 implies that $O_{k}\in\langle\langle\mathsf{RLM}(T)\rangle\rangle\vee\mathbf{RZ}=\langle\langle\mathsf{RLM}(T)\rangle\rangle$. Consequently, $O_{k}^{\mathsf{bar}}\in\langle\langle\mathsf{RLM}(T)\rangle\rangle^{\mathsf{bar}}=\langle\langle\mathsf{RLM}(T)\rangle\rangle\subseteq\langle\langle T\rangle\rangle$.

(ii) This holds because for each $k\geq 2$, the semigroup $O_{k}^{\mathsf{bar}}$ satisfies the identity $x^{k+1}\approx x^{k}$ but violates the identity $x^{k}\approx x^{k-1}$. ∎

4.6. A sufficient condition for the join irreducibility of groups

Recall that a normal subgroup $N$ of a group $G$ splits if there exists a subgroup $K$ of $G$ so that $N\cap K=\{1\}$ and $NK=G$.

Theorem 4.20** (G. M. Bergman, private communication, 2014).**

Suppose that $G$ is any finite $\mathsf{sdi}$ group with an abelian monolith $N$ that splits.* Then $G$ is $\mathsf{ji}$*.**

Proof.

By assumption, there exists a subgroup $K$ of $G$ with $N\cap K=\{1\}$ and $NK=G$. Seeking a contradiction, suppose there exist finite groups $G_{1}$ and $G_{2}$ and some surjective homomorphism $f$ from a subgroup $H$ of $G_{1}\times G_{2}$ onto $G$ such that $G\notin\langle\langle G_{1}\rangle\rangle$ and $G\notin\langle\langle G_{2}\rangle\rangle$. Clearly we can assume that $H\pi_{j}=G_{j}$ for the projection maps $\pi_{j}:G_{1}\times G_{2}\twoheadrightarrow G_{j}$, that is, $H$ is a subdirect product of $G_{1}$ and $G_{2}$. Further, we may assume that $G_{1}$, $G_{2}$, and $H$ are chosen so that the order of $H$ is minimal.

Let $H_{2}=\{h_{2}\in G_{2}\mid(1,h_{2})\in H\}\cong\ker(\pi_{1})\cap H$. If $H_{2}$ is trivial, then $\pi_{1}$ is injective on $H$, so that $H\cong G_{1}$, whence the contradiction $G\prec G_{1}$ is obtained. Hence $H_{2}$ is nontrivial. Observe that

1. (†)

if $L$ is a subgroup of $H_{2}$ such that $\{1\}\times L\unlhd H$, then $L\unlhd G_{2}$;

in particular, $H_{2}\unlhd G_{2}$. Indeed, if $\ell\in L$ and $g_{2}\in G_{2}$, then choosing any $g_{1}\in G_{1}$ with $(g_{1},g_{2})\in H$, we have

[TABLE]

by normality of $\{1\}\times L$ in $H$, whence $g_{2}\ell g_{2}^{-1}\in L$.

Suppose that $\ker(f)$ has nontrivial intersection with the subgroup $\{1\}\times H_{2}$ of $H$, say $\ker(f)\cap(\{1\}\cap H_{2})=\{1\}\times L$ for some $L\subseteq G_{2}$. Then $L$ is normal in $H_{2}$ and so also normal in $G_{2}$ by (†). By dividing $G_{2}$ by this intersection, we could contradictorily decrease the order of $H$. Therefore, $\ker(f)$ intersects $\{1\}\times H_{2}$ trivially.

Similarly, defining $H_{1}=\{h_{1}\in G_{1}\mid(h_{1},1)\in H\}$, we have $\{1\}\neq H_{1}\unlhd G_{1}$ and $\ker(f)$ intersects $H_{1}\times\{1\}$ trivially. Then $H_{1}\times\{1\}$, $\{1\}\times H_{2}$, and $\ker(f)$ are all normal in $H$ and have pairwise trivial intersections.

Note that the centralizer of $N$ in $G$ is $N$. Indeed, since $N$ is the unique minimal normal subgroup of $G$, the action of $K$ on $N$ by conjugation is faithful (otherwise, the kernel would be a normal subgroup of $G$ not containing $N$). If $kn$ centralizes $N$ with $k\in K$ and $n\in N$, then since $N$ is abelian, we have that $k$ centralizes $N$ and hence $k=1$ by the previous observation.

From now on, identify $H_{1}$ with $H_{1}\times\{1\}$ and $H_{2}$ with $\{1\}\times H_{2}$. Then $H_{1}$ and $H_{2}$ are normal in $H$ and commute elementwise. We claim now that $H_{1}f=N=H_{2}f$. Indeed, since $f$ is injective on each of these subgroups and these subgroups are normal in $H$, we conclude that $N$ is contained in $H_{1}f\cap H_{2}f$. Since $H_{1}$ and $H_{2}$ commute elementwise, both $H_{1}f$ and $H_{2}f$ are contained in the centralizer of $N$, which is $N$. We conclude that $H_{1}f=N=H_{2}f$ and $f$ restricts to an isomorphism of $H_{1}$ and $H_{2}$ with $N$.

Let $H^{*}=Kf^{-1}$. Then since $N\cap K=\{1\}$, it follows that $H^{*}\cap H_{1}$ is a subgroup of $\ker(f)$. But $\ker(f)\cap H_{1}$ is trivial, so that $H^{*}\cap H_{1}=\{1\}$. Similarly, $H^{*}\cap H_{2}=\{1\}$. Note that $H^{*}H_{1}$ and $H^{*}H_{2}$ are subgroups of $H$ because $H_{1}$ and $H_{2}$ are normal. Also $(H^{*}H_{1})f=KN=G=(H^{*}H_{2})f$ and so by minimality of $H$, we have $H^{*}H_{1}=H=H^{*}H_{2}$. In particular, $G_{2}\cong H\pi_{2}=(H^{*}H_{1})\pi_{2}=H^{*}\pi_{2}$ and so, since $H^{*}\cap H_{1}=\{1\}$, we deduce that $G_{2}\cong H^{*}$. Similarly, $G_{1}\cong H^{*}$. Therefore, $G\prec G_{1}\times G_{1}$ and so $G\in\langle\langle G_{1}\rangle\rangle$, a contradiction. ∎

4.7. Join irreducibility of the pseudovariety $\langle\langle D_{4}\rangle\rangle=\langle\langle Q_{8}\rangle\rangle$

Proposition 4.21**.**

The pseudovariety $\langle\langle D_{4}\rangle\rangle=\langle\langle Q_{8}\rangle\rangle$ is $\mathsf{ji}$.* Further*,* the pseudovariety $\operatorname{\mathsf{Excl}}(Q_{8})$ is the class of all finite semigroups whose $2$-subgroups are abelian*,* that is*,* finite semigroups whose maximal subgroups have abelian $2$-Sylow subgroups*.**

Proof.

Since the variety of $2$-groups is saturated, it follows that the finite semigroups whose $2$-subgroups are abelian form a pseudovariety. To complete the proof, it suffices to observe that the pseudovariety generated by any finite non-abelian $2$-group contains $\langle\langle Q_{8}\rangle\rangle$. A proof can be found in Almeida [1, Theorem 4.5], based on the classification of finite $2$-groups whose proper subgroups are abelian, going back to Miller and Moreno [18]; we are indebted to the anonymous reviewer for pointing this out. Kearnes [5] gave a direct proof that any finite non-abelian $2$-group generates a variety containing $\langle\langle Q_{8}\rangle\rangle$ via a general description of identity bases for finite nilpotent groups of class $2$. ∎

Since the group $Q_{8}$ is $2$-generated as a semigroup, the pseudovariety $\operatorname{\mathsf{Excl}}(Q_{8})$ can be defined by a pseudoidentity over two variables [23, Proposition 7.1.9]. We proceed to describe such a pseudoidentity. Let $\widehat{F}=\widehat{\{x,y\}^{+}}$ be the free profinite semigroup over $\{x,y\}$ and let $\eta$ be an idempotent in the minimal ideal of $\widehat{F}$. Then $\eta\widehat{F}\eta$ is a profinite group and maps onto the free profinite group on two generators under the natural projection. The elements $x^{\prime}=\eta x\eta$ and $y^{\prime}=\eta y\eta$ map onto the free generators and thus freely topologically generate a free profinite subgroup of $\widehat{F}$, which is a retract. This observation was first made by Almeida and Volkov [3]. If $\mathbf{u}(x,y)$ and $\mathbf{v}(x,y)$ are elements of the free profinite group $\widehat{F}_{\mathbf{G}}(\{x,y\})$ over $\{x,y\}$, then $\mathbf{u}(x^{\prime},y^{\prime}),\mathbf{v}(x^{\prime},y^{\prime})\in\eta\widehat{F}\eta$ and it is easy to see that the pseudoidentity $\mathbf{u}(x^{\prime},y^{\prime})\approx\mathbf{v}(x^{\prime},y^{\prime})$ defines the pseudovariety of all semigroups whose maximal subgroups belong to the pseudovariety defined by $\mathbf{u}(x,y)\approx\mathbf{v}(x,y)$; see [3] for details. Thus, it suffices to find a two-variable group pseudoidentity $\mathbf{u}(x,y)\approx\mathbf{v}(x,y)$ defining the pseudovariety of groups with abelian $2$-Sylow subgroups (or, equivalently, $2$-subgroups).

Let $\widehat{F}_{\mathbf{G}_{2}}(\{x,y\})$ be the free pro-$2$ group over $\{x,y\}$. We have a natural continuous surjection $\pi\colon\widehat{F}_{\mathbf{G}}(\{x,y\})\to\widehat{F}_{\mathbf{G}_{2}}(\{x,y\})$. The pseudovariety of $2$-groups is saturated and so by Ribes and Zalesskii [24, Proposition 7.6.7], the group $\widehat{F}_{\mathbf{G}_{2}}(\{x,y\})$ is a projective profinite group. Therefore, there is a continuous splitting of $\pi$, that is, we can find $\mathbf{w}_{1}(x,y),\mathbf{w}_{2}(x,y)\in\widehat{F}_{\mathbf{G}}(\{x,y\})$ which freely topologically generate a free pro-$2$ subgroup with $\pi(\mathbf{w}_{1}(x,y))=\pi(x)$ and $\pi(\mathbf{w}_{2}(x,y))=\pi(y)$; in other words, $\mathbf{w}_{1}(x,y)$ and $\mathbf{w}_{2}(x,y)$ topologically generate a free pro-$2$ retract of $\widehat{F}_{\mathbf{G}}(\{x,y\})$. Then the pseudovariety of groups with abelian $2$-subgroups is defined by the pseudoidentity $\mathbf{w}_{1}(x,y)\mathbf{w}_{2}(x,y)\approx\mathbf{w}_{2}(x,y)\mathbf{w}_{1}(x,y)$. Thus the pseudovariety $\operatorname{\mathsf{Excl}}(Q_{8})$ is defined by $\mathbf{w}_{1}(x^{\prime},y^{\prime})\mathbf{w}_{2}(x^{\prime},y^{\prime})\approx\mathbf{w}_{2}(x^{\prime},y^{\prime})\mathbf{w}_{1}(x^{\prime},y^{\prime})$, where $x^{\prime}$ and $y^{\prime}$ are as given in the previous paragraph.

5. Join irreducible pseudovarieties

The present section contains 15 subsections. Some background results are recorded in the first subsection, while the latter 14 subsections are devoted to the pseudovarieties generated by the following 14 semigroups:

[TABLE]

Each subsection that is concerned with a semigroup $S$ from (5.1) begins with a theorem that establishes the $\mathsf{ji}$ property of $\langle\langle S\rangle\rangle$ by exhibiting a pseudoidentity that defines the pseudovariety $\operatorname{\mathsf{Excl}}(S)$. A basis $\Sigma_{S}$ of identities for the pseudovariety $\langle\langle S\rangle\rangle$ and an identity $\varepsilon_{S}$ that defines its maximal subpseudovariety $\langle\langle S\rangle\rangle\cap\operatorname{\mathsf{Excl}}(S)$ are then given in a proposition. The pair $(\Sigma_{S},\varepsilon_{S})$ can be used to easily test if a finite semigroup generates the $\mathsf{ji}$ pseudovariety $\langle\langle S\rangle\rangle$. Indeed, for any finite semigroup $T$,

[TABLE]

The pairs $(\Sigma_{S},\varepsilon_{S})$, where $S$ ranges over the semigroups from (5.1), will be used in Section 7 to locate all $\mathsf{ji}$ pseudovarieties generated by a semigroup of order up to five.

5.1. Preliminaries

The free semigroup and free monoid over a countably infinite alphabet $\mathscr{A}$ are denoted by $\mathscr{A}^{+}$ and $\mathscr{A}^{*}$, respectively. Elements of $\mathscr{A}$ are called variables while elements of $\mathscr{A}^{*}$ are called words. For any word $\mathbf{w}\in\mathscr{A}^{+}$,

1. $\bullet$

the number of times a variable $x$ occurs in $\mathbf{w}$ is denoted by $\operatorname{\mathsf{occ}}(x,\mathbf{w})$; 2. $\bullet$

the content of $\mathbf{w}$, denoted by $\operatorname{\mathsf{con}}(\mathbf{w})$, is the set of variables occurring in $\mathbf{w}$, that is, $\operatorname{\mathsf{con}}(\mathbf{w})=\{x\in\mathscr{A}\mid\operatorname{\mathsf{occ}}(x,\mathbf{w})\geq 1\}$; 3. $\bullet$

the initial part of $\mathbf{w}$, denoted by $\operatorname{\mathsf{ini}}(\mathbf{w})$, is the word obtained by retaining the first occurrence of each variable in $\mathbf{w}$; 4. $\bullet$

the final part of $\mathbf{w}$, denoted by $\operatorname{\mathsf{fin}}(\mathbf{w})$, is the word obtained by retaining the last occurrence of each variable in $\mathbf{w}$.

Lemma 5.1**.**

Let $\mathbf{u}\approx\mathbf{v}$ be any semigroup identity.* Then*

1. (i)

$\mathbb{Z}_{n}\models\mathbf{u}\approx\mathbf{v}$* if and only if $\operatorname{\mathsf{occ}}(x,\mathbf{u})\equiv\operatorname{\mathsf{occ}}(x,\mathbf{v})\pmod{n}$ for all $x\in\mathscr{A}$;* 2. (ii)

$N_{n}^{I}\models\mathbf{u}\approx\mathbf{v}$* if and only if for all $x\in\mathscr{A}$, either $\operatorname{\mathsf{occ}}(x,\mathbf{u})=\operatorname{\mathsf{occ}}(x,\mathbf{v})$ or $\operatorname{\mathsf{occ}}(x,\mathbf{u}),\operatorname{\mathsf{occ}}(x,\mathbf{v})\geq n$;* 3. (iii)

$L_{2}^{I}\models\mathbf{u}\approx\mathbf{v}$* if and only if $\operatorname{\mathsf{ini}}(\mathbf{u})=\operatorname{\mathsf{ini}}(\mathbf{v})$;* 4. (iv)

$R_{2}^{I}\models\mathbf{u}\approx\mathbf{v}$* if and only if $\operatorname{\mathsf{fin}}(\mathbf{u})=\operatorname{\mathsf{fin}}(\mathbf{v})$.*

Proof.

These results are well known and easily established. For instance, parts (i) and (ii) follow from Almeida [2, Lemma 6.1.4] while parts (iii) and (iv) can be found in Petrich and Reilly [20, Theorem V.1.9, parts (viii) and (ix)]. ∎

The local of a pseudovariety $\mathbf{V}$, denoted by $\mathbb{L}\mathbf{V}$, is the pseudovariety of all finite semigroups $S$ such that $eSe\in\mathbf{V}$ for any idempotent $e\in S$.

Lemma 5.2** (Almeida [2, Exercise 10.10.1]).**

Let $S$ be any finite semigroup that is not a monoid.* If the pseudovariety $\langle\langle S\rangle\rangle$ is $\mathsf{ji}$, then the pseudovariety $\langle\langle S^{I}\rangle\rangle$ is also $\mathsf{ji}$ and $\operatorname{\mathsf{Excl}}(S^{I})=\mathbb{L}\operatorname{\mathsf{Excl}}(S)$*.**

5.2. The pseudovariety $\langle\langle\mathbb{Z}_{n}\rangle\rangle$

For any set $\pi=\{p_{1},p_{2},p_{3},\ldots\}$ of primes, let $\pi^{\prime}$ denote the set of primes complementary to $\pi$. If $p$ is a prime, then simply write $p^{\prime}$ instead of $\{p\}^{\prime}$. For example, $2^{\prime}$ denotes the set of odd primes. Retaining the above notation, recall that in $\widehat{\{x\}^{+}}$, the sequence $x^{(p_{1}p_{2}\cdots p_{n})^{n!}}$ converges to an element (independent of the enumeration of $\pi$), denoted by $x^{\pi^{\omega}}$, with the following property: if $s$ is an element of a finite semigroup $S$, then $s^{\pi^{\omega}}$ is a generator of the $\pi^{\prime}$-primary component of the finite cyclic group generated by $s^{\omega+1}$. Here we recall that for a finite abelian group $A$, the $\pi^{\prime}$-primary component of $A$ is the direct product of the $p$-Sylow subgroups of $A$ with $p\notin\pi$. In this case, $s^{(\pi^{\prime})^{\omega}}$ will then be a generator of the $\pi$-primary component of $\langle s^{\omega+1}\rangle$; see [23, Proposition 7.1.16].

Theorem 5.3**.**

For any prime $p$ with $n\geq 1$,* the pseudovariety $\langle\langle\mathbb{Z}_{p^{n}}\rangle\rangle$ is $\mathsf{ji}$ and*

[TABLE]

Proof.

The cyclic group $\mathbb{Z}_{p^{n}}=\langle\mathsf{g}\mid\mathsf{g}^{p^{n}}=1\rangle$ violates the pseudoidentity in (5.2) because $(\mathsf{g}^{(p^{\prime})^{\omega}})^{p^{n-1}}=\mathsf{g}^{p^{n-1}}\neq 1=\mathsf{g}^{\omega}$. Therefore, if $\mathbb{Z}_{p^{n}}$ belongs to some pseudovariety $\mathbf{V}$, then $\mathbf{V}$ violates the pseudoidentity in (5.2).

Conversely, suppose that the pseudoidentity in (5.2) is violated by $\mathbf{V}$, say it is violated by $S\in\mathbf{V}$. Generality is not lost by assuming that $S$ is generated by an element $s$ such that $(s^{(p^{\prime})^{\omega}})^{p^{n-1}}\neq s^{\omega}$. Replacing $s$ by $s^{\omega+1}$, we may assume that $S$ is, in fact, a cyclic group generated by $s$ such that $(s^{(p^{\prime})^{\omega}})^{p^{n-1}}\neq 1$. But then the $p$-primary component of $S$ is a cyclic group of order $p^{m}$ with $m\geq n$. Therefore, $\mathbb{Z}_{p^{n}}$ divides $S$, whence $\mathbb{Z}_{p^{n}}\in\mathbf{V}$. ∎

Proposition 5.4**.**

Let $n\geq 1$.**

1. (i)

The identities satisfied by the group $\mathbb{Z}_{n}$ are axiomatized by

[TABLE] 2. (ii)

The maximal subpseudovarieties of $\langle\langle\mathbb{Z}_{n}\rangle\rangle$ are precisely $\langle\langle\mathbb{Z}_{d}\rangle\rangle$, where $d$ ranges over all maximal proper divisors of $n$. Consequently,* for any prime $p$ with $k\geq 1$, the subpseudovariety of $\langle\langle\mathbb{Z}_{p^{k}}\rangle\rangle$ defined by*

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle\mathbb{Z}_{p^{k}}\rangle\rangle$.

Proof.

These results are well known and easily established. For instance, part (i) follows from Almeida [2, Corollary 6.1.5] while part (ii) follows from Petrich and Reilly [20, Lemma VIII.6.14]. ∎

5.3. The pseudovariety $\langle\langle\mathbb{Z}_{2}^{\mathsf{bar}}\rangle\rangle$

Theorem 5.5**.**

The pseudovariety $\langle\langle\mathbb{Z}_{2}^{\mathsf{bar}}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

where $\mathbf{e}$ is an idempotent in the minimal ideal of $\widehat{\{x,y\}^{+}}$.

Proof.

This follows from Theorems 4.4 and 5.3. ∎

Alternately, Rhodes and Steinberg [23, Example 7.3.20] have shown that

[TABLE]

where $\mathbf{e}$ is an idempotent in the minimal ideal of $\widehat{\{x,y\}^{+}}$.

Proposition 5.6**.**

1. (i)

The identities satisfied by the semigroup $\mathbb{Z}_{2}^{\mathsf{bar}}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle\mathbb{Z}_{2}^{\mathsf{bar}}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle\mathbb{Z}_{2}^{\mathsf{bar}}\rangle\rangle$.

Proof.

This follows from the dual of Tishchenko [27, Proposition 3.16], where the variety generated by $(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}}$ is denoted by $\mathbf{W}_{2}$. ∎

5.4. The pseudovariety $\langle\langle N_{n}\rangle\rangle$

Theorem 5.7**.**

For each $n\geq 2$,* the pseudovariety $\langle\langle N_{n}\rangle\rangle$ is $\mathsf{ji}$ and*

[TABLE]

Proof.

The semigroup $N_{n}=\langle\mathsf{a}\mid\mathsf{a}^{n}=0\rangle$ violates the pseudoidentity in (5.3) because $\mathsf{a}^{\omega+n-1}=0\neq\mathsf{a}^{n-1}$. Therefore, if $N_{n}$ belongs to some pseudovariety $\mathbf{V}$, then $\mathbf{V}$ violates the pseudoidentity in (5.3).

Conversely, suppose that the pseudoidentity in (5.3) is violated by the pseudovariety $\mathbf{V}$, say it is violated by $S\in\mathbf{V}$. Then there exists some $a\in S$ such that $a^{\omega+n-1}\neq a^{n-1}$. If there exist some $i\leq n-1$ and some $j>i$ such that $a^{i}=a^{j}$, then $a^{n-1}=a^{n-1-i}a^{i}=a^{n-1-i}a^{j}=a^{n-1}a^{j-i}$, so that

[TABLE]

which is a contradiction. Hence the sets $\{a\},\{a^{2}\},\ldots,\{a^{n-1}\},\{a^{i}\mid i\geq n\}$ are pairwise disjoint. It follows that $J=\{a^{i}\mid i\geq n\}$ is an ideal of the monogenic subsemigroup $\langle a\rangle$ of $S$ such that $\langle a\rangle/J\cong N_{n}$. Consequently, $N_{n}\in\langle\langle S\rangle\rangle\subseteq\mathbf{V}$. ∎

For each nonnegative real number $x$, let $\lfloor x\rfloor$ denote the greatest integer bounded from above by $x$.

Proposition 5.8**.**

Let $n\geq 2$.**

1. (i)

The identities satisfied by the semigroup $N_{n}$ are axiomatized by

[TABLE]

where $k=0,1,\ldots,\lfloor(n-1)/3\rfloor$. 2. (ii)

The subpseudovariety of $\langle\langle N_{n}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle N_{n}\rangle\rangle$.

Proof.

(i) This follows from Shevrin and Volkov [26, Proposition 21.3].

(ii) This follows from Theorem 5.7 and part (i). ∎

5.5. The pseudovariety $\langle\langle N_{n}^{I}\rangle\rangle$

Theorem 5.9**.**

For any $n\geq 1$,* the pseudovariety $\langle\langle N_{n}^{I}\rangle\rangle$ is $\mathsf{ji}$ and*

[TABLE]

Proof.

For $n=1$, the result follows from [23, Table 7.2] because $N_{1}^{I}\cong Sl_{2}$. For $n\geq 2$, the result follows from Lemma 5.2 and Theorem 5.7. ∎

Proposition 5.10**.**

Let $n\geq 1$.**

1. (i)

The identities satisfied by the semigroup $N_{n}^{I}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle N_{n}^{I}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle N_{n}^{I}\rangle\rangle$.

Proof.

(i) This easily established result is well known; see, for example, Almeida [2, Corollary 6.1.5].

(ii) This follows from Theorem 5.9 and part (i). ∎

5.6. The pseudovariety $\langle\langle N_{2}^{\mathsf{bar}}\rangle\rangle$

Theorem 5.11**.**

The pseudovariety $\langle\langle N_{2}^{\mathsf{bar}}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

where $\mathbf{e}$ is an idempotent from the minimal ideal of $\widehat{\{x,z\}^{+}}$.

Proof.

This follows from Theorems 4.4 and 5.7. ∎

Proposition 5.12**.**

1. (i)

The identities satisfied by the semigroup $N_{2}^{\mathsf{bar}}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle N_{2}^{\mathsf{bar}}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle N_{2}^{\mathsf{bar}}\rangle\rangle$.

Proof.

(i) This follows from Tishchenko [27, Corollary 2.5(c) and Proposition 4.4].

(ii) This follows from Tishchenko [27, Proposition 3.4]. ∎

5.7. The pseudovariety $\langle\langle(N_{2}^{\mathsf{bar}})^{I}\rangle\rangle$

Theorem 5.13**.**

The pseudovariety $\langle\langle(N_{2}^{\mathsf{bar}})^{I}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

where $\mathbf{e}$ is an idempotent from the minimal ideal of $\widehat{\{x,z\}^{+}}$.

Proof.

This follows from Lemma 5.2 and Theorem 5.11. ∎

Proposition 5.14** (Lee and Li [12, Corollary 6.6 and Lemma 6.7]).**

1. (i)

The identities satisfied by the semigroup $(N_{2}^{\mathsf{bar}})^{I}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle(N_{2}^{\mathsf{bar}})^{I}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle(N_{2}^{\mathsf{bar}})^{I}\rangle\rangle$.

5.8. The pseudovariety $\langle\langle L_{2}\rangle\rangle$

Theorem 5.15**.**

The pseudovariety $\langle\langle L_{2}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

Proof.

This result is dual to [2, Proposition 10.10.2(b)]. ∎

Proposition 5.16** (Rhodes and Steinberg [23, Table 7.1]).**

1. (i)

The identities satisfied by the semigroup $L_{2}$ are axiomatized by

[TABLE] 2. (ii)

The pseudovariety $\langle\langle L_{2}\rangle\rangle$ is an atom in the lattice $\mathbf{PV}$.

5.9. The pseudovariety $\langle\langle L_{2}^{I}\rangle\rangle$

Theorem 5.17**.**

The pseudovariety $\langle\langle L_{2}^{I}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

Proof.

This follows from Lemma 5.2 and Theorem 5.15. ∎

Proposition 5.18**.**

1. (i)

The identities satisfied by the semigroup $L_{2}^{I}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle L_{2}^{I}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle L_{2}^{I}\rangle\rangle$.

Proof.

This can be found in Almeida [2, Figure 5.1], where the pseudovariety $\langle\langle L_{2}^{I}\rangle\rangle$ is denoted by $\mathbf{MK}_{1}$. ∎

5.10. The pseudovariety $\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle$

Theorem 5.19**.**

The pseudovariety $\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

where $\mathbf{e}$ is an idempotent in the minimal ideal of $\widehat{\{x,y,z\}^{+}}$.**

Proof.

This follows from Theorems 4.4 and 5.15. ∎

Alternately, Rhodes and Steinberg [23, Example 7.3.16] have shown that

[TABLE]

where $\mathbf{e}$ is an idempotent in the minimal ideal of $\widehat{\{x,y,z\}^{+}}$.

Proposition 5.20**.**

1. (i)

The identities satisfied by the semigroup $L_{2}^{\mathsf{bar}}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle$.

Proof.

This can be found in Almeida [2, Figure 5.1], where the pseudovariety $\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle$ is denoted by $[[R_{3}^{\rho}=Q_{3}^{\rho}]]_{\mathbf{B}}$. ∎

5.11. The pseudovariety $\langle\langle A_{0}\rangle\rangle$

Theorem 5.21** (Lee [11, Proposition 2.3]).**

The pseudovariety $\langle\langle A_{0}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

Proposition 5.22** (Lee [6, Section 4], Lee and Volkov [15, Theorem 4.1]).**

1. (i)

The identities satisfied by the semigroup $A_{0}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle A_{0}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle A_{0}\rangle\rangle$.

5.12. The pseudovariety $\langle\langle A_{0}^{I}\rangle\rangle$

Theorem 5.23**.**

The pseudovariety $\langle\langle A_{0}^{I}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

Proof.

This follows from Lemma 5.2 and Theorem 5.21. ∎

Proposition 5.24** (Lee [9, Propositions 1.1 and 1.5(ii)]).**

1. (i)

The identities satisfied by the semigroup $A_{0}^{I}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle A_{0}^{I}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle A_{0}^{I}\rangle\rangle$.

5.13. The pseudovariety $\langle\langle A_{2}\rangle\rangle$

Theorem 5.25** (Lee [7]).**

The pseudovariety $\langle\langle A_{2}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

Proposition 5.26** (Lee [6, Theorem 2.7], Trahtman [31]).**

1. (i)

The identities satisfied by the semigroup $A_{2}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle A_{2}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle A_{2}\rangle\rangle$.

5.14. The pseudovariety $\langle\langle B_{2}\rangle\rangle$

Theorem 5.27** (Rhodes and Steinberg [23, Example 7.3.4]).**

The pseudovariety $\langle\langle B_{2}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

Proposition 5.28** (Lee [6, Theorem 3.6], Trahtman [29]).**

1. (i)

The identities satisfied by the semigroup $B_{2}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle B_{2}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle B_{2}\rangle\rangle$.

5.15. The pseudovariety $\langle\langle\ell_{3}^{\mathsf{bar}}\rangle\rangle$

Theorem 5.29**.**

The pseudovariety $\langle\langle\ell_{3}^{\mathsf{bar}}\rangle\rangle$ is $\mathsf{ji}$ and

[TABLE]

where $\mathbf{e}$ is an idempotent in the minimal ideal of $\widehat{\{x,y,z\}^{+}}$.**

Proof.

This is a special case of Theorem 4.19 since $\ell_{3}^{\mathsf{bar}}\cong O_{2}^{\mathsf{bar}}$. ∎

The remainder of this subsection is devoted to establishing a basis for the identities satisfied by $\ell_{3}^{\mathsf{bar}}$. It turns out that it is notationally simpler to consider the dual semigroup $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}=\{a,b,c,d,e\}$ given in Table 8.

Proposition 5.30**.**

1. (i)

The identities satisfied by the semigroup $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal subpseudovariety of $\langle\langle(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$.

Remark 5.31**.**

It is routinely shown that the semigroup $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$ satisfies the identities (5.4) but violates the identity (5.5).

In this subsection, a word $\mathbf{w}$ is said to be in canonical form if either

1. (CF1)

$\mathbf{w}=x_{0}x_{1}\cdots x_{m}$ or 2. (CF2)

$\mathbf{w}=x_{0}x_{1}\cdots x_{k}\cdot x_{0}\cdot x_{k+1}x_{k+2}\cdots x_{m}$,

where $x_{0},x_{1},\ldots,x_{m}$ are distinct variables with $0\leq k\leq m$.

Remark 5.32**.**

Note the extreme cases for the word $\mathbf{w}$ in (CF2):

1. (i)

if $0=k=m$, then $\mathbf{w}=x_{0}^{2}$; 2. (ii)

if $0=k<m$, then $\mathbf{w}=x_{0}^{2}x_{1}\cdots x_{m}$; 3. (iii)

if $0<k=m$, then $\mathbf{w}=x_{0}x_{1}x_{2}\cdots x_{m}x_{0}$.

Lemma 5.33**.**

Given any word $\mathbf{w}$,* the identities (5.4) can be used to convert $\mathbf{w}$ into some word $\mathbf{w}^{\prime}$ in canonical form with $\operatorname{\mathsf{ini}}(\mathbf{w})=\operatorname{\mathsf{ini}}(\mathbf{w}^{\prime})$*.**

Proof.

Suppose that $\operatorname{\mathsf{ini}}(\mathbf{w})=x_{0}x_{1}\cdots x_{m}$. Then $\mathbf{w}$ can be written as

[TABLE]

where $\mathbf{w}_{i}\in\{x_{0},x_{1},\ldots,x_{i}\}^{*}$ for all $i$. The identities (5.4) can be used to eliminate all occurrences of $x_{1},x_{2},\ldots,x_{m}$ from each $\mathbf{w}_{i}$, resulting in the word

[TABLE]

where $e_{0},e_{1},\ldots,e_{m}\geq 0$. If $e_{0}=e_{1}=\cdots=e_{m}=0$, then the word $\mathbf{w}^{\prime}$ is in canonical form (CF1) such that $\operatorname{\mathsf{ini}}(\mathbf{w})=\operatorname{\mathsf{ini}}(\mathbf{w}^{\prime})$. If $k\geq 0$ is the least index such that $e_{k}\geq 1$, then $e_{0}=e_{1}=\cdots=e_{k-1}=0$, so that

[TABLE]

The word $\mathbf{w}^{\prime\prime}$ is in canonical form (CF2) with $\operatorname{\mathsf{ini}}(\mathbf{w})=\operatorname{\mathsf{ini}}(\mathbf{w}^{\prime\prime})$. ∎

Proof of Proposition 5.30(ii).

As observed in Remark 5.31, the semigroup $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$ violates the identity (5.5). Hence $\langle\langle(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle\cap[[\eqref{id: el3 max}]]$ is a proper subpseudovariety of $\langle\langle(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$. It remains to show that each proper subpseudovariety $\mathbf{V}$ of $\langle\langle(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$ satisfies the identity (5.5). Since $\mathbf{V}\neq\langle\langle(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$, there exists an identity $\mathbf{u}\approx\mathbf{v}$ of $\mathbf{V}$ that is violated by $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$. Further, since the identities (5.4) are satisfied by $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$ and so also by $\mathbf{V}$, it follows from Lemma 5.33 that the words $\mathbf{u}$ and $\mathbf{v}$ can be chosen to be in canonical form. There are two cases.

Case 1

$\operatorname{\mathsf{ini}}(\mathbf{u})\neq\operatorname{\mathsf{ini}}(\mathbf{v})$. Then by Theorem 5.17, the pseudovariety $\mathbf{V}$ satisfies the pseudoidentity that defines $\operatorname{\mathsf{Excl}}(L_{2}^{I})$. Since

[TABLE]

the pseudovariety $\mathbf{V}$ satisfies the identity $\alpha:h^{2}xy\approx h^{2}yx$. Since

[TABLE]

the pseudovariety $\mathbf{V}$ satisfies the identity (5.5).

Case 2

$\operatorname{\mathsf{ini}}(\mathbf{u})=\operatorname{\mathsf{ini}}(\mathbf{v})$ and $\mathbf{u}\neq\mathbf{v}$. If the words $\mathbf{u}$ and $\mathbf{v}$ are both of the form (CF1), then they are contradictorily equal. Hence either $\mathbf{u}$ or $\mathbf{v}$ is of the form (CF2). By symmetry, there are two subcases.

1. 2.1.

$\mathbf{u}$ and $\mathbf{v}$ are both of the form (CF2). Then

[TABLE]

where $0\leq j,k\leq m$. Since $j\neq k$, it suffices to assume by symmetry that $0\leq j<k\leq m$. Let $\varphi$ denote the substitution given by $x_{0}\mapsto xy$, $x_{i}\mapsto y$ for all $i\in\{1,2,\ldots,j\}$, and $x_{i}\mapsto z$ otherwise. Then

[TABLE]

Therefore, the identity (5.5) is deducible from (5.4) and $\mathbf{u}\approx\mathbf{v}$. The pseudovariety $\mathbf{V}$ thus satisfies the identity (5.5). 2. 2.2.

$\mathbf{u}$ is of the form (CF1) while $\mathbf{v}$ is of the form (CF2). Then

[TABLE]

Since

[TABLE]

the pseudovariety $\mathbf{V}$ satisfies the identity $\mathbf{u}^{\prime}\approx\mathbf{v}^{\prime}$. Now $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$ are distinct words in canonical form (CF2) such that $\operatorname{\mathsf{ini}}(\mathbf{u}^{\prime})=\operatorname{\mathsf{ini}}(\mathbf{v}^{\prime})$. Thus the arguments in Subcase 2.1 can be repeated to show that $\mathbf{V}$ satisfies the identity (5.5). ∎

Proof of Proposition 5.30(i).

As noted in Remark 5.31, the identities (5.4) are satisfied by the semigroup $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$. Conversely, suppose that $\mathbf{u}\approx\mathbf{v}$ is any identity satisfied by $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$. By Lemma 5.33, the identities (5.4) can be used to convert $\mathbf{u}$ and $\mathbf{v}$ into words $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$ in canonical form. Since the subsemigroup $\{a,c,e\}$ of $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$ and the semigroup $L_{2}^{I}$ are isomorphic, it follows from Lemma 5.1(iii) that $\operatorname{\mathsf{ini}}(\mathbf{u}^{\prime})=\operatorname{\mathsf{ini}}(\mathbf{v}^{\prime})$. Suppose that $\mathbf{u}^{\prime}\neq\mathbf{v}^{\prime}$. Then by repeating the arguments in Case 2 of the proof of Proposition 5.30(ii), the identity (5.5) is deducible from (5.4) and $\mathbf{u}^{\prime}\approx\mathbf{v}^{\prime}$. Since the semigroup $(\ell_{3}^{\mathsf{bar}})^{\mathsf{op}}$ satisfies the identities (5.4) and $\mathbf{u}^{\prime}\approx\mathbf{v}^{\prime}$, it also satisfies (5.5); but this is impossible by Remark 5.31. Therefore, $\mathbf{u}^{\prime}=\mathbf{v}^{\prime}$. Since

[TABLE]

the identity $\mathbf{u}\approx\mathbf{v}$ is deducible from (5.4). ∎

6. Non-$\mathsf{ji}$ pseudovarieties

This section contains nine subsections, each of which establishes one or more sufficient conditions for a finite semigroup to generate a non-$\mathsf{ji}$ pseudovariety. Each of these sufficient conditions, given as a corollary of some general result, presents some finite set $\Sigma$ of identities and some identities $\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}$ with the property that for any finite semigroup $S$,

[TABLE]

In most cases, $\Sigma$ will be a basis of identities for some join $\mathbf{V}=\bigvee_{i=1}^{k}\mathbf{V}_{i}$ of compact pseudovarieties $\mathbf{V}_{1},\mathbf{V}_{2},\ldots,\mathbf{V}_{k}$ that satisfy the pseudoidentities $\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}$, respectively.

Sufficient conditions developed in this section will be used in Section 7 to locate all non-$\mathsf{ji}$ pseudovarieties generated by a semigroup of order up to five.

6.1. The pseudovariety $\langle\langle\mathbb{Z}_{3},\mathbb{Z}_{4},\mathbb{Z}_{2}^{\mathsf{bar}},(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}},N_{3}^{I}\rangle\rangle$

In this subsection, it is convenient to write

[TABLE]

Proposition 6.1** (Lee and Li [13]).**

The identities satisfied by the semigroup $\mathbb{Z}_{3}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}^{\mathsf{bar}}\times(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}}\times N_{3}^{I}$ are axiomatized by

[TABLE]

Corollary 6.2**.**

Suppose that $S$ is any finite semigroup that satisfies the identities (6.1) but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle\mathscr{K}\rangle\rangle$ that is not $\mathsf{ji}$.**

Proof.

By Proposition 6.1, the inclusion

[TABLE]

holds. But the two identities in (6.2) are satisfied by $\mathbb{Z}_{2}^{\mathsf{bar}}\times(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}}$ and $\mathbb{Z}_{3}\times\mathbb{Z}_{4}\times N_{3}^{I}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle\mathbb{Z}_{2}^{\mathsf{bar}},(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle\mathbb{Z}_{3},\mathbb{Z}_{4},N_{3}^{I}\rangle\rangle$. ∎

Corollary 6.3**.**

Suppose that $S$ is any finite semigroup that satisfies the identities (6.1) but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle\mathscr{K}\rangle\rangle$ that is not $\mathsf{ji}$.**

Proof.

By Proposition 6.1, the inclusion

[TABLE]

holds. Since the three identities in (6.3) are satisfied by $\mathbb{Z}_{3}\times\mathbb{Z}_{4}\times N_{3}^{I}$, $(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}}$, and $\mathbb{Z}_{2}^{\mathsf{bar}}$, respectively, the exclusions $\langle\langle S\rangle\rangle\nsubseteq\langle\langle\mathbb{Z}_{3},\mathbb{Z}_{4},N_{3}^{I}\rangle\rangle$, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$, and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle\mathbb{Z}_{2}^{\mathsf{bar}}\rangle\rangle$ follow. ∎

Corollary 6.4**.**

Suppose that $S$ is any finite semigroup that satisfies the identities (6.1) but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle\mathscr{K}\rangle\rangle$ that is not $\mathsf{ji}$.**

Proof.

The inclusion $\langle\langle S\rangle\rangle\subseteq\bigvee\{\langle\langle T\rangle\rangle\mid T\in\mathscr{K}\}$ holds by Proposition 6.1. But the five identities in (6.4) are satisfied by $N_{3}^{I}$, $\mathbb{Z}_{4}$, $\mathbb{Z}_{3}$, $(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}}$, and $\mathbb{Z}_{2}^{\mathsf{bar}}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle T\rangle\rangle$ for all $T\in\mathscr{K}$. ∎

6.2. The pseudovariety $\langle\langle\mathbb{Z}_{m},N_{n}^{I},L_{2}^{I},R_{2}^{I},A_{0}^{I}\rangle\rangle$

In this subsection, it is convenient to write

[TABLE]

and $T_{m,n}=\mathbb{Z}_{m}\times N_{n}^{I}\times L_{2}^{I}\times R_{2}^{I}\times A_{0}^{I}$.

Proposition 6.5**.**

Let $m\geq 1$ and $n\geq 2$.* Then the identities satisfied by the semigroup $T_{m,n}$ are axiomatized by*

[TABLE]

Remark 6.6**.**

1. (i)

Since $N_{2}$ is isomorphic to the subsemigroup $\{0,\mathsf{f}\mathsf{e}\}$ of $A_{0}$, the monoid $N_{2}^{I}$ belongs to $\langle\langle A_{0}^{I}\rangle\rangle$. Therefore, $\langle\langle T_{m,1}\rangle\rangle=\langle\langle T_{m,2}\rangle\rangle$. This is the reason for the assumption $n\geq 2$ in Proposition 6.5. 2. (ii)

The basic case $(m,n)=(1,2)$ for Proposition 6.5 was first established in Lee [10, Proposition 2.3(i)].

Suppose that a word $\mathbf{w}$ can be written in the form

[TABLE]

where $x\in\mathscr{A}$, $\mathbf{w}_{0},\mathbf{w}_{r}\in\mathscr{A}^{*}$, and $\mathbf{w}_{1},\mathbf{w}_{2},\ldots,\mathbf{w}_{r-1}\in\mathscr{A}^{+}$ are such that $x\notin\operatorname{\mathsf{con}}(\mathbf{w}_{i})$ for all $i$, and $e_{1},e_{2},\ldots,e_{r}\in\{1,2,3,\ldots\}$. Then the factors $x^{e_{1}},x^{e_{2}},\ldots,x^{e_{r}}$ are call $x$-stacks, or simply stacks, of $\mathbf{w}$. The weight of the $x$-stack $x^{e_{i}}$ is $e_{i}$.

It is easily shown that the identities (6.5) can be used to convert any word into a word $\mathbf{w}$ such that for each $x\in\mathscr{A}$,

1. (I)

the number of $x$-stacks in $\mathbf{w}$ is at most two; 2. (II)

if $\mathbf{w}$ has one $x$-stack, then its weight is at most $m+n-1$; 3. (III)

if $\mathbf{w}$ has two $x$-stacks, then the weight of the first $x$-stack is at most $m+n-2$ while the weight of the second $x$-stack is one.

In this subsection, a word $\mathbf{w}$ that satisfies (I)–(III) is said to be in canonical form. Note that if $\mathbf{w}$ is a word in canonical form, then $\operatorname{\mathsf{occ}}(x,\mathbf{w})\leq m+n-1$ for any $x\in\mathscr{A}$.

Lemma 6.7**.**

Let $\mathbf{u}$ and $\mathbf{v}$ be any words in canonical form such that the identity $\mathbf{u}\approx\mathbf{v}$ is satisfied by the semigroup $T_{m,n}$.* Then for any $x\in\mathscr{A}$*,**

1. (i)

$\operatorname{\mathsf{occ}}(x,\mathbf{u})\equiv\operatorname{\mathsf{occ}}(x,\mathbf{v})\pmod{m}$;** 2. (ii)

either $\operatorname{\mathsf{occ}}(x,\mathbf{u})=\operatorname{\mathsf{occ}}(x,\mathbf{v})\leq n$ or $n<\operatorname{\mathsf{occ}}(x,\mathbf{u}),\operatorname{\mathsf{occ}}(x,\mathbf{v})\leq m+n-1$.

Proof.

This follows from Lemma 5.1 parts (i) and (ii). ∎

For any word $\mathbf{w}$ and distinct variables $x_{1},x_{2},\ldots,x_{r}$, let $\mathbf{w}_{\{x_{1},x_{2},\ldots,x_{r}\}}$ denote the word obtained from $\mathbf{w}$ by retaining only the variables $x_{1},x_{2},\ldots,x_{r}$. Any monoid that satisfies an identity $\mathbf{u}\approx\mathbf{v}$ also satisfies $\mathbf{u}_{\{x_{1},x_{2},\ldots,x_{r}\}}\approx\mathbf{v}_{\{x_{1},x_{2},\ldots,x_{r}\}}$ for any distinct variables $x_{1},x_{2},\ldots,x_{r}$.

Lemma 6.8**.**

Let $\mathbf{u}$ and $\mathbf{v}$ be any words in canonical form such that the identity $\mathbf{u}\approx\mathbf{v}$ is satisfied by the semigroup $T_{m,n}$.* Then*

1. (i)

for any distinct $x,y\in\mathscr{A}$, the identity $\mathbf{u}_{\{x,y\}}\approx\mathbf{v}_{\{x,y\}}$ cannot be any of

[TABLE]

where $e_{1},f_{1},e_{2},f_{2},e_{3},f_{3}\geq 1$; 2. (ii)

$\mathbf{u}$* has two $x$-stacks if and only if $\mathbf{v}$ has two $x$-stacks*;** 3. (iii)

$x^{e}$* is the first $x$-stack of $\mathbf{u}$ if and only if $x^{e}$ is the first $x$-stack of $\mathbf{v}$.*

Proof.

(i) The three identities in (6.6) are violated by the semigroups $R_{2}^{I}$, $L_{2}^{I}$, and $A_{0}^{I}$, respectively.

(ii) Suppose that $\mathbf{u}$ has two $x$-stacks. Then by (III),

[TABLE]

for some $\mathbf{u}_{1},\mathbf{u}_{3}\in\mathscr{A}^{*}$ and $\mathbf{u}_{2}\in\mathscr{A}^{+}$ with $x\notin\operatorname{\mathsf{con}}(\mathbf{u}_{1}\mathbf{u}_{2}\mathbf{u}_{3})$ and $2\leq e\leq m+n-1$. Seeking a contradiction, suppose that $\mathbf{v}$ has only one $x$-stack. Then by (II) and part (i),

[TABLE]

for some $\mathbf{v}_{1},\mathbf{v}_{2}\in\mathscr{A}^{*}$ with $x\notin\operatorname{\mathsf{con}}(\mathbf{v}_{1}\mathbf{v}_{2})$ and $1\leq f\leq m+n-1$. Since the word $\mathbf{u}_{2}$ is nonempty, it contains some $y$-stack. Since $\operatorname{\mathsf{con}}(\mathbf{u})=\operatorname{\mathsf{con}}(\mathbf{v})$ by Lemma 5.1(ii), it follows that $y\in\operatorname{\mathsf{con}}(\mathbf{v}_{1}\mathbf{v}_{2})$. By symmetry, it suffices to assume that $y\in\operatorname{\mathsf{con}}(\mathbf{v}_{1})$, so that $\operatorname{\mathsf{ini}}(\mathbf{v})=\cdots y\cdots x\cdots$. Since $\operatorname{\mathsf{ini}}(\mathbf{u})=\operatorname{\mathsf{ini}}(\mathbf{v})$ by Lemma 5.1(iii), it follows that $y\in\operatorname{\mathsf{con}}(\mathbf{u}_{1})$. Hence the word $\mathbf{u}$ contains two $y$-stacks, the first of which occurs in $\mathbf{u}_{1}$ while the second occurs in $\mathbf{u}_{2}$. Thus $\operatorname{\mathsf{fin}}(\mathbf{u})=\cdots y\cdots x\cdots$. Since $\operatorname{\mathsf{fin}}(\mathbf{u})=\operatorname{\mathsf{fin}}(\mathbf{v})$ by Lemma 5.1(iv), it follows that $y\notin\operatorname{\mathsf{con}}(\mathbf{v}_{2})$. Therefore, the identity $\mathbf{u}_{\{x,y\}}\approx\mathbf{v}_{\{x,y\}}$ is $y^{r}x^{e-1}yx\approx y^{s}x^{f}$ for some $r,s\geq 1$, but this contradicts part (i).

(iii) Let $x^{e}$ be a first $x$-stack of $\mathbf{u}$. By part (ii), there are two cases.

Case 1

$\mathbf{u}$ and $\mathbf{v}$ each has only one $x$-stack. Then by (II),

[TABLE]

for some $\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{v}_{1},\mathbf{v}_{2}\in\mathscr{A}^{*}$ with $x\notin\operatorname{\mathsf{con}}(\mathbf{u}_{1}\mathbf{u}_{2}\mathbf{v}_{1}\mathbf{v}_{2})$ and $1\leq e,f\leq m+n-1$. Since $e=\operatorname{\mathsf{occ}}(x,\mathbf{u})$ and $f=\operatorname{\mathsf{occ}}(x,\mathbf{v})$, it follows from part (ii) that either $e=f\leq n$ or $n<e,f\leq m+n-1$. If $n<e,f\leq m+n-1$, then $e=f$ by part (i).

Case 2

$\mathbf{u}$ and $\mathbf{v}$ each has two $x$-stacks. Then by (III),

[TABLE]

for some $\mathbf{u}_{1},\mathbf{u}_{3},\mathbf{v}_{1},\mathbf{v}_{3}\in\mathscr{A}^{*}$ and $\mathbf{u}_{2},\mathbf{v}_{2}\in\mathscr{A}^{+}$ with $x\notin\operatorname{\mathsf{con}}(\mathbf{u}_{1}\mathbf{u}_{2}\mathbf{u}_{3}\mathbf{v}_{1}\mathbf{v}_{2}\mathbf{v}_{3})$ and $2\leq e,f\leq m+n-1$. Since $e=\operatorname{\mathsf{occ}}(x,\mathbf{u})$ and $f=\operatorname{\mathsf{occ}}(x,\mathbf{v})$, it follows from the same argument in Case 1 that $e=f$. ∎

Lemma 6.9**.**

Let $\mathbf{u}$ and $\mathbf{v}$ be any words in canonical form such that the identity $\mathbf{u}\approx\mathbf{v}$ is satisfied by the semigroup $T_{m,n}$.* Then the following are equivalent*:**

1. (a)

$\mathbf{u}\in\mathscr{A}^{*}x^{e}y^{f}\mathscr{A}^{*}$* where $x^{e}$ and $y^{f}$ are stacks of $\mathbf{u}$;* 2. (b)

$\mathbf{v}\in\mathscr{A}^{*}x^{e}y^{f}\mathscr{A}^{*}$* where $x^{e}$ and $y^{f}$ are stacks of $\mathbf{v}$.*

Further,* $x^{e}$ is the first $x$-stack of $\mathbf{u}$ if and only if $x^{e}$ is the first $x$-stack of $\mathbf{v}$, and $y^{f}$ is the first $y$-stack of $\mathbf{u}$ if and only if $y^{f}$ is the first $y$-stack of $\mathbf{v}$*.**

Proof.

First, note that $\operatorname{\mathsf{ini}}(\mathbf{u})=\operatorname{\mathsf{ini}}(\mathbf{v})$ and $\operatorname{\mathsf{fin}}(\mathbf{u})=\operatorname{\mathsf{fin}}(\mathbf{v})$ by Lemma 5.1. Suppose that (a) holds. Then

[TABLE]

for some $\mathbf{u}_{1},\mathbf{u}_{2}\in\mathscr{A}^{*}$ such that $\mathbf{u}_{1}$ does not end with $x$ while $\mathbf{u}_{2}$ does not begin with $y$. There are four cases depending on which of $x^{e}$ and $y^{f}$ are first stacks in $\mathbf{u}$.

Case 1

$x^{e}$ is the first $x$-stack in $\mathbf{u}$ and $y^{f}$ is the first $y$-stack in $\mathbf{u}$. Then clearly $x,y\notin\operatorname{\mathsf{con}}(\mathbf{u}_{1})$, so that $\operatorname{\mathsf{ini}}(\mathbf{u})=\cdots xy\cdots$. By Lemma 6.8(iii), $x^{e}$ is the first $x$-stack of $\mathbf{v}$ and $y^{f}$ is the first $y$-stack of $\mathbf{v}$. Since $\operatorname{\mathsf{ini}}(\mathbf{v})=\operatorname{\mathsf{ini}}(\mathbf{u})=\cdots xy\cdots$,

[TABLE]

for some $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\in\mathscr{A}^{*}$ such that $x\notin\operatorname{\mathsf{con}}(\mathbf{v}_{1})$ and $y\notin\operatorname{\mathsf{con}}(\mathbf{v}_{1}\mathbf{v}_{2})$, and that any stack of $\mathbf{v}$ that occurs in $\mathbf{v}_{2}$ cannot be a first stack. Suppose that $\mathbf{v}_{2}\neq\emptyset$. Then the first variable $z$ of $\mathbf{v}_{2}$ constitutes the second $z$-stack of $\mathbf{v}$. Hence

[TABLE]

where $z^{r}$ is the first $z$-stack of $\mathbf{v}$, and $\operatorname{\mathsf{ini}}(\mathbf{v})=\cdots z\cdots xy\cdots$. By Lemma 6.8(ii), the word $\mathbf{u}$ contains two $z$-stacks; by part (iii) of the same lemma, the first $z$-stack of $\mathbf{u}$ is $z^{r}$. Since $\operatorname{\mathsf{ini}}(\mathbf{u})=\operatorname{\mathsf{ini}}(\mathbf{v})=\cdots z\cdots xy\cdots$, the $z$-stack $z^{r}$ of $\mathbf{u}$ occurs in $\mathbf{u}_{1}$:

[TABLE]

The second $z$-stack of $\mathbf{u}$ occurs in either $\mathbf{u}_{1}$ or $\mathbf{u}_{2}$. There are two subcases.

1. 1.1.

The second $z$-stack of $\mathbf{u}$ occurs in $\mathbf{u}_{1}$. Then $\operatorname{\mathsf{fin}}(\mathbf{v})=\operatorname{\mathsf{fin}}(\mathbf{u})=\cdots z\cdots x\cdots$, so that $\mathbf{v}$ must contain a second $x$-stack occurring in either $\mathbf{v}_{2}$ or $\mathbf{v}_{3}$. The identity $\mathbf{u}_{\{x,z\}}\approx\mathbf{v}_{\{x,z\}}$ is thus $z^{r+1}x^{e+1}\approx z^{r}x^{e}zx$, which is impossible by Lemma 6.8(i). 2. 1.2.

The second $z$-stack of $\mathbf{u}$ occurs in $\mathbf{u}_{2}$. Then $\operatorname{\mathsf{fin}}(\mathbf{u})=\operatorname{\mathsf{fin}}(\mathbf{v})=\cdots z\cdots y\cdots$, so that $\mathbf{u}$ must contain a second $y$-stack occurring after the second $z$-stack:

[TABLE]

The identity $\mathbf{u}_{\{y,z\}}\approx\mathbf{v}_{\{y,z\}}$ is thus $z^{r}y^{f}zy\approx z^{r+1}y^{f+1}$, which is impossible by Lemma 6.8(i).

Since both subcases are impossible, $\mathbf{v}_{2}=\emptyset$. Hence (b) holds.

Case 2

$x^{e}$ is the first $x$-stack in $\mathbf{u}$ and $y^{f}$ is the second $y$-stack in $\mathbf{u}$. Then $f=1$ by (III) and

[TABLE]

where $y^{r}$ is the first $y$-stack of $\mathbf{u}$. Since $\operatorname{\mathsf{ini}}(\mathbf{v})=\operatorname{\mathsf{ini}}(\mathbf{u})=\cdots y\cdots x\cdots$, it follows from Lemma 6.8 parts (i) and (iii) that

[TABLE]

for some $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4}\in\mathscr{A}^{*}$, where $y^{r}$ is the first $y$-stack of $\mathbf{v}$ and $x^{e}$ is the first $x$-stack of $\mathbf{v}$. Suppose that $\mathbf{v}_{3}\neq\emptyset$. Then $\mathbf{v}_{3}$ contains some $z$-stack $z^{s}$:

[TABLE]

There are two subcases depending on whether $z^{s}$ is the first or second $z$-stack in $\mathbf{v}$.

1. 2.1.

$z^{s}$ is the first $z$-stack in $\mathbf{v}$. Then $\operatorname{\mathsf{ini}}(\mathbf{u})=\operatorname{\mathsf{ini}}(\mathbf{v})=\cdots y\cdots x\cdots z\cdots$, so that every $z$ of $\mathbf{u}$ occurs in $\mathbf{u}_{2}$. Hence $\mathbf{u}_{\{y,z\}}\in y^{r+1}\{z\}^{+}$ and

[TABLE]

But this is impossible by Lemma 6.8(i). 2. 2.2.

$z^{s}$ is the second $z$-stack in $\mathbf{v}$. Then $\operatorname{\mathsf{fin}}(\mathbf{u})=\operatorname{\mathsf{fin}}(\mathbf{v})=\cdots z\cdots y\cdots$, so that every $z$ of $\mathbf{u}$ occurs in $\mathbf{u}_{1}$. Hence $\mathbf{u}_{\{x,z\}}\in\{z\}^{+}\{x\}^{+}$ and

[TABLE]

But this is impossible by Lemma 6.8(i).

Since both subcases are impossible, $\mathbf{v}_{3}=\emptyset$. Hence (b) holds.

Case 3

$x^{e}$ is the second $x$-stack in $\mathbf{u}$ and $y^{f}$ is the first $y$-stack in $\mathbf{u}$. Then $e=1$ by (III) and

[TABLE]

where $x^{r}$ is the first $x$-stack of $\mathbf{u}$ with and $y\notin\operatorname{\mathsf{con}}(\mathbf{u}_{1})$ and $x\notin\operatorname{\mathsf{con}}(\mathbf{u}_{2})$. Since $\operatorname{\mathsf{ini}}(\mathbf{v})=\operatorname{\mathsf{ini}}(\mathbf{u})=\cdots x\cdots y\cdots$, it follows from Lemma 6.8 that

[TABLE]

for some $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4}\in\mathscr{A}^{*}$ with $x\notin\operatorname{\mathsf{con}}(\mathbf{v}_{1}\mathbf{v}_{2}\mathbf{v}_{3}\mathbf{v}_{4})$ and $y\notin\operatorname{\mathsf{con}}(\mathbf{v}_{1}\mathbf{v}_{2}\mathbf{v}_{3})$. Suppose that $\mathbf{v}_{3}\neq\emptyset$. Then $\mathbf{v}_{3}$ contains some $z$-stack $z^{s}$:

[TABLE]

There are two subcases depending on whether or not $z^{s}$ is the first $z$-stack of $\mathbf{v}$.

1. 3.1.

$z^{s}$ is the first $z$-stack of $\mathbf{v}$. Since $\operatorname{\mathsf{ini}}(\mathbf{u})=\operatorname{\mathsf{ini}}(\mathbf{v})=\cdots z\cdots y\cdots$, the first $z$-stack $z^{s}$ of $\mathbf{u}$ occurs in $\mathbf{u}_{1}$. But since $\operatorname{\mathsf{fin}}(\mathbf{u})=\operatorname{\mathsf{fin}}(\mathbf{v})=\cdots x\cdots z\cdots$, the word $\mathbf{u}$ must contain a second $z$-stack in $\mathbf{u}_{2}$, whence $\mathbf{v}$ must also contain a second $z$-stack by Lemma 6.8(ii). Hence $\mathbf{v}_{\{x,z\}}=x^{r+1}z^{s+1}$ and

[TABLE]

But this is impossible by Lemma 6.8(i). 2. 3.2.

$z^{s}$ is the second $z$-stack of $\mathbf{v}$. Then the identity $\mathbf{u}_{\{y,z\}}\approx\mathbf{v}_{\{y,z\}}$ obtained by an argument symmetrical to the one in Subcase 3.1 produces a similar contradiction.

Since both subcases are impossible, $\mathbf{v}_{3}=\emptyset$. Hence (b) holds.

Case 4

$x^{e}$ is the second $x$-stack in $\mathbf{u}$ and $y^{f}$ is the second $y$-stack in $\mathbf{u}$. Then (b) holds by an argument symmetrical to Case 1.

Therefore, (b) holds in all four cases. By symmetry, (b) implies (a). ∎

Proof of Proposition 6.5.

It is routinely verified that the semigroup $T_{m,n}$ satisfies the identities (6.5). Conversely, suppose that $\mathbf{u}\approx\mathbf{v}$ is any identity satisfied by the semigroup $T_{m,n}$. As observed earlier, the identities (6.5) can be used to convert $\mathbf{u}$ and $\mathbf{v}$ into words $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$ in canonical form. By Lemma 6.7 parts (ii) and (iii), the words $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$ share the same set of stacks. By Lemma 6.9, two stacks are adjacent in $\mathbf{u}^{\prime}$ if and only if they are adjacent in $\mathbf{v}^{\prime}$. Therefore, $\mathbf{u}^{\prime}=\mathbf{v}^{\prime}$. Since

[TABLE]

the identity $\mathbf{u}\approx\mathbf{v}$ is deducible from (6.5). ∎

Corollary 6.10**.**

Suppose that $S$ is any finite semigroup that satisfies the identities

[TABLE]

but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle\mathscr{T}_{6,5}\rangle\rangle$ that is not $\mathsf{ji}$.

Proof.

By Proposition 6.5 with $m=6$ and $n=5$, the identities satisfied by $T_{6,5}$ are axiomatized by (6.7). Hence the inclusion

[TABLE]

holds. But the two identities in (6.8) are satisfied by $L_{2}^{I}\times R_{2}^{I}$ and $\mathbb{Z}_{6}\times N_{5}^{I}\times A_{0}^{I}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle L_{2}^{I},R_{2}^{I}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle\mathbb{Z}_{6},N_{5}^{I},A_{0}^{I}\rangle\rangle$. ∎

Corollary 6.11**.**

Suppose that $S$ is any finite semigroup that satisfies the identities (6.7) but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle\mathscr{T}_{6,5}\rangle\rangle$ that is not $\mathsf{ji}$.

Proof.

The argument in the proof of Corollary 6.10 implies the inclusion

[TABLE]

The two identities in (6.9) are satisfied by $N_{5}^{I}\times L_{2}^{I}\times R_{2}^{I}\times A_{0}^{I}$ and $\mathbb{Z}_{6}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle N_{5}^{I},L_{2}^{I},R_{2}^{I},A_{0}^{I}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle\mathbb{Z}_{6}\rangle\rangle$. ∎

6.3. Pseudovarieties of noncommutative nilpotent semigroups

Proposition 6.12**.**

Any $\mathsf{ji}$ pseudovariety of nilpotent semigroups is commutative.

Proof.

Let $\mathbf{V}$ be any $\mathsf{ji}$ pseudovariety of nilpotent semigroups. Then the inclusion $\mathbf{V}\subseteq\mathbf{Com}\vee\mathbf{G}$ holds [2, Figure 9.1]. Since $\mathbf{V}$ is $\mathsf{ji}$ and $\mathbf{V}\nsubseteq\mathbf{G}$, the inclusion $\mathbf{V}\subseteq\mathbf{Com}$ follows. ∎

Corollary 6.13**.**

Suppose that $S$ is any finite semigroup that satisfies the identity

[TABLE]

but violates the identity

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a pseudovariety of nilpotent semigroups that is not $\mathsf{ji}$.

Proof.

By assumption, the semigroup $S$ is nilpotent and noncommutative. The result then holds by Proposition 6.12. ∎

6.4. The pseudovariety $\langle\langle N_{n+r},N_{n}^{I}\rangle\rangle$

Proposition 6.14**.**

Let $n,r\geq 1$. Then the identities satisfied by the semigroup $N_{n+r}\times N_{n}^{I}$ are axiomatized by

[TABLE]

for all $m\geq 1$ and $e_{1},e_{2},\ldots,e_{m},f_{1},f_{2},\ldots,f_{m}\geq 1$ such that

1. (a)

$e=f<n+r$,* where $e=\sum_{i=1}^{m}e_{i}$ and $f=\sum_{i=1}^{m}f_{i}$;* 2. (b)

for each $k\in\{1,2,\ldots,m\}$, either

1. $\bullet$

$e_{k}=f_{k}$* or* 2. $\bullet$

$e_{k},f_{k}\geq n$* and $e+e_{k},f+f_{k}\geq n+r$.*

Proof.

It is straightforwardly verified that the semigroup $N_{n+r}\times N_{n}^{I}$ satisfies the identities (6.10). Conversely, let $\mathbf{u}\approx\mathbf{v}$ be any identity satisfied by the semigroup $N_{n+r}\times N_{n}^{I}$. In view of Lemma 5.1(ii), the identity (6.10a) can be used to convert $\mathbf{u}$ and $\mathbf{v}$ into

[TABLE]

respectively, where $e_{i}=\operatorname{\mathsf{occ}}(x_{i},\mathbf{u})$ and $f_{i}=\operatorname{\mathsf{occ}}(x_{i},\mathbf{v})$ are such that either $e_{i}=f_{i}$ or $e_{i},f_{i}\geq n$. Let $e=\sum_{i=1}^{m}e_{i}$ and $f=\sum_{i=1}^{m}f_{i}$. Generality is not lost by assuming that $e\leq f$. There are four cases to consider.

Case 1

$n+r\leq e\leq f$. Choose any $i\in\{1,2,\ldots,m\}$. Suppose that $e_{i}\neq f_{i}$. Then as observed earlier, $e_{i},f_{i}\geq n$. Hence

[TABLE]

Similarly, $\mathbf{v}^{\prime}\stackrel{{\scriptstyle\eqref{id: Nn+rjNnI basis}}}{{\approx}}x_{1}^{f_{1}}x_{2}^{f_{2}}\cdots x_{i-1}^{f_{i-1}}x_{i}^{f_{i}+e_{i}}x_{i+1}^{f_{i+1}}\cdots x_{m}^{f_{m}}$. Therefore, the identities (6.10) can be used to convert $\mathbf{u}^{\prime}$ into $\mathbf{v}^{\prime}$. It follows that $\mathbf{u}\approx\mathbf{v}$ is deducible from (6.10).

Case 2

$e<n+r\leq f$. Let $\varphi:\mathscr{A}\rightarrow N_{n+r}$ be the substitution that maps all variables to $\mathsf{a}$. Then $\mathbf{u}^{\prime}\varphi=\mathsf{a}^{e}\neq 0$ and $\mathbf{v}^{\prime}\varphi=\mathsf{a}^{f}=0$ imply the contradiction $\mathbf{u}^{\prime}\varphi\neq\mathbf{v}^{\prime}\varphi$. The present case is thus impossible.

Case 3

$e<f<n+r$. Then the contradiction $\mathbf{u}^{\prime}\varphi=\mathsf{a}^{e}\neq\mathsf{a}^{f}=\mathbf{v}^{\prime}\varphi$ is obtained. Hence the present case is impossible.

Case 4

$e=f<n+r$. Suppose that $e_{k}\neq f_{k}$ for some $k$ so that $e+e_{k}\neq f+f_{k}$. Then as observed earlier, $e_{k},f_{k}\geq n$. Let $\psi:\mathscr{A}\rightarrow N_{n+r}$ be the substitution that maps $x_{k}$ to $\mathsf{a}^{2}$ and all other variables to $\mathsf{a}$. Then $\mathbf{u}^{\prime}\psi=\mathbf{v}^{\prime}\psi$ in $N_{n}$, where

[TABLE]

and $\mathbf{v}^{\prime}\psi=\mathsf{a}^{f+f_{k}}$ similarly. Thus $\mathsf{a}^{e+e_{k}}=\mathsf{a}^{f+f_{k}}$. But $e+e_{k}\neq f+f_{k}$ implies that $e+e_{k},f+f_{k}\geq n$. Hence the identity $\mathbf{u}^{\prime}\approx\mathbf{v}^{\prime}$ also satisfies (ii) and is deducible from (6.10c). The identity $\mathbf{u}\approx\mathbf{v}$ is thus deducible from (6.10). ∎

Corollary 6.15**.**

Suppose that $S$ is any finite semigroup that satisfies the identities

[TABLE]

but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle N_{4},N_{2}^{I}\rangle\rangle$ that is not $\mathsf{ji}$.

Proof.

By Proposition 6.14 with $n=r=2$, the identities satisfied by $N_{4}\times N_{2}^{I}$ are axiomatized by (6.11). Hence the inclusion

[TABLE]

holds. But the two identities in (6.12) are satisfied by $N_{2}^{I}$ and $N_{4}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle N_{4}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle N_{2}^{I}\rangle\rangle$. ∎

Corollary 6.16**.**

Suppose that $S$ is any finite semigroup that satisfies the identities

[TABLE]

but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle N_{5},N_{1}^{I}\rangle\rangle$ that is not $\mathsf{ji}$.

Proof.

By Proposition 6.14 with $n=1$ and $r=4$, the identities satisfied by $N_{5}\times N_{1}^{I}$ are axiomatized by (6.13). Hence the inclusion

[TABLE]

holds. But the two identities in (6.14) are satisfied by $N_{1}^{I}$ and $N_{5}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle N_{5}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle N_{1}^{I}\rangle\rangle$. ∎

6.5. The pseudovariety $\langle\langle N_{n}^{I},N_{2}^{\mathsf{bar}}\rangle\rangle$

It turns out to be notationally simpler to find a basis of identities for $N_{n}^{I}\times(N_{2}^{\mathsf{bar}})^{\mathsf{op}}$ instead of $N_{n}^{I}\times N_{2}^{\mathsf{bar}}$.

Proposition 6.17**.**

Let $n\geq 2$. Then the identities satisfied by the semigroup $N_{n}^{I}\times(N_{2}^{\mathsf{bar}})^{\mathsf{op}}$ are axiomatized by

[TABLE]

In this subsection, a word of length at least two is said to be in canonical form if it is either

1. (CF1)

$x^{2}\cdot x^{e}z_{1}^{f_{1}}z_{2}^{f_{2}}\cdots z_{k}^{f_{k}}$ or 2. (CF2)

$xy\cdot x^{e_{1}}y^{e_{2}}z_{1}^{f_{1}}z_{2}^{f_{2}}\cdots z_{k}^{f_{k}}$,

where

1. (I)

$x,y,z_{1},z_{2},\ldots,z_{k}$ are distinct variables with $k\geq 0$; 2. (II)

$z_{1},z_{2},\ldots,z_{k}$ are in alphabetical order; 3. (III)

$e\in\{0,1,\ldots,n-2\}$, $e_{i}\in\{0,1,\ldots,n-1\}$, and $f_{i}\in\{1,2,\ldots,n\}$.

Lemma 6.18**.**

The identities (6.15) can be used to convert any word of length at least two into a word in canonical form.

Proof.

Let $\mathbf{w}$ be any word of length at least two. Then $\mathbf{w}=x^{2}\mathbf{u}$ or $\mathbf{w}=xy\mathbf{u}$ for some distinct $x,y\in\mathscr{A}$ and $\mathbf{u}\in\mathscr{A}^{*}$. The identity (6.15b) can first be used to rearrange the variables of the suffix $\mathbf{u}$ until $\mathbf{w}$ becomes a word of the form (CF1) or (CF2) with (I) and (II) satisfied. The identities (6.15a) can then be used to reduce the exponents $e,e_{i},f_{i}$ so that (III) is satisfied. ∎

Lemma 6.19**.**

The semigroup $N_{2}^{\mathsf{bar}}$ satisfies an identity $\mathbf{u}\approx\mathbf{v}$ if and only if the words $\mathbf{u}$ and $\mathbf{v}$ share the same suffix of length two.

Proof.

This is routinely established and its dual result for $(N_{2}^{\mathsf{bar}})^{\mathsf{op}}$ was observed by Lee and Li [12, Remark 6.2(i)]. ∎

Proof of Proposition 6.17.

It is easily verified, either directly or by Lemmas 5.1(ii) and 6.19, that the identities (6.15) are satisfied by the semigroup $N_{n}^{I}\times(N_{2}^{\mathsf{bar}})^{\mathsf{op}}$. Hence it remains to show that any identity $\mathbf{u}\approx\mathbf{v}$ satisfied by the semigroup $N_{n}^{I}\times(N_{2}^{\mathsf{bar}})^{\mathsf{op}}$ is deducible from the identities (6.15). It is easily shown that if either $\mathbf{u}$ or $\mathbf{v}$ is a single variable, then the identity $\mathbf{u}\approx\mathbf{v}$ is trivial by Lemma 5.1(ii) and so is vacuously deducible from the identities (6.15). Therefore, assume that $\mathbf{u}$ and $\mathbf{v}$ are words of length at least two. By Lemma 6.18, the identities (6.15) can be used to convert $\mathbf{u}$ and $\mathbf{v}$ into words $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$ in canonical form. By Lemma 6.19, the words $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$ share the same prefix of length two. Therefore, $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$ are both of the form (CF1) or both of the form (CF2). In any case, it is routinely verified by Lemma 5.1(ii) that $\mathbf{u}^{\prime}=\mathbf{v}^{\prime}$. Since

[TABLE]

the identity $\mathbf{u}\approx\mathbf{v}$ is deducible from (6.15). ∎

Corollary 6.20**.**

Suppose that $S$ is any finite semigroup that satisfies the identities

[TABLE]

but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle N_{5}^{I},N_{2}^{\mathsf{bar}}\rangle\rangle$ that is not $\mathsf{ji}$.

Proof.

By Proposition 6.17 with $n=5$, the identities satisfied by $N_{5}^{I}\times N_{2}^{\mathsf{bar}}$ are axiomatized by (6.16). Hence the inclusion

[TABLE]

holds. But the two identities in (6.17) are satisfied by $N_{5}^{I}$ and $N_{2}^{\mathsf{bar}}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle N_{5}^{I}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle N_{2}^{\mathsf{bar}}\rangle\rangle$. ∎

6.6. The pseudovariety $\langle\langle N_{2}^{I},(L_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$

Proposition 6.21**.**

The identities satisfied by the semigroup $N_{2}^{I}\times(L_{2}^{\mathsf{bar}})^{\mathsf{op}}$ are axiomatized by

[TABLE]

Proof.

Let $T_{6}=\{a,b,c,d,e,f\}$ be the semigroup given in Table 9. The identities satisfied by $T_{6}$ are axiomatized by (6.18) [16, Proposition 26.1]. It is easily deduced from the proof of this result that any identity violated by $T_{6}$ is also violated by one of the following subsemigroups of $T_{6}$:

[TABLE]

Since $L_{2}^{I},R_{2}\in\langle\langle(L_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$, any identity violated by $T_{6}$ is violated by $N_{2}^{I}$ or $(L_{2}^{\mathsf{bar}})^{\mathsf{op}}$. Therefore, the semigroup $N_{2}^{I}\times(L_{2}^{\mathsf{bar}})^{\mathsf{op}}$ does not generate any proper subpseudovariety of $\langle\langle T_{6}\rangle\rangle$, whence $\langle\langle N_{2}^{I}\times(L_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle=\langle\langle T_{6}\rangle\rangle$. ∎

Corollary 6.22**.**

Suppose that $S$ is any finite semigroup that satisfies the identities (6.18) but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle N_{2}^{I},(L_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$ that is not $\mathsf{ji}$.

Proof.

By Proposition 6.21, the inclusion

[TABLE]

holds. But the two identities in (6.19) are satisfied by $(L_{2}^{\mathsf{bar}})^{\mathsf{op}}$ and $N_{2}^{I}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle N_{2}^{I}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle(L_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle$. ∎

6.7. The pseudovariety $\langle\langle L_{2}^{I},\ell_{3},\ell_{3}^{\mathsf{op}}\rangle\rangle$

Proposition 6.23** (Zhang and Luo [32, Proposition 3.2(3) and Figure 5]).**

The identities satisfied by the semigroup $L_{2}^{I}\times\ell_{3}\times\ell_{3}^{\mathsf{op}}$ are axiomatized by

[TABLE]

Corollary 6.24**.**

Suppose that $S$ is any finite semigroup that satisfies the identities (6.20) but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle L_{2}^{I},\ell_{3},\ell_{3}^{\mathsf{op}}\rangle\rangle$ that is not $\mathsf{ji}$.**

Proof.

By Proposition 6.23, the inclusion

[TABLE]

holds. But the two identities in (6.21) are satisfied by $L_{2}^{I}\times\ell_{3}$ and $\ell_{3}^{\mathsf{op}}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle L_{2}^{I},\ell_{3}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle\ell_{3}^{\mathsf{op}}\rangle\rangle$. ∎

6.8. The pseudovariety $\langle\langle A_{0},\ell_{3}^{I},(\ell_{3}^{\mathsf{op}})^{I}\rangle\rangle$

Proposition 6.25** (Lee [9, Proposition 2.8]).**

The identities satisfied by the semigroup $A_{0}\times\ell_{3}^{I}\times(\ell_{3}^{\mathsf{op}})^{I}$ are axiomatized by

[TABLE]

Proof.

The identities satisfied by the semigroup $A_{0}\times B_{0}^{I}$ are axiomatized by the identities (6.22) [9, Proposition 2.8]. Since $\langle\langle B_{0}^{I}\rangle\rangle=\langle\langle\ell_{3}^{I},(\ell_{3}^{\mathsf{op}})^{I}\rangle\rangle$ [9, Figure 4], the result follows. ∎

Corollary 6.26**.**

Suppose that $S$ is any finite semigroup that satisfies the identities (6.22) but violates all of the identities

[TABLE]

Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle A_{0},\ell_{3}^{I},(\ell_{3}^{\mathsf{op}})^{I}\rangle\rangle$ that is not $\mathsf{ji}$.**

Proof.

By Proposition 6.25, the inclusion

[TABLE]

holds. But the three identities in (6.23) are satisfied by $A_{0}$, $(\ell_{3}^{\mathsf{op}})^{I}$, and $\ell_{3}^{I}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle A_{0}\rangle\rangle$, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle\ell_{3}^{I}\rangle\rangle$, and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle(\ell_{3}^{\mathsf{op}})^{I}\rangle\rangle$. ∎

6.9. The pseudovariety $\langle\langle(N_{2}^{\mathsf{bar}})^{I},L_{2}^{\mathsf{bar}}\rangle\rangle$

The semigroup $W=\{a,b,c,d,e\}$ given in Table 10 is required in this subsection.

Proposition 6.27**.**

1. (i)

The identities satisfied by the semigroup $W$ are axiomatized by

[TABLE] 2. (ii)

The subpseudovariety of $\langle\langle W\rangle\rangle$ defined by the identity

[TABLE]

is the unique maximal proper subpseudovariety of $\langle\langle W\rangle\rangle$.

Remark 6.28**.**

Proposition 6.27(i) was first established in Tishchenko and Volkov [28, Theorem 2]. But since its proof is not long and an understanding of the identities satisfied by $W$ is fundamental to the proof of Proposition 6.27(ii), it is given below for the sake of completeness.

In this subsection, a word of the form

[TABLE]

where $x_{1},x_{2},\ldots,x_{m}$ are distinct variables and $e_{1},e_{2},\ldots,e_{m}\in\{1,2\}$, is said to be in canonical form.

Remark 6.29**.**

It is easily shown that the identities (6.24) can be used to convert any word into one in canonical form.

Proof of Proposition 6.27.

(i) It is routinely shown that the semigroup $W$ satisfies the identities (6.24). Conversely, suppose that $\mathbf{u}\approx\mathbf{v}$ is any identity satisfied by $W$. By Remark 6.29, the identities (6.24) can be used to convert $\mathbf{u}$ and $\mathbf{v}$ into some words $\mathbf{u}^{\prime}$ and $\mathbf{v}^{\prime}$ in canonical form. Since the subsemigroup $\{a,c,d\}$ of $W$ and the semigroup $R_{2}^{I}$ are isomorphic, $\operatorname{\mathsf{fin}}(\mathbf{u}^{\prime})=\operatorname{\mathsf{fin}}(\mathbf{v}^{\prime})$ by Lemma 5.1(iv). Hence

[TABLE]

for some distinct $x_{1},x_{2},\ldots,x_{m}\in\mathscr{A}$ and $e_{1},e_{2},\ldots,e_{m},f_{1},f_{2},\ldots,f_{m}\in\{1,2\}$. If $e_{k}\neq f_{k}$, then by making the substitution $\varphi$ given by $x_{k}\mapsto b$, $x_{i}\mapsto e$ for any $i<k$, and $x_{i}\mapsto c$ for any $i>k$, the contradiction $\mathbf{u}^{\prime}\varphi\neq\mathbf{v}^{\prime}\varphi$ is obtained. Therefore, $e_{i}=f_{i}$ for all $i$, so that $\mathbf{u}^{\prime}=\mathbf{v}^{\prime}$. Consequently, the identity $\mathbf{u}\approx\mathbf{v}$ is deducible from the identities (6.24).

(ii) The semigroup $W$ violates the identity (6.25) because $e^{2}b^{2}c^{2}\neq e^{2}bc^{2}$. Therefore, $\langle\langle W\rangle\rangle\cap[[\eqref{id: W max}]]$ is a proper subpseudovariety of $\langle\langle W\rangle\rangle$. It remains to verify that every proper subpseudovariety $\mathbf{V}$ of $\langle\langle W\rangle\rangle$ satisfies the identity (6.25). Since $\mathbf{V}\neq\langle\langle W\rangle\rangle$, there exists an identity $\mathbf{u}\approx\mathbf{v}$ of $\mathbf{V}$ that is violated by $W$. Further, since the identities (6.24) are satisfied by $\mathbf{V}$, it follows from Remark 6.29 that the words $\mathbf{u}$ and $\mathbf{v}$ can be chosen to be in canonical form. There are two cases.

Case 1

$\operatorname{\mathsf{fin}}(\mathbf{u})\neq\operatorname{\mathsf{fin}}(\mathbf{v})$. Then by Lemma 5.1(iv) and the dual of Theorem 5.17, the pseudovariety $\mathbf{V}$ satisfies the pseudoidentity

[TABLE]

Since

[TABLE]

the pseudovariety $\mathbf{V}$ satisfies the identity

[TABLE]

Since

[TABLE]

the pseudovariety $\mathbf{V}$ satisfies the identity (6.25).

Case 2

$\operatorname{\mathsf{fin}}(\mathbf{u})=\operatorname{\mathsf{fin}}(\mathbf{v})$ and $\mathbf{u}\neq\mathbf{v}$. Then

[TABLE]

for some distinct $x_{1},x_{2},\ldots,x_{m}\in\mathscr{A}$ and $e_{1},e_{2},\ldots,e_{m},f_{1},f_{2},\ldots,f_{m}\in\{1,2\}$ such that $e_{k}\neq f_{k}$ for some $k$, say $(e_{k},f_{k})=(2,1)$. Let $\varphi$ denote the substitution given by $x_{k}\mapsto y$, $x_{i}\mapsto x^{2}$ for any $i<k$, and $x_{i}\mapsto z^{2}$ for any $i>k$. Then

[TABLE]

so that the pseudovariety $\mathbf{V}$ satisfies the identity (6.25). ∎

Proposition 6.30**.**

The pseudovariety $\langle\langle W\rangle\rangle$ is $\mathsf{sji}$ but not $\mathsf{ji}$.**

Proof.

The pseudovariety $\langle\langle W\rangle\rangle$ is $\mathsf{sji}$ by Proposition 6.27(ii). To show that $\langle\langle W\rangle\rangle$ is not $\mathsf{ji}$, the semigroups $N_{2}^{\mathsf{bar}}=\{\overline{0},\mathsf{a},\overline{\mathsf{a}},\overline{I}\}$ and $L_{2}^{\mathsf{bar}}=\{\mathsf{e},\mathsf{f},\overline{\mathsf{e}},\overline{\mathsf{f}},\overline{I}\}$ from Subsections 3.2 and 3.3 are required. Let $T_{11}$ denote the subsemigroup of $(N_{2}^{\mathsf{bar}})^{I}\times L_{2}^{\mathsf{bar}}$ generated by $\mathsf{E}=(1,\mathsf{e})$, $\mathsf{X}=(\mathsf{a},\mathsf{f})$, and $\mathsf{Y}=(\overline{I},\overline{I})$. It is routinely checked that

[TABLE]

so that the semigroup $T_{11}$ consists of the following 11 elements:

[TABLE]

see Table 11. Identifying the elements $\{b,c,e,h,i,j,k\}$ in $T_{11}$ results in a semigroup that is isomorphic to $W$. Therefore,

[TABLE]

But the semigroup $W$ violates the identities

[TABLE]

and these identities are satisfied by $(N_{2}^{\mathsf{bar}})^{I}$ and $L_{2}^{\mathsf{bar}}$, respectively. Consequently, $\langle\langle W\rangle\rangle\nsubseteq\langle\langle(N_{2}^{\mathsf{bar}})^{I}\rangle\rangle$ and $\langle\langle W\rangle\rangle\nsubseteq\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle$. ∎

Corollary 6.31**.**

Suppose that $S$ is any finite semigroup that satisfies the identities (6.24) but violates all the identities in (6.27).* Then $\langle\langle S\rangle\rangle$ is a subpseudovariety of $\langle\langle(N_{2}^{\mathsf{bar}})^{I},L_{2}^{\mathsf{bar}}\rangle\rangle$ that is not $\mathsf{ji}$*.**

Proof.

The inclusions

[TABLE]

hold by Proposition 6.27(i) and the proof of Proposition 6.30. But the identities in (6.27) are satisfied by $(N_{2}^{\mathsf{bar}})^{I}$ and $L_{2}^{\mathsf{bar}}$, respectively. Therefore, $\langle\langle S\rangle\rangle\nsubseteq\langle\langle(N_{2}^{\mathsf{bar}})^{I}\rangle\rangle$ and $\langle\langle S\rangle\rangle\nsubseteq\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle$. ∎

7. Pseudovarieties generated by a semigroup of order up to five

Theorem 7.1**.**

Let $S$ be any nontrivial semigroup of order up to five.* Suppose that the pseudovariety $\langle\langle S\rangle\rangle$ is $\mathsf{ji}$. Then $\langle\langle S\rangle\rangle$ coincides with one of the following $30$ pseudovarieties*:**

[TABLE]

Proof.

The 30 pseudovarieties are $\mathsf{ji}$ by results in Sec. 5; see Table 12. Up to isomorphism and anti-isomorphism, there exist $1308$ nontrivial semigroups of order up to five. With the aid of a computer, it is routinely determined, using the sufficient conditions given in Subsections 7.1 and 7.2 below, which of these semigroups generate $\mathsf{ji}$ pseudovarieties. Specifically, by Conditions A1–A23 and their dual conditions, $241$ of the $1308$ semigroups generate the 30 $\mathsf{ji}$ pseudovarieties, while by Conditions B1–B13 and their dual conditions, the remaining $1067$ semigroups generate pseudovarieties that are not $\mathsf{ji}$; see Table 13. The proof of Theorem 7.1 is thus complete. ∎

7.1. Conditions sufficient for join irreducibility

The following conditions and their dual conditions are sufficient for a finite semigroup $S$ to generate a $\mathsf{ji}$ pseudovariety in Theorem 7.1.

Condition A​​ 1** (Proposition 5.4).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle\mathbb{Z}_{2}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{2}y\approx y\}$,** 2. $\bullet$

$S\not\models x\approx y$.**

Condition A​​ 2** (Proposition 5.4).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle\mathbb{Z}_{3}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{3}y\approx y\}$,** 2. $\bullet$

$S\not\models x\approx y$.**

Condition A​​ 3** (Proposition 5.4).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle\mathbb{Z}_{4}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{4}y\approx y\}$,** 2. $\bullet$

$S\not\models x^{3}\approx x$.**

Condition A​​ 4** (Proposition 5.4).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle\mathbb{Z}_{5}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{5}y\approx y\}$,** 2. $\bullet$

$S\not\models x\approx y$.**

Condition A​​ 5** (Proposition 5.6).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle\mathbb{Z}_{2}^{\mathsf{bar}}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{3}\approx x,\,xyxy\approx yx^{2}y\}$,** 2. $\bullet$

$S\not\models xyx\approx yx^{2}$.**

Condition A​​ 6** (Proposition 5.8).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{2}\rangle\rangle}$ holds if

1. $\bullet$

$S\models x^{2}\approx y_{1}y_{2}$,** 2. $\bullet$

$S\not\models x^{2}\approx x$.**

Condition A​​ 7** (Proposition 5.8).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{3}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{3}\approx y_{1}y_{2}y_{3}\}$,** 2. $\bullet$

$S\not\models x^{3}\approx x^{2}$.**

Condition A​​ 8** (Proposition 5.8).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{4}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{2}y\approx xy^{2},\,x^{4}\approx y_{1}y_{2}y_{3}y_{4}\}$,** 2. $\bullet$

$S\not\models x^{4}\approx x^{3}$.**

Condition A​​ 9** (Proposition 5.8).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{5}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{2}yz\approx xy^{2}z,\,x^{5}\approx y_{1}y_{2}y_{3}y_{4}y_{5}\}$,** 2. $\bullet$

$S\not\models x^{5}\approx x^{4}$.**

Condition A​​ 10** (Proposition 5.10).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{1}^{I}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{2}\approx x,\,xy\approx yx\}$,** 2. $\bullet$

$S\not\models x\approx y$.**

Condition A​​ 11** (Proposition 5.10).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{2}^{I}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{3}\approx x^{2},\,xy\approx yx\}$,** 2. $\bullet$

$S\not\models x^{2}y\approx xy^{2}$.**

Condition A​​ 12** (Proposition 5.10).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{3}^{I}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{4}\approx x^{3},\,xy\approx yx\}$,** 2. $\bullet$

$S\not\models x^{3}y^{2}\approx x^{2}y^{3}$.**

Condition A​​ 13** (Proposition 5.10).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{4}^{I}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{5}\approx x^{4},\,xy\approx yx\}$,** 2. $\bullet$

$S\not\models x^{4}y^{3}\approx x^{3}y^{4}$.**

Condition A​​ 14** (Proposition 5.12).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle N_{2}^{\mathsf{bar}}\rangle\rangle}$ holds if

1. $\bullet$

$S\models xyz\approx yz$,** 2. $\bullet$

$S\not\models xy\approx y^{2}$.**

Condition A​​ 15** (Proposition 5.14).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle(N_{2}^{\mathsf{bar}})^{I}\rangle\rangle}$ holds if

1. $\bullet$

S\models\left\{\begin{array}[c]{l}x^{3}\approx x^{2},\,x^{2}hx\approx xhx,\,xhx^{2}\approx hx^{2},\,xyxy\approx yx^{2}y,\\ xyhxy\approx yxhxy,\,xyxty\approx yx^{2}ty,\,xyhxty\approx yxhxty\end{array}\right\},** 2. $\bullet$

$S\not\models xyxyh^{2}\approx x^{2}y^{2}h^{2}$.**

Condition A​​ 16** (Proposition 5.16).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle L_{2}\rangle\rangle}$ holds if

1. $\bullet$

$S\models xy\approx x$,** 2. $\bullet$

$S\not\models x\approx y$.**

Condition A​​ 17** (Proposition 5.18).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle L_{2}^{I}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{2}\approx x,\,xyx\approx xy\}$,** 2. $\bullet$

$S\not\models xyz\approx xzy$.**

Condition A​​ 18** (Proposition 5.20).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle L_{2}^{\mathsf{bar}}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{2}\approx x,\,xyz\approx xzxyz\}$,** 2. $\bullet$

$S\not\models xyz\approx xzyz$.**

Condition A​​ 19** (Proposition 5.22).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle A_{0}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{3}\approx x^{2},\,x^{2}yx^{2}\approx yxy\}$,** 2. $\bullet$

$S\not\models x^{2}y^{2}\approx y^{2}x^{2}$.**

Condition A​​ 20** (Proposition 5.24).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle A_{0}^{I}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{3}\approx x^{2},\,x^{2}yx^{2}\approx xyx,\,xyxy\approx yxyx,\,xyxzx\approx xyzx\}$,** 2. $\bullet$

$S\not\models hx^{2}y^{2}h\approx hy^{2}x^{2}h$.**

Condition A​​ 21** (Proposition 5.26).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle A_{2}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{3}\approx x^{2},\,xyxyx\approx xyx,\,xyxzx\approx xzxyx\}$,** 2. $\bullet$

$S\not\models x^{2}y^{2}x^{2}\approx x^{2}yx^{2}$.**

Condition A​​ 22** (Proposition 5.28).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle B_{2}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{3}\approx x^{2},\,xyxyx\approx xyx,\,x^{2}y^{2}\approx y^{2}x^{2}\}$,** 2. $\bullet$

$S\not\models xy^{2}x\approx xyx$.**

Condition A​​ 23** (Proposition 5.30).**

The equality ${\langle\langle S\rangle\rangle=\langle\langle\ell_{3}^{\mathsf{bar}}\rangle\rangle}$ holds if

1. $\bullet$

$S\models\{x^{2}y\approx xy,\,xyz\approx yxyz\}$,** 2. $\bullet$

$S\not\models xyzx\approx yxzx$.**

7.2. Conditions sufficient for non-join irreducibility

The following conditions and their dual conditions are sufficient for a finite semigroup $S$ to generate a pseudovariety that is not $\mathsf{ji}$.

Condition B​​ 1** (Corollary 6.2).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle\mathbb{Z}_{3},\mathbb{Z}_{4},\mathbb{Z}_{2}^{\mathsf{bar}},(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}},N_{3}^{I}\rangle\rangle}$ if

1. $\bullet$

S\models\left\{\begin{array}[c]{l}x^{15}\approx x^{3},\,x^{14}hx\approx x^{2}hx,\,x^{13}hx^{2}\approx xhx^{2},\,x^{13}hxtx\approx xhxtx,\\ x^{3}hx\approx xhx^{3},\,xhx^{2}tx\approx x^{3}htx,\,xhx^{2}y^{2}ty\approx xhy^{2}x^{2}ty,\\ xhykxytxdy\approx xhykyxtxdy,\,xhykxytydx\approx xhykyxtydx\end{array}\right\},** 2. $\bullet$

$S\not\models x^{3}\approx x$,*  $S\not\models xy\approx yx$.*

Condition B​​ 2** (Corollary 6.3).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle\mathbb{Z}_{3},\mathbb{Z}_{4},\mathbb{Z}_{2}^{\mathsf{bar}},(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}},N_{3}^{I}\rangle\rangle}$ if

1. $\bullet$

S\models\left\{\begin{array}[c]{l}x^{15}\approx x^{3},\,x^{14}hx\approx x^{2}hx,\,x^{13}hx^{2}\approx xhx^{2},\,x^{13}hxtx\approx xhxtx,\\ x^{3}hx\approx xhx^{3},\,xhx^{2}tx\approx x^{3}htx,\,xhx^{2}y^{2}ty\approx xhy^{2}x^{2}ty,\\ xhykxytxdy\approx xhykyxtxdy,\,xhykxytydx\approx xhykyxtydx\end{array}\right\},** 2. $\bullet$

$S\not\models xy\approx yx$,*  $S\not\models xyx^{2}\approx xy$,  $S\not\models x^{2}yx\approx yx$.*

Condition B​​ 3** (Corollary 6.4).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle\mathbb{Z}_{3},\mathbb{Z}_{4},\mathbb{Z}_{2}^{\mathsf{bar}},(\mathbb{Z}_{2}^{\mathsf{bar}})^{\mathsf{op}},N_{3}^{I}\rangle\rangle}$ if

1. $\bullet$

S\models\left\{\begin{array}[c]{l}x^{15}\approx x^{3},\,x^{14}hx\approx x^{2}hx,\,x^{13}hx^{2}\approx xhx^{2},\,x^{13}hxtx\approx xhxtx,\\ x^{3}hx\approx xhx^{3},\,xhx^{2}tx\approx x^{3}htx,\,xhx^{2}y^{2}ty\approx xhy^{2}x^{2}ty,\\ xhykxytxdy\approx xhykyxtxdy,\,xhykxytydx\approx xhykyxtydx\end{array}\right\},** 2. $\bullet$

$S\not\models x^{4}\approx x^{3}$,*  $S\not\models x^{4}y\approx y$,  $S\not\models x^{3}y\approx y$,  $S\not\models xyx^{2}\approx xy$,  $S\not\models x^{2}yx\approx yx$.*

Condition B​​ 4** (Corollary 6.10).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle\mathbb{Z}_{6},N_{5}^{I},L_{2}^{I},R_{2}^{I},A_{0}^{I}\rangle\rangle}$ if

1. $\bullet$

$S\models\{x^{11}\approx x^{5},\,x^{10}yx\approx x^{4}yx,\,x^{2}yx\approx xyx^{2},\,xyxzx\approx x^{2}yzx\}$,** 2. $\bullet$

$S\not\models x^{2}\approx x$,*  $S\not\models xyxy\approx yxyx$.*

Condition B​​ 5** (Corollary 6.11).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle\mathbb{Z}_{6},N_{5}^{I},L_{2}^{I},R_{2}^{I},A_{0}^{I}\rangle\rangle}$ if

1. $\bullet$

$S\models\{x^{11}\approx x^{5},\,x^{10}yx\approx x^{4}yx,\,x^{2}yx\approx xyx^{2},\,xyxzx\approx x^{2}yzx\}$,** 2. $\bullet$

$S\not\models x^{6}\approx x^{5}$,*  $S\not\models x^{6}y\approx y$.*

Condition B​​ 6** (Corollary 6.13).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ pseudovariety of nilpotent semigroups if

1. $\bullet$

$S\models x^{6}\approx y_{1}y_{2}y_{3}y_{4}y_{5}y_{6}$,** 2. $\bullet$

$S\not\models xy\approx yx$.**

Condition B​​ 7** (Corollary 6.15).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle N_{4},N_{2}^{I}\rangle\rangle}$ if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{3}y_{1}y_{2}\approx x^{2}y_{1}y_{2}\}$,** 2. $\bullet$

$S\not\models x^{3}\approx x^{2}$,*  $S\not\models x^{2}y\approx xy^{2}$.*

Condition B​​ 8** (Corollary 6.16).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle N_{5},N_{1}^{I}\rangle\rangle}$ if

1. $\bullet$

$S\models\{xy\approx yx,\,x^{2}yz\approx xy^{2}z,\,x^{2}y_{1}y_{2}y_{3}y_{4}\approx xy_{1}y_{2}y_{3}y_{4}\}$,** 2. $\bullet$

$S\not\models x^{2}\approx x$,*  $S\not\models x^{5}\approx y^{5}$.*

Condition B​​ 9** (Corollary 6.20).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle N_{5}^{I},N_{2}^{\mathsf{bar}}\rangle\rangle}$ if

1. $\bullet$

$S\models\{x^{6}\approx x^{5},\,x^{5}yx\approx x^{4}yx,\,xyzt\approx yxzt\}$,** 2. $\bullet$

$S\not\models xy\approx yx$,*  $S\not\models xyz\approx yz$.*

Condition B​​ 10** (Corollary 6.22).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle N_{2}^{I},(L_{2}^{\mathsf{bar}})^{\mathsf{op}}\rangle\rangle}$ if

1. $\bullet$

$S\models\{x^{3}\approx x^{2},\,x^{2}yx^{2}\approx xyx,\,xhytxy\approx x^{2}hyty,\,xhytyx\approx xhy^{2}tx\}$,** 2. $\bullet$

$S\not\models x^{2}\approx x$,*  $S\not\models xy\approx yx$.*

Condition B​​ 11** (Corollary 6.24).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle L_{2}^{I},\ell_{3},\ell_{3}^{\mathsf{op}}\rangle\rangle}$ if

1. $\bullet$

$S\models\{x^{3}\approx x^{2},\,xyx\approx x^{2}y^{2},\,xy^{2}z\approx xyz\}$,** 2. $\bullet$

$S\not\models x^{2}y\approx xy$,*  $S\not\models xy^{2}\approx xy$.*

Condition B​​ 12** (Corollary 6.26).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of ${\langle\langle A_{0},\ell_{3}^{I},(\ell_{3}^{\mathsf{op}})^{I}\rangle\rangle}$ if

1. $\bullet$

S\models\left\{\begin{array}[c]{l}x^{3}\approx x^{2},\,x^{2}yx^{2}\approx xyx,\,xyxy\approx yxyx,\\ xyxzx\approx xyzx,\,xy^{2}z^{2}x\approx xz^{2}y^{2}x\end{array}\right\},** 2. $\bullet$

$S\not\models xyx\approx yxy$,*  $S\not\models xyx\approx x^{2}y$,  $S\not\models xyx\approx yx^{2}$.*

Condition B​​ 13** (Corollary 6.31).**

A pseudovariety $\langle\langle S\rangle\rangle$ is a non-$\mathsf{ji}$ subpseudovariety of $\langle\langle(N_{2}^{\mathsf{bar}})^{I},L_{2}^{\mathsf{bar}}\rangle\rangle$ if

1. $\bullet$

$S\models\{x^{3}\approx x^{2},\,xyx\approx y^{2}x\}$,** 2. $\bullet$

$S\not\models x^{2}\approx x$,*  $S\not\models xyx^{2}\approx yx^{2}$.*

Acknowledgments

We are very grateful to the following colleagues: the anonymous reviewer, for a careful reading of the entire paper and a number of useful suggestions; Norman Reilly, for a helpful discussion on $\mathsf{sji}$ pseudovarieties of bands; George Bergman, for allowing us to include Theorem 4.20 and its proof in the paper; and Wendy Wong, for checking the sufficient conditions in Sec. 7, with a computer, against all semigroups of order up to five. We also thank Keith Kearnes and the reviewer for information on Proposition 4.21.

John Rhodes was supported by Simons Foundation Collaboration Grants for Mathematicians #313548. Benjamin Steinberg was supported by Simons Foundation #245268, United States–Israel Binational Science Foundation #2012080, and NSA MSP #H98230-16-1-0047.