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
01
Characterization of semigroups generating join irreducible pseudovarieties.
02
Introduction of an operator that preserves join irreducibility.
03
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 AĂB of finite semigroups, then S divides either An or Bn for some nâĽ1. We present a new operator VâŚVbar 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 suchâŚ
Tables13
Table 1. Table 1. Some generators of primitive đđ đđ \mathsf{ji} pseudovarieties
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Full text
Join irreducible semigroups
Edmond W. H. Lee
Department of Mathematics, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, FLÂ 33314, USA
Dedicated to the 80th birthday of Norman Reilly on 30 Jan 2020
and the 65th birthday of Mikhail Volkov on 27 May 2020
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 AĂB of finite semigroups, then S divides either An or Bn for some nâĽ1.
We present a new operator VâŚVbar 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)].
Key words and phrases:
Semigroup, pseudovariety, join irreducible
2000 Mathematics Subject Classification:
20M07
The second author was supported by Simons Foundation Collaboration Grants for Mathematicians #313548.
The third author was supported by Simons Foundation #245268, United StatesâIsrael Binational Science Foundation #2012080, and NSA MSP #H98230-16-1-0047.
In the 1970s, Eilenberg [4] highlighted the importance of PV, the algebraic lattice of all pseudovarieties of finite semigroups, via his research with Schßtzenberger, by providing a correspondence between PV and varieties of regular languages.
Specifically, they proved that the lattice 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 q-theory of finite semigroups focuses on 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 PV, continuous functions with Cnt(PV), and differentiable functions with GMC(PV); see [23, Chapter 2].
From a number of points of view, 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 approachâemployed by Almeida in his work and monograph [2]âthat 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 PV going back to Stone with lattice theoretic foundations going back to Birkhoff; see [23, Chapter 7].
Since PV is a lattice, it is natural to investigate its elements that satisfy important lattice properties.
For any element â in a lattice L,
(1)
â is compact if, for any XâL,
[TABLE]
2. (2)
â is join irreducible (ji) if, for any XâL,
[TABLE]
3. (3)
â is finite join irreducible (fji) if, for any finite FâL,
[TABLE]
4. (4)
â is meet irreducible (mi) if, for any set XâL,
[TABLE]
5. (5)
â is finite meet irreducible (fmi) if, for any finite FâL,
[TABLE]
6. (6)
â is strictly join irreducible (sji) if, for any set XâL,
[TABLE]
7. (7)
â is strictly finite join irreducible (sfji) if, for any finite FâL,
[TABLE]
8. (8)
â is strictly meet irreducible (smi) if, for any XâL,
[TABLE]
9. (9)
â is strictly finite meet irreducible (sfmi) if, for any finite FâL,
[TABLE]
An algebraic lattice is a complete lattice that is join generated by its compact elements.
The compact elements of PV are the finitely generated pseudovarieties.
The pseudovariety generated by a finite semigroup S is denoted by â¨â¨SâŠâŠ.
It is clear that for any VâPV,
[TABLE]
The abstract spectral theory of a lattice is closely connected to the computation of its maximal distributive image, which is determined by the latticeâs fji and fmi elements; see [23, Chapter 7] and the references therein.
The fji and fmi elements of PV are thus very important.
The ji pseudovarieties are just the compact fji pseudovarieties, as is easy to see, so we are interested in finite semigroups that generate pseudovarieties that are fji or equivalently ji.
By abuse of terminology, we say that a finite semigroup S is join irreducible (ji) if the pseudovariety â¨â¨SâŠâŠ is ji; finite semigroups that satisfy the properties in (3)â(9) are similarly defined.
A finite semigroup S is ji if and only if for all finite semigroups T1â and T2â,
[TABLE]
where AâşB means that A is a homomorphic image of a subsemigroup of B, and An=AĂAĂâŻĂA is the direct product of n copies of A.
For finite semigroups, there are several properties stronger than being ji: a finite semigroup S is Ă-prime [2, Section 9.3] if for all finite semigroups T1â and T2â,
[TABLE]
a semigroup S is KovĂĄcsâNewman (KN) if whenever f:Tâ S is a surjective homomorphism where T is a subsemigroup of T1âĂT2â for some finite semigroups T1â and T2â, subdirectly embedded, then f factors through one of the projections.
Semigroups that are 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 KN, any cyclic group Zpâ of prime order p is Ă-prime but not KN.
The well-known Brandt semigroup B2â of order five is ji but not Ă-prime [23, Example 7.4.3].
Since the lattice PV is algebraic, it follows from a well-known theorem of Birkhoff that its smi elements constitute the unique minimal set of meet generators [23, Section 7.1].
It easily follows from Reitermanâs theorem [23, Section 3.2] that each smi pseudovariety is defined by a single pseudoidentity but not conversely.
Now the reverse of the lattice PV is not algebraic but is locally dually algebraic, so the sji elements
of PV constitute the unique minimal set of join generators for PV [23, Section 7.2].
The sji pseudovarieties are precisely those having a unique proper maximal subpseudovariety.
Every ji pseudovariety is 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 ji pseudovarieties do not join generate the lattice PV.
This prompts the following tantalizing question.
Question 1.1**.**
What do the ji elements in 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 ji decidable, that is, is the above function computable?
If ji is not decidable, then a systematic study of ji semigroups seems doomed in general.
But even if ji is decidable, then it is probably hopeless, in practice, to find all ji semigroups.
In any case, an important step is to find methods to produce new ji semigroups and methods to identify and eliminate finite semigroups that are not 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 ji is the âfive for one phenomenonâ related to the exclusion classExcl(S) of S, the class of all finite semigroups T for which Sâ/â¨â¨TâŠâŠ.
Indeed, a finite semigroup S is ji if and only if Excl(S) is a pseudovariety [23, Theorem 7.1.2].
In this case, Excl(S) is mi and so is defined by a single pseudoidentity, and since Excl(S) is also smi, it has Excl(S)â¨â¨â¨SâŠâŠ as a unique cover.
Further, â¨â¨SâŠâŠâŠExcl(S) is the unique maximal subpseudovariety of â¨â¨SâŠâŠ, and so Excl(S) determines â¨â¨SâŠâŠ; see [23, Section 7.1].
For example, the Brandt semigroup B2â is ji, the exclusion class Excl(B2â) coincides with the pseudovariety
[TABLE]
of finite semigroups whose J-classes are subsemigroups [23, Example 7.3.4], and â¨â¨B2ââŠâŠâŠDS=â¨â¨B0ââŠâŠ is the unique maximal subpseudovariety of â¨â¨B2ââŠâŠ, where B0â is a subsemigroup of B2â 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 ji semigroups.
One approachâand a very important problem in its own rightâis to find new operators on PV that preserve the property of being ji.
The following are some known examples.
Example 1.3**.**
For any semigroup S, the opposite semigroup Sop of S is obtained by reversing the multiplication on S.
Then the dual operator
For any semigroup S, let SI denote the monoid obtained by adjoining an external identity element I to S, and define
[TABLE]
Then the operator VâŚVâ={SââŁSâV} on PV preserves the property of being 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 Sop with S.
The situation for the operator VâŚVâ, however, can be different because it is possible that no new ji pseudovariety is produced.
Indeed, if a pseudovariety V is generated by some monoid, then Vâ=V cannot be a new example of ji pseudovariety.
But if V=â¨â¨SâŠâŠ is a ji pseudovariety that is not generated by any monoid, then Vâ=â¨â¨SIâŠâŠ is a ji pseudovariety properly containing V.
Note that the operator VâŚVI={SIâŁSâV} does not preserve the property of being ji.
For example, the cyclic group Zpâ of any prime order p generates a ji pseudovariety, but the pseudovariety â¨â¨ZpââŠâŠI=â¨â¨ZpIââŠâŠ is not ji because â¨â¨ZpIââŠâŠ=â¨â¨ZpââŠâŠâ¨â¨â¨Sl2ââŠâŠ, where Sl2â is the semilattice of order two.
On the other hand, it is possible for VI to be ji even though V is not ji.
For example, if S=Sl2âĂR2â, where R2â is the right zero semigroup of order two, then the pseudovariety â¨â¨SâŠâŠ=â¨â¨Sl2ââŠâŠâ¨â¨â¨R2ââŠâŠ is not ji but â¨â¨SâŠâŠI=â¨â¨Sl2IâĂR2IââŠâŠ=â¨â¨R2IââŠâŠ is ji [23, Example 7.3.1].
Remark 1.5**.**
It is clear that the operator VâŚVop also preserves the property of being sji, but the operator VâŚVâ does not preserve this property.
For instance, the pseudovariety â¨â¨B0ââŠâŠ is sji while â¨â¨B0ââŠâŠâ=â¨â¨B0IââŠâŠ is not sji; see Proposition 3.1.
Given a finite semigroup S, consider the right regular representation (Sâ,S) of S acting on Sâ by right multiplication.
Then Sbar is defined by adding all constant maps on Sâ 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 Sbar and hence we sometimes (abusively) denote this latter semigroup by Sbar as well.
Some small examples of Sbar can be found in Section 3.
It turns out that the operator VâŚVbar=â¨â¨SbarâŁSâVâŠâŠ on PV preserves the property of being ji.
This result, the details of which are given in Subsection 4.3, is important: for any finite nontrivial ji semigroup S, the pseudovarieties
[TABLE]
where Xâ=((Xop)bar)op, constitute an infinite increasing chain of ji pseudovarieties (Corollary 4.11) whose complete union is an fji pseudovariety that is not compact [23, Chapter 7].
Unsurprisingly, irregularities do show up when the operator VâŚVbar is applied.
For instance, it is sometimes possible for â¨â¨SbarâŠâŠ=â¨â¨SâŠâŠ, so that no new ji pseudovariety is obtained.
Further, it is possible for â¨â¨SbarâŠâŠ to be ji even though â¨â¨SâŠâŠ is not 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 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 SâŚSop, SâŚSâ, and SâŚSbar, and their iterations such as (((Sop)bar)op and (((((Sbar)op)â)bar)op)bar.
If we have a list of ji semigroups, applying the three operators and their iterations give ji semigroups that may or may not generate new ji pseudovarieties.
A ji pseudovariety V is primitive if Vî =â¨â¨SââŠâŠ and Vî =â¨â¨SbarâŠâŠ for any finite semigroup S that generates a ji proper subpseudovariety of V.
Now we are only interested in knowing the primitive ji pseudovarieties up to isomorphism and anti-isomorphism of members since the others can be obtained by applying the operators.
Therefore when describing 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 ji pseudovariety is â3barâ; all the other semigroups were previously known to be ji.
Note that â3barâ is ji but â3â 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 âi=1kââ¨â¨SiââŠâŠ, and advanced algebraic theory of finite semigroups [23].
The following are all other 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 ji pseudovariety of semigroups if and only if it generates a ji pseudovariety of groups; see [23, Chapter 7].
A pseudovariety V of groups is called saturated if whenever Ď:GâH is a homomorphism of finite groups with HâV, there exists a subgroup Kâ¤G such that KâV and KĎ=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 V is a saturated pseudovariety of groups, then a group GâV generates a ji pseudovariety in the lattice of all semigroup pseudovarieties if and only if it generates a ji member of the lattice of subpseudovarieties of V.
In particular, a p-group G is ji if and only if whenever G divides a direct product AĂB of p-groups, then G divides either An or Bn for some nâĽ1.
Abelian groups
The following statements on any directly indecomposable finite abelian group A are equivalent: A is ji, A is Ă-prime, and Aâ Zpnâ for some prime p and nâĽ1.
This result follows from the Fundamental Theorem of Finite Abelian Groups and that Zpnâlifts in the sense that whenever Zpnâ is a homomorphic image of some semigroup S, then Zpn+râ embeds into S for some râĽ0.
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 Nâ Hn for some simple group H and nâĽ1.
A finite group is monolithic if and only if it is subdirectly indecomposable;
recall that a semigroup S is a subdirect product of S1â and S2â, written SâŞS1âĂS2â, if S is a subsemigroup of S1âĂS2â mapping onto both S1â and S2â via the projections Ďiâ.
A semigroup S is subdirectly indecomposable (sdi) if SâŞS1âĂS2â implies that at least one of the projections Ďiâ:Sâ Siâ is an isomorphism.
Therefore when locating ji groups from among finite groups, it suffices to concentrate on those that are monolithic.
Groups with non-abelian monolith
KovĂĄcs and Newman proved that any monolithic group with non-abelian monolith is KN [23, Section 7.4] and so also Ă-prime and ji.
Therefore, all simple non-abelian groups are ji.
Groups with abelian monolith
An abelian monolith N of a finite group Gsplits if there exists a subgroup K of G so that NâŠK={1} and NK=G.
A finite subdirectly indecomposable group with an abelian monolith that splits is ji; this result is due to G.âM. Bergman and its proof is given in Subsection 4.6.
Therefore, the symmetric group Sym3â over three symbols is ji.
Groups of small order
The ji pseudovarieties generated by a group of order seven or less are â¨â¨Z2ââŠâŠ, â¨â¨Z3ââŠâŠ, â¨â¨Z4ââŠâŠ, â¨â¨Z5ââŠâŠ, â¨â¨Sym3ââŠâŠ, and â¨â¨Z7ââŠâŠ.
Regarding groups of order eight that generate other ji pseudovarieties, besides Z8â, there are two nontrivial cases: the dihedral group D4â of the square and the quaternion group Q8â.
Let Gâ{D4â,Q8â}.
Then forming GĂG and dividing out the two centers identified, (GĂG)/{(1,1),(a,a)} gives isomorphic groups, denoted by GâG.
Since Gâ¤GâG, it follows that â¨â¨D4ââŠâŠ=â¨â¨Q8ââŠâŠ.
Therefore, the groups D4â and Q8â are not Ă-prime and so also not KN.
However, the pseudovariety â¨â¨D4ââŠâŠ=â¨â¨Q8ââŠâŠ is ji; see Subsection 4.7.
This result is due independently to Kearnes [5] and the anonymous reviewer.
Other ji pseudovarieties generated by a group of order up to 11 are â¨â¨Z9ââŠâŠ, â¨â¨D5ââŠâŠ, and â¨â¨Z11ââŠâŠ.
1.2. J-trivial semigroups
Presently, the only ji pseudovarieties of J-trivial semigroups known in the literature are generated by the following:
The pseudoidentity defining the pseudovariety Excl(Nnâ) is given in Subsection 5.4, while the pseudovarieties Excl(Hnâ) and Excl(Knâ) are defined by the pseudoidentities
The pseudovariety Com of finite commutative semigroups can be decomposed as
[TABLE]
where G is the pseudovariety of finite groups and A is the pseudovariety of finite aperiodic semigroups [2, Figure 9.1].
Therefore any ji pseudovariety of commutative semigroups is contained in either ComâŠG or ComâŠA.
As noted in Subsection 1.1, the ji subpseudovarieties of ComâŠG are each generated by a cyclic group Zpnâ of prime power order.
As for ComâŠA, each of its finite semigroups satisfies the identity xn+1âxn for all sufficiently large nâĽ1 and so belongs to â¨â¨NnIââŠâŠ; see Proposition 5.10(i).
A complete description of ji subpseudovarieties of Com is thus dependent on the answer to the following question.
Question 1.6**.**
For each nâĽ1, what are the ji subpseudovarieties of â¨â¨NnIââŠâŠ?
Presently, the only known examples of ji subpseudovarieties of â¨â¨NnIââŠâŠ are â¨â¨NkââŠâŠ and â¨â¨NkIââŠâŠ, where 1â¤kâ¤n.
1.4. Bands
The pseudovariety B of finite bands is fji (Corollary 4.12).
Each proper subpseudovariety of B is compact and a complete description of the lattice of subpseudovarieties of B is well known; see, for example, Almeida [2, Section 5.5].
The atoms of this lattice are Sl=â¨â¨Sl2ââŠâŠ, LZ=â¨â¨L2ââŠâŠ, and RZ=â¨â¨R2ââŠâŠ; see Subsection 3.3.
Let LNB=Slâ¨LZ.
For any pseudovariety V, define the Malâcev 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 sji pseudovarieties of bands are as follows:
â
LZ, RZ, and Sl;
2. â
(ιβâ)nSl and βâ(ιβâ)nSl, nâĽ1, and their duals;
3. â
(βâÎą)n+1LNB and Îą(βâÎą)nLNB, nâĽ0, and their duals.
However, we observe that ÎąLNB=ÎąLZ.
Since SâŚSbar preserves join irreducibility, it follows that the pseudovariety generated by a finite band is ji if and only if it is sji; see Theorem 4.14.
As observed after Question 1.1, it is decidable if a finite semigroup generates a sji pseudovariety.
Therefore, Question 1.2 is affirmatively answered for bands.
1.5. KovĂĄcsâNewman semigroups
All 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 sdi homomorphic images, we can restrict our search for new ji semigroups to 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 ji pseudovarieties, we clearly need only to find finite semigroups S such that â¨â¨SâŠâŠ is ji and there exist no semigroups T with âŁTâŁ<âŁS⣠and â¨â¨TâŠâŠ=â¨â¨SâŠâŠ.
Such a semigroup S is called a minimal order generator for the compact pseudovariety â¨â¨SâŠâŠ.
Now the minimal order generators of ji pseudovarieties, in fact of sji pseudovarieties, must be sdi.
To see this, suppose that S is any finite semigroup that is not sdi.
Then SâŞS1âĂS2âĂâŻĂSkâ for some homomorphic images Sjâ of S such that âŁSjââŁ<âŁSâŁ.
But since â¨â¨SâŠâŠ=â¨â¨S1ââŠâŠâ¨â¨â¨S2ââŠâŠâ¨âŻâ¨â¨â¨SkââŠâŠ and â¨â¨SâŠâŠ is sji, it follows that â¨â¨SâŠâŠ=â¨â¨SjââŠâŠ for some j, whence S is not a minimal order generator
If a finite semigroup S is Ă-prime (e.g. KN), then S is a minimal order generator and any minimal order generator for â¨â¨SâŠâŠ is isomorphic to S.
The proof is clear.
However, minimal order generators for the same ji pseudovariety need not be isomorphic; for example, â¨â¨Q8ââŠâŠ=â¨â¨D4ââŠâŠ is ji and Q8ââD4â.
It should be pointed out that a finite semigroup S being sdi does not imply that the pseudovariety â¨â¨SâŠâŠ is ji or even sji.
For example, the Rees matrix semigroup
[TABLE]
is sdi, but â¨â¨SâŠâŠ=â¨â¨B2ââŠâŠâ¨â¨â¨Z2ââŠâŠ is not sji; see [23, Section 4.7].
1.7. Organization
The article is organized as follows.
In Section 2, the operator VâŚVbar 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 ji pseudovarieties are established.
In Section 5, some explicit pseudovarieties are shown to be 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-ji pseudovariety are established.
Results in Sections 4â6 are then employed in Section 7 to prove that among all pseudovarieties generated by a semigroup of order up to five, only 30 are 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 (X,S)â=(X,SâŞX) where X is the set of constant maps on X.
The constant map to a fixed element xâX is denoted by 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 ⺠of transformation semigroups.
Let (X,S) and (Y,T) be any transformation semigroups.*
Then*
(i)
(X,S)âş(Y,T)* implies that (X,S)ââş(Y,T)â;*
2. (ii)
(X,S)âş(Y,T)* implies that SâşT;*
3. (iii)
(X,S)Ă(Y,T)ââş(X,S)âĂ(Y,T)â.**
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 (Sâ,S)âş(X,S)âŁXâŁ*.**
Following [23, Chapter 4], write (Sâ,S)â=(Sâ,Sbar) and call Sbar the augmentation of S.
Note that if (X,S)âş(Sâ,S), then (X,S)âş(Sâ,S)âş(X,S)âŁX⣠by Lemma 2.2 and hence
[TABLE]
Thus if Sâ˛=SâŞX, then Sâ˛âşSbarâş(Sâ˛)âŁXâŁ, yielding the following result.
Corollary 2.3**.**
If (X,S) is a transformation semigroup such that (X,S)âş(Sâ,S),* then â¨â¨SâŞXâŠâŠ=â¨â¨SbarâŠâŠ.
In particular*,* if S is any semigroup and J is any right ideal of S on which it acts faithfully*,* then â¨â¨SbarâŠâŠ=â¨â¨SâŞJâŠâŠ*.**
The following are some elementary properties enjoyed by augmentation.
Proposition 2.4**.**
Let S and T be any finite semigroups.*
Then*
(i)
SâşT* implies that SbarâşTbar;*
2. (ii)
(SĂT)barâşSbarĂTbar.**
Proof.
(i) Suppose that SâşT, so that (Sâ,S)âş(Tâ,T) by Eilenberg [4, Proposition I.5.8].
Then by Lemma 2.1(i),
(ii) First note that ((SĂT)â,SĂT)âş(SâĂTâ,SĂT).
Then
[TABLE]
Therefore, (SĂT)barâşSbarĂTbar by Lemma 2.1(ii).
â
In the following, augmentation is viewed as a continuous operator on the lattice PV of pseudovarieties.
An operator is continuous if it preserves order and directed joins [23].
For any pseudovariety V, define
[TABLE]
Recall that RZ=â¨â¨R2ââŠâŠ is the pseudovariety of right zero semigroups.
Proposition 2.5**.**
The operator on PV defined by VâŚVbar is continuous,* non-decreasing*,* and idempotent*.*
Further*,**
(i)
â¨â¨SâŠâŠbar=â¨â¨SbarâŠâŠ* for any finite semigroup S;*
2. (ii)
RZâVbar* for any nontrivial pseudovariety V.*
Consequently,* if â¨â¨SâŠâŠ=â¨â¨TâŠâŠ, then â¨â¨SbarâŠâŠ=â¨â¨TbarâŠâŠ*.**
Proof.
Clearly augmentation is order preserving.
Let {VδââŁÎ´âD} be any directed set of pseudovarieties, so that the complete join V=âδâDâVδâ is a union.
The inclusion VδbarââVbar clearly holds for all δâD, so that âδâDâVδbarââVbar.
Conversely, if SâVbar, say SâşT1barâĂT2barâĂâŻĂTkbarâ for some T1â,T2â,âŚ,TkââV, then due to directedness, there exists δâD with T1â,T2â,âŚ,TkââVδâ, whence SâVδbarâ.
Therefore, augmentation is continuous.
Since SâşSbar, it is obvious that augmentation is non-decreasing and the inclusion Vbarâ(Vbar)bar holds.
To establish the reverse inclusion, it suffices to prove that (Sbar)barâVbar for all SâV.
But Sbar acts faithfully on the right of its minimal ideal Sâ and it contains all the constant mappings.
Thus (Sâ,Sbar)=(Sâ,Sbar)â and (Sâ,Sbar)âş((Sbar)â,Sbar).
It follows from Corollary 2.3 that Sbar=SbarâŞSâ generates the same pseudovariety as (Sbar)bar.
This shows that (Sbar)barâVbar, so that augmentation is idempotent.
It remains to establish parts (i) and (ii).
(i) The inclusion â¨â¨SbarâŠâŠââ¨â¨SâŠâŠbar holds trivially.
To establish the reverse inclusion, suppose that Tââ¨â¨SâŠâŠbar, so that TâşUbar for some Uââ¨â¨SâŠâŠ.
Then UâşSn for some nâĽ0 and so TâşUbarâş(Sn)barâş(Sbar)n by Proposition 2.4(ii).
Therefore, Tââ¨â¨SbarâŠâŠ.
Consequently, â¨â¨SbarâŠâŠ=â¨â¨SâŠâŠbar.
(ii) If S is a nontrivial semigroup in V, then the right zero semigroup R2â is a subsemigroup of Sbar, whence RZâV.
â
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 â¨â¨SâŠâŠbar=â¨â¨SâŠâŠ*.**
Proof.
By Proposition 2.5, it suffices to prove that â¨â¨SbarâŠâŠ=â¨â¨SâŠâŠ.
But since (J,S)â=(J,S), it follows that S=SâŞJ.
The desired conclusion then follows from Corollary 2.3.
â
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 e and f are exclusively reserved for idempotent elements.
3.1. Cyclic groups
The cyclic group of order nâĽ1 is
[TABLE]
The augmentation of Z2â={1,g} is the semigroup Z2barâ={1,g,1,gâ} given in Table 2.
The semigroup Z2barâ is isomorphic to the semigroup of transformations of the set {1,2}.
Information on identities satisfied by the semigroups Znâ and Z2barâ is given in Subsections 5.2 and 5.3, respectively.
3.2. Nilpotent semigroups
The monogenic nilpotent semigroup of order nâĽ1 is
[TABLE]
The augmentation of N2â={0,a} is the semigroup N2barâ={0,a,a,I} given in Table 2.
Information on identities satisfied by the semigroups Nnâ, NnIâ, N2barâ, and (N2barâ)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 Sl2â={0,1} and the left zero and right zero semigroups of order two:
[TABLE]
see Table 3.
Note that Sl2ââ N1Iâ and L2opââ R2â.
It is well known that Sl2â generates the pseudovariety Sl of semilattices, L2â generates the pseudovariety LZ of left zero semigroups, and R2â generates the pseudovariety RZ of right zero semigroups.
The augmentation of L2â is the semigroup L2barâ={e,f,e,f,I} given in Table 4.
Information on identities satisfied by the semigroups L2â, L2Iâ, and L2barâ 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 A2â and B2â contain subsemigroups isomorphic to
and its augmentation â3barâ={0,a,e,a,e} are given in Table 7.
Note that
[TABLE]
Information on identities satisfied by the semigroups A0â, A0Iâ, A2â, B2â, and â3barâ 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 â3â belongs to an infinite class of semigroups S with the property that â¨â¨SâŠâŠ is not ji but â¨â¨SâŠâŠbar is ji.
The semigroup B0â serves as a counterexample to the implications
[TABLE]
mentioned in the introduction.
Proposition 3.1**.**
(i)
The pseudovariety â¨â¨B0ââŠâŠ is **sji.**
2. (ii)
The pseudovariety â¨â¨B0ââŠâŠ is not **ji.**
3. (iii)
The pseudovariety â¨â¨B0IââŠâŠ is not **sji.**
Proof.
The pseudovariety â¨â¨B0ââŠâŠ is sji since it has a unique maximal proper subpseudovariety [8, Lemma 5(b)].
The pseudovariety â¨â¨B0IââŠâŠ is not sji since it has two maximal proper subpseudovarieties [8, Lemma 6(b)].
In particular, the pseudovariety â¨â¨B0IââŠâŠ is not ji, whence the pseudovariety â¨â¨B0ââŠâŠ is also not ji; see Lemma 5.2.
â
4. Some general results on join irreducibility
The pseudovariety defined by a class Σ of pseudoidentities is denoted by [[ÎŁ]], while the pseudovariety generated by a class K of finite semigroups is denoted by â¨â¨KâŠâŠ.
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={s1â,s2â,âŚ,snâ} be any finite semigroup and let V denote the variety generated by S.
Since the lattice of subvarieties of V is isomorphic to the lattice of subpseudovarieties of â¨â¨SâŠâŠ, it suffices to show that V contains positively and only finitely many maximal subvarieties.
Given any identity uâv such that Sî â¨uâv, there exists some substitutionÂ Ď into S such that uĎî =vĎ.
ThenÂ Ď induces a substitution ĎⲠinto the set Xnâ={x1â,x2â,âŚ,xnâ} given by xĎâ˛=xiâ if xĎ=siâ.
Therefore uĎâ˛âvĎⲠis an identity over Xnâ such that uâvâ˘uĎâ˛âvĎⲠand Sî â¨uĎâ˛âvĎâ˛.
It follows that every proper subvariety of V satisfies some identity over Xnâ.
Modulo the equational theory of the semigroup S, there can only be finitely many identities over Xnâ that are violated by S; these identities form a finite preordered set P under equational deduction â˘.
Each greatest element of (P,â˘) defines within V a maximal subvariety.
â
The exclusion classExcl(S) of a finite semigroup S is the class of all finite semigroups T for which Sâ/â¨â¨TâŠâŠ.
Recall that a finite semigroup S is ji if and only if Excl(S) is a pseudovariety [23, Theorem 7.1.2].
In this section, some results on the property of being ji are established.
There are seven subsections.
The main result of Subsection 4.1 demonstrates that many exclusion classes of ji semigroups in this paper are not definable by a certain type of pseudoidentities.
In Subsection 4.2, the notion of a âlargeâ pseudovariety is introduced.
It turns out that the exclusion class of a 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 VâŚVbar on PV preserves the property of being ji.
More specifically, if uâv is a pseudoidentity that defines the exclusion class Excl(S) of a ji semigroup S, then it is shown how a pseudoidentity that defines Excl(Sbar) can be obtained from uâv.
In Subsection 4.4, it is shown that alternately performing the operators VâŚVbar and VâŚVâ=â¨â¨((Sop)bar)opâŁSâVâŠâŠ on a nontrivial pseudovariety â¨â¨SâŠâŠ results in an infinite increasing chain of pseudovarieties; if the semigroup S is ji to begin with, then the pseudovarieties are all ji.
In Subsection 4.5, an infinite class {OkââŁkâĽ2} of finite semigroups is introduced and shown to satisfy the following property: for each kâĽ2, the pseudovariety â¨â¨OkââŠâŠ is not ji, while the pseudovariety â¨â¨OkââŠâŠbar is ji.
In Subsection 4.6, a sufficient condition, due to G.âM. Bergman, is presented under which a finite sdi group is ji.
In Subsection 4.7, the pseudovariety â¨â¨Q8ââŠâŠ=â¨â¨D4ââŠâŠ is shown to be 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 A is denoted by A+.
A pseudoidentity uâv is simple if u and v belong to the smallest subsemigroup F(A) of A+ containing A that is closed under product and unary implicit operations; the latter condition means that {w}+ââF(A) for all wâF(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 V is any proper pseudovariety of semigroups containing all semigroups with abelian maximal subgroups.*
Then V cannot be defined by simple pseudoidentities*.**
Proof.
Let A be a fixed countably infinite set and for any m,nâĽ1, let Bm,nâ be the variety of semigroups defined by the identity xmâxm+n.
Then the free semigroup B(1,m,n) on one-generator in Bm,nâ is finite and if xΡâ{x}+â, then there exists an integer nΡââ¤m+nâ1 such that xΡ=xnΡâ in B(1,m,n).
Thus each implicit operation in F(A) has a natural interpretation on any semigroup in Bm,nâ which agrees with its usual interpretation in finite semigroups (namely interpret wΡ as wnΡâ for every element w of a semigroup SâBm,nâ).
Suppose that V is defined by a set Σ of simple pseudoidentities.
Let W be the variety of universal algebras defined by Σ in the signatureÂ Ď consisting of multiplication and all unary implicit operations and let T be a finite semigroup.
Then there exist mâĽ6 and nâĽ1 such that T belongs to Bm,nâ.
As discussed above, Bm,nâ can be viewed as a variety in the signatureÂ Ď such that the unary implicit operations have their usual interpretations in all finite semigroups in Bm,nâ.
Now McCammond [17] has shown that for each integer kâĽ1, 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 W contains all finite semigroups with abelian maximal subgroups, it follows that B(k,m,n)âW, whence Bm,nââW.
Therefore, T belongs to W and so satisfies the pseudoidentities Σ.
Consequently, TâV and hence V is the pseudovariety of all finite semigroups.
â
In this paper, pseudoidentities involving idempotents from the minimal ideal of a free profinite semigroup are often used to define the exclusion pseudovarieties of 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 V and W are pseudovarieties of semigroups, then their Malâcev 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 Ď:SâT such that TâW and eĎâ1âV for all idempotents eâT.
A remarkable property of the Malâcev product is that
Let 1 denote the pseudovariety of trivial semigroups.
For any ji semigroup S, we say that Excl(S) is large if
[TABLE]
If Excl(S) is large and {VÎąââŁÎąâA} is a collection of pseudovarieties such that âÎąâAâVÎąâ=1, then it follows from (4.1) and the fact that Excl(S) is 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 ÎąâA.
In particular, either
[TABLE]
where A is the pseudovariety of finite aperiodic semigroups and 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:
â
I2=0 (the null case);
2. â
S acts faithfully on the right of the set of L-classes of I (the left letter mapping case);
3. â
S acts faithfully on the left of the set of L-classes of I (the right letter mapping case);
4. â
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 ji semigroup of semisimple type (left letter mapping, right letter mapping,* or group mapping*).*
Then 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 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 SâV for any pseudovariety 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).
â
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 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 VâŚVbar preserves the property of being ji.*
In particular*,* if a pseudovariety â¨â¨SâŠâŠ is ji, then the pseudovariety â¨â¨SbarâŠâŠ is also ji.
Further*,* if Excl(S)=[[uâv]] where u,vâA+, then*
[TABLE]
where zâ/A and e is an idempotent in the minimal ideal of (AâŞ{z})+â.**
Proof.
First note that since Sâ/Excl(S), there exists some homomorphism Ď:A+âS such that uĎî =vĎ.
Let 1 denote the identity element of Sâ, and extendÂ Ď to a homomorphism (AâŞ{z})+ââSbar by sending z to 1.
Then (ezu)ĎĎ=uĎâî =vĎâ=(ezv)ĎĎ and so Sbarâ/[[(ezu)Ďâ(ezv)Ď]].
To complete the proof, it suffices to assume that Tâ/[[(ezu)Ďâ(ezv)Ď]], and show that Sbarââ¨â¨TâŠâŠ.
Replacing T by a subsemigroup if necessary, generality is not lost by assuming the existence of a surjective homomorphism Ď:(AâŞ{z})+ââT such that (ezu)ĎĎî =(ezv)ĎĎ.
Now the semigroup T acts on the right of the set B of L-classes of its minimal ideal J; let (B,RLM(T)) denote the resulting faithful transformation semigroup.
Note that (B,RLM(T))=(B,RLM(T))â because if bâB, then any element of T in the 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 â¨â¨RLM(T)âŠâŠbar=â¨â¨RLM(T)âŠâŠ, since the constant mappings form the minimal ideal of RLM(T).
Since (ez)Ď is in the minimal ideal J of T, the elements ((ez)Ď)(uĎ) and ((ez)Ď)(vĎ) are R-equivalent.
However, they are not L-equivalent because otherwise they would be H-equivalent and hence have the same idempotent power, as J is completely simple.
Thus uĎ and vĎ have distinct images under the quotient map TâRLM(T).
Consequently, there is a homomorphism Ď:A+âRLM(T) such that uĎî =vĎ, that is, RLM(T)â/Excl(S).
Therefore, Sââ¨â¨RLM(T)âŠâŠ, whence Sbarââ¨â¨RLM(T)âŠâŠbar=â¨â¨RLM(T)âŠâŠââ¨â¨TâŠâŠ as required.
â
Corollary 4.5**.**
Let (X,S) be any transformation semigroup with (X,S)âş(Sâ,S).*
Suppose that the pseudovariety â¨â¨SâŠâŠ is ji.
Then the pseudovariety â¨â¨SâŞXâŠâŠ is also ji*.**
Proof.
This follows from Corollary 2.3 and Theorem 4.4.
â
Note that if S is ji, then Excl(Sbar) 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, Sâ is obtained by considering the left action of S on Sâ and adjoining constant maps.
For any pseudovariety V, define
[TABLE]
By symmetry, VâŚVâ is a continuous idempotent operator that preserves join irreducibility; see [23, Chapter 2].
Define the operators Îą,β:PVâPV by ÎąV=Vbar and βV=Vâ.
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 ι and β.
An important observation is that βιâ¨â¨SâŠâŠ is a compact pseudovariety containing Sl2â that is generated by (Sbar)â, which is left mapping with respect to its minimal ideal.
Thus it suffices to handle the case that Slââ¨â¨SâŠâŠ and S is left mapping with respect to its minimal ideal.
Proposition 4.6**.**
For any finite semigroup S,**
[TABLE]
Proof.
Clearly, Sbar/Sâ divides the semigroup S0 obtained by adjoining an external zero element [math] to S.
Since Sâ is a right zero semigroup and â¨â¨S0âŠâŠââ¨â¨SâŠâŠâ¨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.
â
Define the operators Îą,βâ:PVâPV 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 Sl2â as a subsemigroup, define the hierarchy
[TABLE]
Observe that VnââUnâ for all nâĽ0 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 V is any pseudovariety such that SlâV.*
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 SâV*.**
Proof.
If SâV, 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 Sbar 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 D is the pseudovariety of semigroups whose idempotents are right zeroes; this occurs if and only if the quotient of Sbar by the intersection LM of all its left mapping congruences belongs to 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 Sbar contains no elements of S.
Then Sbar/LMâ S0 because Sbar acts trivially on the left of its minimal ideal and acts as S does on the left of its other J-classes.
Since Sl2ââV by assumption, it follows that Sbar/LMâV if and only if SâV.
â
Corollary 4.8**.**
Suppose that S is any nontrivial band that is left mapping with respect to its minimal ideal and that V is any pseudovariety such that SlâV.*
Then (Sbar)ââβâÎąV if and only if SâV*.**
Proof.
Since Sbar 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 SâV.
â
The hierarchies (4.2) and (4.3) for the case S=Sl2â are now analyzed.
Recall that B denotes the pseudovariety of finite bands.
Lemma 4.9**.**
Consider the hierarchies (4.2) and (4.3) with S=Sl2â.*
Then*
(i)
VnââUnâ1â* for all nâĽ1;*
2. (ii)
the hierarchies (4.2) and (4.3) are strict;**
3. (iii)
ânâĽ0âUnâ=ânâĽ0âVnâ=B.**
Proof.
(i) This is established by induction on n.
The exclusion V1ââU0â holds since SbarâV1â while Sbarâ/Sl=U0â due to R2ââSbar.
Suppose that VnââUnâ1â for some nâĽ2.
Note that Vnâ is generated by a band of the form T=Râ and so T is left mapping with respect to its minimal ideal.
Since Tâ/Unâ1â, it follows from Corollary 4.8 that (Tbar)ââ/Unâ.
Therefore, (Tbar)ââVn+1ââUnâ, whence Vn+1ââUnâ.
(ii)
Since VnââUnâ1â by part (i) and Vnâ1ââUnâ1â, the hierarchy (4.2) is strict.
Similarly, VnââUnâ and VnââUnâ1â 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 B.
â
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 (Sbar)â if necessary, S can be assumed to contain Sl2â as a subsemigroup.
It then follows from Lemma 4.9 that ânâĽ0âVnâ contains the pseudovariety B.
But since B is not contained in any compact pseudovariety [25], the union ânâĽ0âVnâ is not compact.
Since each pseudovariety Vnâ is compact, the hierarchy is strict.
â
Corollary 4.11**.**
If â¨â¨SâŠâŠ is ji,* then the pseudovarieties*
[TABLE]
are ji;* these pseudovarieties are all distinct except possibly for â¨â¨SâŠâŠ=â¨â¨SbarâŠâŠ.
A dual result holds when â is first applied before bar*.**
Corollary 4.12**.**
The pseudovariety B is fji.**
Proof.
Since the pseudovariety Sl is ji, each step in the hierarchy (4.2) is ji with S=Sl2â.
As the union of a chain of ji pseudovarieties is fji [23, Lemma 6.1.13], it follows from Lemma 4.9 that B is fji.
â
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 sji band is ji.
Recall that LNB=Slâ¨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 L2barâ.**
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}.
â
By Proposition 4.13 and results from Pastijn [19], a description of proper sji pseudovarieties of bands can be given as follows.
Let S=L2barâ and T=R2ââ.
Then the proper nontrivial sji pseudovarieties of bands are LZ, RZ, and those pseudovarieties that can be obtained by applying an alternating word w(Îą,βâ) over {Îą,βâ} to the pseudovarieties generated by S, T, or Sl2â (where the last letter of w should be βâ when starting from â¨â¨SâŠâŠ and should be Îą when starting from â¨â¨TâŠâŠ).
Further, there are no sji pseudovarieties strictly in between any successive iterations of these operators.
Since ÎąVâ¤ÎąV and βVâ¤Î˛âV for any pseudovariety V containing Sl, and each successive iteration of ι and β starting from the pseudovariety generated by one of S, T, or Sl2â (where the rightmost operator applied must be β for S and ι for T) results in a new ji pseudovariety, it follows that if w(x,y) is any alternating word over {x,y}, then w(Îą,β)V=w(Îą,βâ)V whenever V is one of the pseudovarieties generated by S, T, or Sl2â.
Consequently, each sji proper pseudovariety of bands is, in fact, ji by Corollary 4.11.
The following result is thus established.
Theorem 4.14**.**
Any sji band is ji,* that is*,* a proper pseudovariety of bands is sji if and only if it is ji*.**
In particular, since sji is a decidable property, ji is also decidable for finite bands.
The answer to Question 1.2 is thus affirmative for bands.
4.5. From non-ji pseudovarieties to ji pseudovarieties
For each kâĽ2, define the semigroup
[TABLE]
The main goal of this subsection is to show that the pseudovariety â¨â¨OkââŠâŠ is not ji whereas the pseudovariety â¨â¨OkââŠâŠbar is ji.
It is also shown that the pseudovarieties â¨â¨O2ââŠâŠbar,â¨â¨O3ââŠâŠbar,â¨â¨O4ââŠâŠbar,⌠are all distinct.
It is easily seen that the semigroups O2â and â3â are isomorphic by referring to their presentations.
Since the semigroup â3barâ is of order five (Subsection 3.4), the ji pseudovariety â¨â¨â3barââŠâŠ=â¨â¨O2ââŠâŠbar is required later in the paper (Theorem 5.29).
Lemma 4.15**.**
For each kâĽ2,* the semigroup Okâ consists precisely of the following 2kâ1 distinct elements*:**
[TABLE]
Proof.
It is routinely checked that (4.4) are all the elements of Okâ.
Therefore, it remains to verify that the elements in (4.4) are distinct.
Recall that the right zero semigroup of order two is R2â={e,f} and that the monogenic nilpotent semigroup of order k is
[TABLE]
Consider the subsemigroup T=(NkIâĂR2â)â{(I,e)} of NkIâĂR2â and the ideal J={(0,e),(0,f),(akâ1,f)} of T.
Define Ď:{x,e}+âT/J by xĎ=(a,e) and eĎ=(I,f).
Then
[TABLE]
It follows thatÂ Ď induces a homomorphism OkââŚT/J that separates the elements in (4.4).
â
Proposition 4.16**.**
The pseudovariety â¨â¨OkââŠâŠ is not ji.**
Proof.
Since OkââşT/JâşNkIâĂR2â by the proof of Lemma 4.15 (where we retain the notation of that proof), the inclusion â¨â¨OkââŠâŠââ¨â¨NkIââŠâŠâ¨RZ holds.
But â¨â¨NkIââŠâŠ consists of commutative semigroups while RZ consists of bands.
Therefore, â¨â¨OkââŠâŠââ¨â¨NkIââŠâŠ and â¨â¨OkââŠâŠâRZ.
â
It remains to prove that the pseudovariety â¨â¨OkââŠâŠbar is ji.
Lemma 4.17**.**
Suppose that U is any semigroup generated by two elements f and y such that f2=f,* fy=y, and ykâ1â/{ynâŁnâĽk}.
Then*
(i)
y,y2,âŚ,ykâ1* are distinct and not in {ynâŁnâĽk};*
2. (ii)
f,yf,y2f,âŚ,ykâ2f* are distinct and not in {ymfâŁmâĽkâ1};*
3. (iii)
yi=yjf* implies that either i=j or i,jâĽkâ1.*
Proof.
(i) This follows from the structure of monogenic semigroups.
(ii) Suppose that yif=yjf for some i,jâĽ0.
Then yi+1=yify=yjfy=yj+1.
Therefore, by part (i), either i=j or i,jâĽkâ1.
(iii) Suppose that yi=yjf for some iâĽ1 and jâĽ0.
Then yi+1=yjfy=yj+1.
Therefore, by part (i), either i=j or i,jâĽkâ1.
â
Recall that the inclusion â¨â¨OkââŠâŠââ¨â¨NkIââŠâŠâ¨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 d2=d,* dz=z, and zkâ1â/{znâŁnâĽk}.
Then â¨â¨OkââŠâŠââ¨â¨TâŠâŠâ¨RZ*.**
Proof.
Consider the semigroup TĂR2â and its subsemigroup U=â¨y,f⊠generated by y=(z,e) and f=(d,f).
Then it is routinely checked that
(a)
f2=f, fy=y,
2. (b)
yn=(zn,e) for all nâĽ1,
3. (c)
ynf=(znd,f) for all nâĽ1.
It follows from (b) and the assumption zkâ1â/{znâŁnâĽk} that
(d)
ykâ1â/{ynâŁnâĽk}.
It is clear from (a) that U={yi,yjfâŁiâĽ1,jâĽ0}.
In fact, it follows from (a)â(d) and Lemma 4.17 that
(e)
the elements y,y2,y3,âŚ,yk,f,yf,y2f,âŚ,ykâ1f of U are distinct.
Now it is routinely checked that the set
[TABLE]
is an ideal of U.
By (e), the set UâJ consists of the elements
[TABLE]
Therefore, Okââ U/J by Lemma 4.15, whence â¨â¨OkââŠâŠââ¨â¨TâŠâŠâ¨RZ.
â
Theorem 4.19**.**
(i)
For each kâĽ2, the pseudovariety â¨â¨OkââŠâŠbar is ji and
[TABLE]
where e is an idempotent in the minimal ideal of {a,b,c}+â.
2. (ii)
The pseudovarieties â¨â¨O2ââŠâŠbar,â¨â¨O3ââŠâŠbar,â¨â¨O4ââŠâŠbar,⌠are all distinct.**
Proof.
(i) LetÂ Ď denote the substitution into Okbarâ given by aâŚe, bâŚx, and câŚ1.
Then (ec(aĎb)kâ1)ĎĎ=xkâ1 and (ec((aĎb)kâ1)Ď+1)ĎĎ=xk, and these are different elements of Okbarâ.
Therefore, the semigroup Okbarâ violates the pseudoidentity in (4.5).
It remains to assume that a semigroup T violates the pseudoidentity in (4.5), and then show that Okbarâââ¨â¨TâŠâŠ.
Replacing T by a subsemigroup if necessary, generality is not lost by assuming the existence of a surjective homomorphism Ď:{a,b,c}+ââT such that
[TABLE]
Put f=aĎĎ and y=(aĎb)Ď and note that f2=f and fy=y.
The semigroup T acts on the right of the set B of L-classes of its minimal ideal J; denote the corresponding faithful transformation semigroup by (B,RLM(T)).
Note that (B,RLM(T))=(B,RLM(T))â because if bâB, then any element of T in the L-class of b acts on B as a constant map to b by the structure of completely simple semigroups.
Consequently, â¨â¨RLM(T)âŠâŠbar=â¨â¨RLM(T)âŠâŠ by Corollary 2.6 since the constant mappings form the minimal ideal of RLM(T).
Since (ec)Ď is in the minimal ideal J of T, it follows that the elements ((ec)Ď)ykâ1 and ((ec)Ď)(ykâ1)Ď+1 are R-equivalent.
However, they are not L-equivalent because otherwise they would be H-equivalent and hence have the same idempotent power, as J is completely simple. Consequently, RLM(T) is nontrivial and so it follows from Proposition 2.5 that â¨â¨RLM(T)âŠâŠ=â¨â¨RLM(T)âŠâŠâ¨RZ.
Also, if z denotes the image of y under the quotient map TâRLM(T) and d denotes the image of f under this map, then d2=d, dz=z, and zkâ1 is not a group element (as zkâ1 and (zkâ1)Ď+1 act differently on the L-class of (ec)Ď).
Thus Lemma 4.18 implies that Okâââ¨â¨RLM(T)âŠâŠâ¨RZ=â¨â¨RLM(T)âŠâŠ.
Consequently, Okbarâââ¨â¨RLM(T)âŠâŠbar=â¨â¨RLM(T)âŠâŠââ¨â¨TâŠâŠ.
(ii) This holds because for each kâĽ2, the semigroup Okbarâ satisfies the identity xk+1âxk but violates the identity xkâxkâ1.
â
4.6. A sufficient condition for the join irreducibility of groups
Recall that a normal subgroup N of a group Gsplits if there exists a subgroup K of G so that NâŠK={1} and NK=G.
Suppose that G is any finite sdi group with an abelian monolith N that splits.*
Then G is ji*.**
Proof.
By assumption, there exists a subgroup K of G with NâŠK={1} and NK=G.
Seeking a contradiction, suppose there exist finite groups G1â and G2â and some surjective homomorphism f from a subgroup H of G1âĂG2â onto G such that Gâ/â¨â¨G1ââŠâŠ and Gâ/â¨â¨G2ââŠâŠ.
Clearly we can assume that HĎjâ=Gjâ for the projection maps Ďjâ:G1âĂG2ââ Gjâ, that is, H is a subdirect product of G1â and G2â.
Further, we may assume that G1â, G2â, and H are chosen so that the order of H is minimal.
Let H2â={h2ââG2ââŁ(1,h2â)âH}â ker(Ď1â)âŠH.
If H2â is trivial, then Ď1â is injective on H, so that Hâ G1â, whence the contradiction GâşG1â is obtained.
Hence H2â is nontrivial.
Observe that
(â )
if L is a subgroup of H2â such that {1}ĂLâ´H, then Lâ´G2â;
in particular, H2ââ´G2â.
Indeed, if ââL and g2ââG2â, then choosing any g1ââG1â with (g1â,g2â)âH, we have
[TABLE]
by normality of {1}ĂL in H, whence g2ââg2â1ââL.
Suppose that ker(f) has nontrivial intersection with the subgroup {1}ĂH2â of H, say ker(f)âŠ({1}âŠH2â)={1}ĂL for some LâG2â.
Then L is normal in H2â and so also normal in G2â by (â ).
By dividing G2â by this intersection, we could contradictorily decrease the order of H.
Therefore, ker(f) intersects {1}ĂH2â trivially.
Similarly, defining H1â={h1ââG1ââŁ(h1â,1)âH}, we have {1}î =H1ââ´G1â and ker(f) intersects H1âĂ{1} trivially.
Then H1âĂ{1}, {1}ĂH2â, 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 kâK and nâN, then since N is abelian, we have that k centralizes N and hence k=1 by the previous observation.
From now on, identify H1â with H1âĂ{1} and H2â with {1}ĂH2â.
Then H1â and H2â are normal in H and commute elementwise.
We claim now that H1âf=N=H2âf.
Indeed, since f is injective on each of these subgroups and these subgroups are normal in H, we conclude that N is contained in H1âfâŠH2âf.
Since H1â and H2â commute elementwise, both H1âf and H2âf are contained in the centralizer of N, which is N.
We conclude that H1âf=N=H2âf and f restricts to an isomorphism of H1â and H2â with N.
Let Hâ=Kfâ1.
Then since NâŠK={1}, it follows that HââŠH1â is a subgroup of ker(f).
But ker(f)âŠH1â is trivial, so that HââŠH1â={1}.
Similarly, HââŠH2â={1}.
Note that HâH1â and HâH2â are subgroups of H because H1â and H2â are normal.
Also (HâH1â)f=KN=G=(HâH2â)f and so by minimality of H, we have HâH1â=H=HâH2â.
In particular, G2ââ HĎ2â=(HâH1â)Ď2â=HâĎ2â and so, since HââŠH1â={1}, we deduce that G2ââ Hâ.
Similarly, G1ââ Hâ.
Therefore, GâşG1âĂG1â and so Gââ¨â¨G1ââŠâŠ, a contradiction.
â
4.7. Join irreducibility of the pseudovariety â¨â¨D4ââŠâŠ=â¨â¨Q8ââŠâŠ
Proposition 4.21**.**
The pseudovariety â¨â¨D4ââŠâŠ=â¨â¨Q8ââŠâŠ is ji.*
Further*,* the pseudovariety Excl(Q8â) 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 â¨â¨Q8ââŠâŠ.
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 â¨â¨Q8ââŠâŠ via a general description of identity bases for finite nilpotent groups of class 2.
â
Since the group Q8â is 2-generated as a semigroup, the pseudovariety Excl(Q8â) can be defined by a pseudoidentity over two variables [23, Proposition 7.1.9].
We proceed to describe such a pseudoidentity.
Let F={x,y}+â be the free profinite semigroup over {x,y} and let Ρ be an idempotent in the minimal ideal of F.
Then ΡFΡ is a profinite group and maps onto the free profinite group on two generators under the natural projection.
The elements xâ˛=ΡxΡ and yâ˛=ΡyΡ map onto the free generators and thus freely topologically generate a free profinite subgroup of F, which is a retract.
This observation was first made by Almeida and Volkov [3].
If u(x,y) and v(x,y) are elements of the free profinite group FGâ({x,y}) over {x,y}, then u(xâ˛,yâ˛),v(xâ˛,yâ˛)âΡFΡ and it is easy to see that the pseudoidentity u(xâ˛,yâ˛)âv(xâ˛,yâ˛) defines the pseudovariety of all semigroups whose maximal subgroups belong to the pseudovariety defined by u(x,y)âv(x,y); see [3] for details.
Thus, it suffices to find a two-variable group pseudoidentity u(x,y)âv(x,y) defining the pseudovariety of groups with abelian 2-Sylow subgroups (or, equivalently, 2-subgroups).
Let FG2ââ({x,y}) be the free pro-2 group over {x,y}.
We have a natural continuous surjection Ď:FGâ({x,y})âFG2ââ({x,y}).
The pseudovariety of 2-groups is saturated and so by Ribes and Zalesskii [24, Proposition 7.6.7], the group FG2ââ({x,y}) is a projective profinite group.
Therefore, there is a continuous splitting of Ď, that is, we can find w1â(x,y),w2â(x,y)âFGâ({x,y}) which freely topologically generate a free pro-2 subgroup with Ď(w1â(x,y))=Ď(x) and Ď(w2â(x,y))=Ď(y); in other words, w1â(x,y) and w2â(x,y) topologically generate a free pro-2 retract of FGâ({x,y}).
Then the pseudovariety of groups with abelian 2-subgroups is defined by the pseudoidentity w1â(x,y)w2â(x,y)âw2â(x,y)w1â(x,y).
Thus the pseudovariety Excl(Q8â) is defined by w1â(xâ˛,yâ˛)w2â(xâ˛,yâ˛)âw2â(xâ˛,yâ˛)w1â(xâ˛,yâ˛), where xⲠand yⲠare as given in the previous paragraph.
5. Join irreducible pseudovarieties
The present section contains 15Â subsections.
Some background results are recorded in the first subsection, while the latter 14Â subsections are devoted to the pseudovarieties generated by the following 14Â semigroups:
[TABLE]
Each subsection that is concerned with a semigroup S from (5.1) begins with a theorem that establishes the ji property of â¨â¨SâŠâŠ by exhibiting a pseudoidentity that defines the pseudovariety Excl(S).
A basis ΣSâ of identities for the pseudovariety â¨â¨SâŠâŠ and an identity ξSâ that defines its maximal subpseudovariety â¨â¨SâŠâŠâŠExcl(S) are then given in a proposition.
The pair (ÎŁSâ,ÎľSâ) can be used to easily test if a finite semigroup generates the ji pseudovariety â¨â¨SâŠâŠ.
Indeed, for any finite semigroup T,
[TABLE]
The pairs (ÎŁSâ,ÎľSâ), where S ranges over the semigroups from (5.1), will be used in Section 7 to locate all 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 A are denoted by A+ and Aâ, respectively.
Elements of A are called variables while elements of Aâ are called words.
For any word wâA+,
â
the number of times a variable x occurs in w is denoted by occ(x,w);
2. â
the content of w, denoted by con(w), is the set of variables occurring in w, that is, con(w)={xâAâŁocc(x,w)âĽ1};
3. â
the initial part of w, denoted by ini(w), is the word obtained by retaining the first occurrence of each variable in w;
4. â
the final part of w, denoted by fin(w), is the word obtained by retaining the last occurrence of each variable in w.
Lemma 5.1**.**
Let uâv be any semigroup identity.*
Then*
(i)
Znââ¨uâv* if and only if occ(x,u)âĄocc(x,v)(modn) for all xâA;*
2. (ii)
NnIââ¨uâv* if and only if for all xâA, either occ(x,u)=occ(x,v) or occ(x,u),occ(x,v)âĽn;*
3. (iii)
L2Iââ¨uâv* if and only if ini(u)=ini(v);*
4. (iv)
R2Iââ¨uâv* if and only if fin(u)=fin(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)].
â
The local of a pseudovariety V, denoted by LV, is the pseudovariety of all finite semigroups S such that eSeâV for any idempotent eâS.
Let S be any finite semigroup that is not a monoid.*
If the pseudovariety â¨â¨SâŠâŠ is ji, then the pseudovariety â¨â¨SIâŠâŠ is also ji and Excl(SI)=LExcl(S)*.**
5.2. The pseudovariety â¨â¨ZnââŠâŠ
For any set Ď={p1â,p2â,p3â,âŚ} of primes, let ĎⲠdenote the set of primes complementary to Ď.
If p is a prime, then simply write pⲠinstead of {p}â˛.
For example, 2Ⲡdenotes the set of odd primes.
Retaining the above notation, recall that in {x}+â, the sequence x(p1âp2ââŻpnâ)n! converges to an element (independent of the enumeration of Ď), denoted by xĎĎ, with the following property: if s is an element of a finite semigroup S, then sĎĎ is a generator of the Ďâ˛-primary component of the finite cyclic group generated by sĎ+1.
Here we recall that for a finite abelian group A, the Ďâ˛-primary component of A is the direct product of the p-Sylow subgroups of A with pâ/Ď.
In this case, s(Ďâ˛)Ď will then be a generator of the Ď-primary component of â¨sĎ+1âŠ; see [23, Proposition 7.1.16].
Theorem 5.3**.**
For any prime p with nâĽ1,* the pseudovariety â¨â¨ZpnââŠâŠ is ji and*
[TABLE]
Proof.
The cyclic group Zpnâ=â¨gâŁgpn=1⊠violates the pseudoidentity in (5.2) because (g(pâ˛)Ď)pnâ1=gpnâ1î =1=gĎ.
Therefore, if Zpnâ belongs to some pseudovariety V, then V violates the pseudoidentity in (5.2).
Conversely, suppose that the pseudoidentity in (5.2) is violated by V, say it is violated by SâV.
Generality is not lost by assuming that S is generated by an element s such that (s(pâ˛)Ď)pnâ1î =sĎ.
Replacing s by sĎ+1, we may assume that S is, in fact, a cyclic group generated by s such that (s(pâ˛)Ď)pnâ1î =1.
But then the p-primary component of S is a cyclic group of order pm with mâĽn.
Therefore, Zpnâ divides S, whence ZpnââV.
â
Proposition 5.4**.**
Let nâĽ1.**
(i)
The identities satisfied by the group Znâ are axiomatized by
[TABLE]
2. (ii)
The maximal subpseudovarieties of â¨â¨ZnââŠâŠ are precisely â¨â¨ZdââŠâŠ, where d ranges over all maximal proper divisors of n.
Consequently,* for any prime p with kâĽ1, the subpseudovariety of â¨â¨ZpkââŠâŠ defined by*
[TABLE]
is the unique maximal subpseudovariety of â¨â¨ZpkââŠâŠ.
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].
â
5.3. The pseudovariety â¨â¨Z2barââŠâŠ
Theorem 5.5**.**
The pseudovariety â¨â¨Z2barââŠâŠ is ji and
[TABLE]
where e is an idempotent in the minimal ideal of {x,y}+â.
Alternately, Rhodes and Steinberg [23, Example 7.3.20] have shown that
[TABLE]
where e is an idempotent in the minimal ideal of {x,y}+â.
Proposition 5.6**.**
(i)
The identities satisfied by the semigroup Z2barâ are axiomatized by
[TABLE]
2. (ii)
The subpseudovariety of â¨â¨Z2barââŠâŠ defined by the identity
[TABLE]
is the unique maximal subpseudovariety of â¨â¨Z2barââŠâŠ.
Proof.
This follows from the dual of Tishchenko [27, Proposition 3.16], where the variety generated by (Z2barâ)op is denoted by W2â.
â
5.4. The pseudovariety â¨â¨NnââŠâŠ
Theorem 5.7**.**
For each nâĽ2,* the pseudovariety â¨â¨NnââŠâŠ is ji and*
[TABLE]
Proof.
The semigroup Nnâ=â¨aâŁan=0⊠violates the pseudoidentity in (5.3) because aĎ+nâ1=0î =anâ1.
Therefore, if Nnâ belongs to some pseudovariety V, then V violates the pseudoidentity in (5.3).
Conversely, suppose that the pseudoidentity in (5.3) is violated by the pseudovariety V, say it is violated by SâV.
Then there exists some aâS such that aĎ+nâ1î =anâ1.
If there exist some iâ¤nâ1 and some j>i such that ai=aj, then anâ1=anâ1âiai=anâ1âiaj=anâ1ajâi, so that
[TABLE]
which is a contradiction.
Hence the sets {a},{a2},âŚ,{anâ1},{aiâŁiâĽn} are pairwise disjoint.
It follows that J={aiâŁiâĽn} is an ideal of the monogenic subsemigroup â¨a⊠of S such that â¨aâŠ/Jâ Nnâ.
Consequently, Nnâââ¨â¨SâŠâŠâV.
â
For each nonnegative real number x, let âxâ denote the greatest integer bounded from above by x.
Proposition 5.8**.**
Let nâĽ2.**
(i)
The identities satisfied by the semigroup Nnâ are axiomatized by
[TABLE]
where k=0,1,âŚ,â(nâ1)/3â.
2. (ii)
The subpseudovariety of â¨â¨NnââŠâŠ defined by the identity
[TABLE]
is the unique maximal subpseudovariety of â¨â¨NnââŠâŠ.
Proof.
(i) This follows from Shevrin and Volkov [26, Proposition 21.3].
(ii) This follows from Theorem 5.7 and part (i).
â
5.5. The pseudovariety â¨â¨NnIââŠâŠ
Theorem 5.9**.**
For any nâĽ1,* the pseudovariety â¨â¨NnIââŠâŠ is ji and*
[TABLE]
Proof.
For n=1, the result follows from [23, Table 7.2] because N1Iââ Sl2â.
For nâĽ2, the result follows from Lemma 5.2 and Theorem 5.7.
â
Proposition 5.10**.**
Let nâĽ1.**
(i)
The identities satisfied by the semigroup NnIâ are axiomatized by
[TABLE]
2. (ii)
The subpseudovariety of â¨â¨NnIââŠâŠ defined by the identity
[TABLE]
is the unique maximal subpseudovariety of â¨â¨NnIââŠâŠ.
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).
â
5.6. The pseudovariety â¨â¨N2barââŠâŠ
Theorem 5.11**.**
The pseudovariety â¨â¨N2barââŠâŠ is ji and
[TABLE]
where e is an idempotent from the minimal ideal of {x,z}+â.
The identities satisfied by the semigroup B2â are axiomatized by
[TABLE]
2. (ii)
The subpseudovariety of â¨â¨B2ââŠâŠ defined by the identity
[TABLE]
is the unique maximal subpseudovariety of â¨â¨B2ââŠâŠ.
5.15. The pseudovariety â¨â¨â3barââŠâŠ
Theorem 5.29**.**
The pseudovariety â¨â¨â3barââŠâŠ is ji and
[TABLE]
where e is an idempotent in the minimal ideal of {x,y,z}+â.**
Proof.
This is a special case of Theorem 4.19 since â3barââ O2barâ.
â
The remainder of this subsection is devoted to establishing a basis for the identities satisfied by â3barâ.
It turns out that it is notationally simpler to consider the dual semigroup (â3barâ)op={a,b,c,d,e} given in Table 8.
Proposition 5.30**.**
(i)
The identities satisfied by the semigroup (â3barâ)op are axiomatized by
[TABLE]
2. (ii)
The subpseudovariety of â¨â¨(â3barâ)opâŠâŠ defined by the identity
[TABLE]
is the unique maximal subpseudovariety of â¨â¨(â3barâ)opâŠâŠ.
Remark 5.31**.**
It is routinely shown that the semigroup (â3barâ)op satisfies the identities (5.4) but violates the identity (5.5).
In this subsection, a word w is said to be in canonical form if either
where x0â,x1â,âŚ,xmâ are distinct variables with 0â¤kâ¤m.
Remark 5.32**.**
Note the extreme cases for the word w in (CF2):
(i)
if 0=k=m, then w=x02â;
2. (ii)
if 0=k<m, then w=x02âx1ââŻxmâ;
3. (iii)
if 0<k=m, then w=x0âx1âx2ââŻxmâx0â.
Lemma 5.33**.**
Given any word w,* the identities (5.4) can be used to convert w into some word wⲠin canonical form with ini(w)=ini(wâ˛)*.**
Proof.
Suppose that ini(w)=x0âx1ââŻxmâ.
Then w can be written as
[TABLE]
where wiââ{x0â,x1â,âŚ,xiâ}â for all i.
The identities (5.4) can be used to eliminate all occurrences of x1â,x2â,âŚ,xmâ from each wiâ, resulting in the word
[TABLE]
where e0â,e1â,âŚ,emââĽ0.
If e0â=e1â=âŻ=emâ=0, then the word wⲠis in canonical form (CF1) such that ini(w)=ini(wâ˛).
If kâĽ0 is the least index such that ekââĽ1, then e0â=e1â=âŻ=ekâ1â=0, so that
[TABLE]
The word wâ˛â˛ is in canonical form (CF2) with ini(w)=ini(wâ˛â˛).
â
As observed in Remark 5.31, the semigroup (â3barâ)op violates the identity (5.5).
Hence â¨â¨(â3barâ)opâŠâŠâŠ[[\eqrefid:el3max]] is a proper subpseudovariety of â¨â¨(â3barâ)opâŠâŠ.
It remains to show that each proper subpseudovariety V of â¨â¨(â3barâ)opâŠâŠ satisfies the identity (5.5).
Since Vî =â¨â¨(â3barâ)opâŠâŠ, there exists an identity uâv of V that is violated by (â3barâ)op.
Further, since the identities (5.4) are satisfied by (â3barâ)op and so also by V, it follows from Lemma 5.33 that the words u and v can be chosen to be in canonical form.
There are two cases.
Case 1
ini(u)î =ini(v).
Then by Theorem 5.17, the pseudovariety V satisfies the pseudoidentity that defines Excl(L2Iâ).
Since
[TABLE]
the pseudovariety V satisfies the identity Îą:h2xyâh2yx.
Since
[TABLE]
the pseudovariety V satisfies the identity (5.5).
Case 2
ini(u)=ini(v) and uî =v.
If the words u and v are both of the form (CF1), then they are contradictorily equal.
Hence either u or v is of the form (CF2).
By symmetry, there are two subcases.
2.1.
u and v are both of the form (CF2). Then
[TABLE]
where 0â¤j,kâ¤m.
Since jî =k, it suffices to assume by symmetry that 0â¤j<kâ¤m.
LetÂ Ď denote the substitution given by x0ââŚxy, xiââŚy for all iâ{1,2,âŚ,j}, and xiââŚz otherwise.
Then
[TABLE]
Therefore, the identity (5.5) is deducible from (5.4) and uâv.
The pseudovariety V thus satisfies the identity (5.5).
2. 2.2.
u is of the form (CF1) while v is of the form (CF2).
Then
[TABLE]
Since
[TABLE]
the pseudovariety V satisfies the identity uâ˛âvâ˛.
Now uⲠand vⲠare distinct words in canonical form (CF2) such that ini(uâ˛)=ini(vâ˛).
Thus the arguments in Subcase 2.1 can be repeated to show that V satisfies the identity (5.5).
â
As noted in Remark 5.31, the identities (5.4) are satisfied by the semigroup (â3barâ)op.
Conversely, suppose that uâv is any identity satisfied by (â3barâ)op.
By Lemma 5.33, the identities (5.4) can be used to convert u and v into words uⲠand vⲠin canonical form.
Since the subsemigroup {a,c,e} of (â3barâ)op and the semigroup L2Iâ are isomorphic, it follows from Lemma 5.1(iii) that ini(uâ˛)=ini(vâ˛).
Suppose that uâ˛î =vâ˛.
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 uâ˛âvâ˛.
Since the semigroup (â3barâ)op satisfies the identities (5.4) and uâ˛âvâ˛, it also satisfies (5.5); but this is impossible by Remark 5.31.
Therefore, uâ˛=vâ˛.
Since
This section contains nine subsections, each of which establishes one or more sufficient conditions for a finite semigroup to generate a non-ji pseudovariety.
Each of these sufficient conditions, given as a corollary of some general result, presents some finite set Σ of identities and some identities Îľ1â,Îľ2â,âŚ,Îľkâ with the property that for any finite semigroup S,
[TABLE]
In most cases, ÎŁ will be a basis of identities for some join V=âi=1kâViâ of compact pseudovarieties V1â,V2â,âŚ,Vkâ that satisfy the pseudoidentities Îľ1â,Îľ2â,âŚ,Îľkâ, respectively.
Sufficient conditions developed in this section will be used in Section 7 to locate all non-ji pseudovarieties generated by a semigroup of order up to five.
6.1. The pseudovariety â¨â¨Z3â,Z4â,Z2barâ,(Z2barâ)op,N3IââŠâŠ
holds.
But the two identities in (6.2) are satisfied by Z2barâĂ(Z2barâ)op and Z3âĂZ4âĂN3Iâ, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨Z2barâ,(Z2barâ)opâŠâŠ and â¨â¨SâŠâŠââ¨â¨Z3â,Z4â,N3IââŠâŠ.
â
Corollary 6.3**.**
Suppose that S is any finite semigroup that satisfies the identities (6.1) but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨KâŠâŠ that is not ji.**
holds.
Since the three identities in (6.3) are satisfied by Z3âĂZ4âĂN3Iâ, (Z2barâ)op, and Z2barâ, respectively, the exclusions â¨â¨SâŠâŠââ¨â¨Z3â,Z4â,N3IââŠâŠ, â¨â¨SâŠâŠââ¨â¨(Z2barâ)opâŠâŠ, and â¨â¨SâŠâŠââ¨â¨Z2barââŠâŠ follow.
â
Corollary 6.4**.**
Suppose that S is any finite semigroup that satisfies the identities (6.1) but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨KâŠâŠ that is not ji.**
Proof.
The inclusion â¨â¨SâŠâŠââ{â¨â¨TâŠâŠâŁTâK} holds by Proposition 6.1.
But the five identities in (6.4) are satisfied by N3Iâ, Z4â, Z3â, (Z2barâ)op, and Z2barâ, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨TâŠâŠ for all TâK.
â
6.2. The pseudovariety â¨â¨Zmâ,NnIâ,L2Iâ,R2Iâ,A0IââŠâŠ
In this subsection, it is convenient to write
[TABLE]
and Tm,nâ=ZmâĂNnIâĂL2IâĂR2IâĂA0Iâ.
Proposition 6.5**.**
Let mâĽ1 and nâĽ2.*
Then the identities satisfied by the semigroup Tm,nâ are axiomatized by*
[TABLE]
Remark 6.6**.**
(i)
Since N2â is isomorphic to the subsemigroup {0,fe} of A0â, the monoid N2Iâ belongs to â¨â¨A0IââŠâŠ.
Therefore, â¨â¨Tm,1ââŠâŠ=â¨â¨Tm,2ââŠâŠ.
This is the reason for the assumption nâĽ2 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 w can be written in the form
[TABLE]
where xâA, w0â,wrââAâ, and w1â,w2â,âŚ,wrâ1ââA+ are such that xâ/con(wiâ) for all i, and e1â,e2â,âŚ,erââ{1,2,3,âŚ}.
Then the factors xe1â,xe2â,âŚ,xerâ are call x-stacks, or simply stacks, of w.
The weight of the x-stack xeiâ is eiâ.
It is easily shown that the identities (6.5) can be used to convert any word into a word w such that for each xâA,
(I)
the number of x-stacks in w is at most two;
2. (II)
if w has one x-stack, then its weight is at most m+nâ1;
3. (III)
if w has two x-stacks, then the weight of the first x-stack is at most m+nâ2 while the weight of the second x-stack is one.
In this subsection, a word w that satisfies (I)â(III) is said to be in canonical form.
Note that if w is a word in canonical form, then occ(x,w)â¤m+nâ1 for any xâA.
Lemma 6.7**.**
Let u and v be any words in canonical form such that the identity uâv is satisfied by the semigroup Tm,nâ.*
Then for any xâA*,**
(i)
occ(x,u)âĄocc(x,v)(modm);**
2. (ii)
either occ(x,u)=occ(x,v)â¤n or n<occ(x,u),occ(x,v)â¤m+nâ1.
Proof.
This follows from Lemma 5.1 parts (i) and (ii).
â
For any word w and distinct variables x1â,x2â,âŚ,xrâ, let w{x1â,x2â,âŚ,xrâ}â denote the word obtained from w by retaining only the variables x1â,x2â,âŚ,xrâ.
Any monoid that satisfies an identity uâv also satisfies u{x1â,x2â,âŚ,xrâ}ââv{x1â,x2â,âŚ,xrâ}â for any distinct variables x1â,x2â,âŚ,xrâ.
Lemma 6.8**.**
Let u and v be any words in canonical form such that the identity uâv is satisfied by the semigroup Tm,nâ.*
Then*
(i)
for any distinct x,yâA, the identity u{x,y}ââv{x,y}â cannot be any of
[TABLE]
where e1â,f1â,e2â,f2â,e3â,f3ââĽ1;
2. (ii)
u* has two x-stacks if and only if v has two x-stacks*;**
3. (iii)
xe* is the first x-stack of u if and only if xe is the first x-stack of v.*
Proof.
(i) The three identities in (6.6) are violated by the semigroups R2Iâ, L2Iâ, and A0Iâ, respectively.
(ii) Suppose that u has two x-stacks.
Then by (III),
[TABLE]
for some u1â,u3ââAâ and u2ââA+ with xâ/con(u1âu2âu3â) and 2â¤eâ¤m+nâ1.
Seeking a contradiction, suppose that v has only one x-stack.
Then by (II) and part (i),
[TABLE]
for some v1â,v2ââAâ with xâ/con(v1âv2â) and 1â¤fâ¤m+nâ1.
Since the word u2â is nonempty, it contains some y-stack.
Since con(u)=con(v) by Lemma 5.1(ii), it follows that yâcon(v1âv2â).
By symmetry, it suffices to assume that yâcon(v1â), so that ini(v)=âŻyâŻxâŻ.
Since ini(u)=ini(v) by Lemma 5.1(iii), it follows that yâcon(u1â).
Hence the word u contains two y-stacks, the first of which occurs in u1â while the second occurs in u2â.
Thus fin(u)=âŻyâŻxâŻ.
Since fin(u)=fin(v) by Lemma 5.1(iv), it follows that yâ/con(v2â).
Therefore, the identity u{x,y}ââv{x,y}â is yrxeâ1yxâysxf for some r,sâĽ1, but this contradicts part (i).
(iii) Let xe be a first x-stack of u.
By part (ii), there are two cases.
Case 1
u and v each has only one x-stack.
Then by (II),
[TABLE]
for some u1â,u2â,v1â,v2ââAâ with xâ/con(u1âu2âv1âv2â) and 1â¤e,fâ¤m+nâ1.
Since e=occ(x,u) and f=occ(x,v), it follows from part (ii) that either e=fâ¤n or n<e,fâ¤m+nâ1.
If n<e,fâ¤m+nâ1, then e=f by part (i).
Case 2
u and v each has two x-stacks.
Then by (III),
[TABLE]
for some u1â,u3â,v1â,v3ââAâ and u2â,v2ââA+ with xâ/con(u1âu2âu3âv1âv2âv3â) and 2â¤e,fâ¤m+nâ1.
Since e=occ(x,u) and f=occ(x,v), it follows from the same argument in Case 1 that e=f.
â
Lemma 6.9**.**
Let u and v be any words in canonical form such that the identity uâv is satisfied by the semigroup Tm,nâ.*
Then the following are equivalent*:**
vâAâxeyfAâ* where xe and yf are stacks of v.*
Further,* xe is the first x-stack of u if and only if xe is the first x-stack of v, and yf is the first y-stack of u if and only if yf is the first y-stack of v*.**
Proof.
First, note that ini(u)=ini(v) and fin(u)=fin(v) by Lemma 5.1.
Suppose that (a) holds. Then
[TABLE]
for some u1â,u2ââAâ such that u1â does not end with x while u2â does not begin with y.
There are four cases depending on which of xe and yf are first stacks in u.
Case 1
xe is the first x-stack in u and yf is the first y-stack in u.
Then clearly x,yâ/con(u1â), so that ini(u)=âŻxyâŻ.
By Lemma 6.8(iii), xe is the first x-stack of v and yf is the first y-stack of v.
Since ini(v)=ini(u)=âŻxyâŻ,
[TABLE]
for some v1â,v2â,v3ââAâ such that xâ/con(v1â) and yâ/con(v1âv2â), and that any stack of v that occurs in v2â cannot be a first stack.
Suppose that v2âî =â .
Then the first variable z of v2â constitutes the second z-stack of v.
Hence
[TABLE]
where zr is the first z-stack of v, and ini(v)=âŻzâŻxyâŻ.
By Lemma 6.8(ii), the word u contains two z-stacks; by part (iii) of the same lemma, the first z-stack of u is zr.
Since ini(u)=ini(v)=âŻzâŻxyâŻ, the z-stack zr of u occurs in u1â:
[TABLE]
The second z-stack of u occurs in either u1â or u2â.
There are two subcases.
1.1.
The second z-stack of u occurs in u1â.
Then fin(v)=fin(u)=âŻzâŻxâŻ, so that v must contain a second x-stack occurring in either v2â or v3â.
The identity u{x,z}ââv{x,z}â is thus zr+1xe+1âzrxezx, which is impossible by Lemma 6.8(i).
2. 1.2.
The second z-stack of u occurs in u2â.
Then fin(u)=fin(v)=âŻzâŻyâŻ, so that u must contain a second y-stack occurring after the second z-stack:
[TABLE]
The identity u{y,z}ââv{y,z}â is thus zryfzyâzr+1yf+1, which is impossible by Lemma 6.8(i).
Since both subcases are impossible, v2â=â .
Hence (b) holds.
Case 2
xe is the first x-stack in u and yf is the second y-stack in u.
Then f=1 by (III) and
[TABLE]
where yr is the first y-stack of u.
Since ini(v)=ini(u)=âŻyâŻxâŻ, it follows from Lemma 6.8 parts (i) and (iii) that
[TABLE]
for some v1â,v2â,v3â,v4ââAâ, where yr is the first y-stack of v and xe is the first x-stack of v.
Suppose that v3âî =â . Then v3â contains some z-stack zs:
[TABLE]
There are two subcases depending on whether zs is the first or second z-stack in v.
2.1.
zs is the first z-stack in v.
Then ini(u)=ini(v)=âŻyâŻxâŻzâŻ, so that every z of u occurs in u2â.
Hence u{y,z}ââyr+1{z}+ and
Since both subcases are impossible, v3â=â .
Hence (b) holds.
Case 3
xe is the second x-stack in u and yf is the first y-stack in u.
Then e=1 by (III) and
[TABLE]
where xr is the first x-stack of u with and yâ/con(u1â) and xâ/con(u2â).
Since ini(v)=ini(u)=âŻxâŻyâŻ, it follows from Lemma 6.8 that
[TABLE]
for some v1â,v2â,v3â,v4ââAâ with xâ/con(v1âv2âv3âv4â) and yâ/con(v1âv2âv3â).
Suppose that v3âî =â . Then v3â contains some z-stack zs:
[TABLE]
There are two subcases depending on whether or not zs is the first z-stack of v.
3.1.
zs is the first z-stack of v.
Since ini(u)=ini(v)=âŻzâŻyâŻ, the first z-stack zs of u occurs in u1â.
But since fin(u)=fin(v)=âŻxâŻzâŻ, the word u must contain a second z-stack in u2â, whence v must also contain a second z-stack by Lemma 6.8(ii).
Hence v{x,z}â=xr+1zs+1 and
zs is the second z-stack of v. Then the identity u{y,z}ââv{y,z}â obtained by an argument symmetrical to the one in Subcase 3.1 produces a similar contradiction.
Since both subcases are impossible, v3â=â . Hence (b) holds.
Case 4
xe is the second x-stack in u and yf is the second y-stack in u.
Then (b) holds by an argument symmetrical to Case 1.
Therefore, (b) holds in all four cases. By symmetry, (b) implies (a).
â
It is routinely verified that the semigroup Tm,nâ satisfies the identities (6.5).
Conversely, suppose that uâv is any identity satisfied by the semigroup Tm,nâ.
As observed earlier, the identities (6.5) can be used to convert u and v into words uⲠand vⲠin canonical form.
By Lemma 6.7 parts (ii) and (iii), the words uⲠand vⲠshare the same set of stacks.
By Lemma 6.9, two stacks are adjacent in uⲠif and only if they are adjacent in vâ˛.
Therefore, uâ˛=vâ˛.
Since
Suppose that S is any finite semigroup that satisfies the identities
[TABLE]
but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨T6,5ââŠâŠ that is not ji.
Proof.
By Proposition 6.5 with m=6 and n=5, the identities satisfied by T6,5â are axiomatized by (6.7).
Hence the inclusion
[TABLE]
holds.
But the two identities in (6.8) are satisfied by L2IâĂR2Iâ and Z6âĂN5IâĂA0Iâ, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨L2Iâ,R2IââŠâŠ and â¨â¨SâŠâŠââ¨â¨Z6â,N5Iâ,A0IââŠâŠ.
â
Corollary 6.11**.**
Suppose that S is any finite semigroup that satisfies the identities (6.7) but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨T6,5ââŠâŠ that is not ji.
Proof.
The argument in the proof of Corollary 6.10 implies the inclusion
[TABLE]
The two identities in (6.9) are satisfied by N5IâĂL2IâĂR2IâĂA0Iâ and Z6â, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨N5Iâ,L2Iâ,R2Iâ,A0IââŠâŠ and â¨â¨SâŠâŠââ¨â¨Z6ââŠâŠ.
â
6.3. Pseudovarieties of noncommutative nilpotent semigroups
Proposition 6.12**.**
Any ji pseudovariety of nilpotent semigroups is commutative.
Proof.
Let V be any ji pseudovariety of nilpotent semigroups.
Then the inclusion VâComâ¨G holds [2, Figure 9.1].
Since V is ji and VâG, the inclusion VâCom follows.
â
Corollary 6.13**.**
Suppose that S is any finite semigroup that satisfies the identity
[TABLE]
but violates the identity
[TABLE]
Then â¨â¨SâŠâŠ is a pseudovariety of nilpotent semigroups that is not ji.
Proof.
By assumption, the semigroup S is nilpotent and noncommutative.
The result then holds by Proposition 6.12.
â
6.4. The pseudovariety â¨â¨Nn+râ,NnIââŠâŠ
Proposition 6.14**.**
Let n,râĽ1.
Then the identities satisfied by the semigroup Nn+râĂNnIâ are axiomatized by
[TABLE]
for all mâĽ1 and e1â,e2â,âŚ,emâ,f1â,f2â,âŚ,fmââĽ1 such that
(a)
e=f<n+r,* where e=âi=1mâeiâ and f=âi=1mâfiâ;*
2. (b)
for each kâ{1,2,âŚ,m}, either
â
ekâ=fkâ* or*
2. â
ekâ,fkââĽn* and e+ekâ,f+fkââĽn+r.*
Proof.
It is straightforwardly verified that the semigroup Nn+râĂNnIâ satisfies the identities (6.10).
Conversely, let uâv be any identity satisfied by the semigroup Nn+râĂNnIâ. In view of Lemma 5.1(ii), the identity (6.10a) can be used to convert u and v into
[TABLE]
respectively, where eiâ=occ(xiâ,u) and fiâ=occ(xiâ,v) are such that either eiâ=fiâ or eiâ,fiââĽn.
Let e=âi=1mâeiâ and f=âi=1mâfiâ.
Generality is not lost by assuming that eâ¤f.
There are four cases to consider.
Case 1
n+râ¤eâ¤f.
Choose any iâ{1,2,âŚ,m}.
Suppose that eiâî =fiâ.
Then as observed earlier, eiâ,fiââĽn.
Hence
[TABLE]
Similarly, vâ˛â\eqrefid:Nn+rjNnIbasisx1f1ââx2f2âââŻxiâ1fiâ1ââxifiâ+eiââxi+1fi+1âââŻxmfmââ.
Therefore, the identities (6.10) can be used to convert uⲠinto vâ˛.
It follows that uâv is deducible from (6.10).
Case 2
e<n+râ¤f.
Let Ď:AâNn+râ be the substitution that maps all variables to a.
Then uâ˛Ď=aeî =0 and vâ˛Ď=af=0 imply the contradiction uâ˛Ďî =vâ˛Ď.
The present case is thus impossible.
Case 3
e<f<n+r.
Then the contradiction uâ˛Ď=aeî =af=vâ˛Ď is obtained.
Hence the present case is impossible.
Case 4
e=f<n+r.
Suppose that ekâî =fkâ for some k so that e+ekâî =f+fkâ.
Then as observed earlier, ekâ,fkââĽn.
Let Ď:AâNn+râ be the substitution that maps xkâ to a2 and all other variables to a.
Then uâ˛Ď=vâ˛Ď in Nnâ, where
[TABLE]
and vâ˛Ď=af+fkâ similarly.
Thus ae+ekâ=af+fkâ.
But e+ekâî =f+fkâ implies that e+ekâ,f+fkââĽn.
Hence the identity uâ˛âvⲠalso satisfies (ii) and is deducible from (6.10c).
The identity uâv is thus deducible from (6.10).
â
Corollary 6.15**.**
Suppose that S is any finite semigroup that satisfies the identities
[TABLE]
but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨N4â,N2IââŠâŠ that is not ji.
Proof.
By Proposition 6.14 with n=r=2, the identities satisfied by N4âĂN2Iâ are axiomatized by (6.11).
Hence the inclusion
[TABLE]
holds.
But the two identities in (6.12) are satisfied by N2Iâ and N4â, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨N4ââŠâŠ and â¨â¨SâŠâŠââ¨â¨N2IââŠâŠ.
â
Corollary 6.16**.**
Suppose that S is any finite semigroup that satisfies the identities
[TABLE]
but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨N5â,N1IââŠâŠ that is not ji.
Proof.
By Proposition 6.14 with n=1 and r=4, the identities satisfied by N5âĂN1Iâ are axiomatized by (6.13).
Hence the inclusion
[TABLE]
holds.
But the two identities in (6.14) are satisfied by N1Iâ and N5â, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨N5ââŠâŠ and â¨â¨SâŠâŠââ¨â¨N1IââŠâŠ.
â
6.5. The pseudovariety â¨â¨NnIâ,N2barââŠâŠ
It turns out to be notationally simpler to find a basis of identities for NnIâĂ(N2barâ)op instead of NnIâĂN2barâ.
Proposition 6.17**.**
Let nâĽ2.
Then the identities satisfied by the semigroup NnIâĂ(N2barâ)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
(CF1)
x2â xez1f1ââz2f2âââŻzkfkââ or
2. (CF2)
x,y,z1â,z2â,âŚ,zkâ are distinct variables with kâĽ0;
2. (II)
z1â,z2â,âŚ,zkâ are in alphabetical order;
3. (III)
eâ{0,1,âŚ,nâ2}, eiââ{0,1,âŚ,nâ1}, and fiââ{1,2,âŚ,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 w be any word of length at least two.
Then w=x2u or w=xyu for some distinct x,yâA and uâAâ.
The identity (6.15b) can first be used to rearrange the variables of the suffix u until 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,eiâ,fiâ so that (III) is satisfied.
â
Lemma 6.19**.**
The semigroup N2barâ satisfies an identity uâv if and only if the words u and v share the same suffix of length two.
Proof.
This is routinely established and its dual result for (N2barâ)op was observed by Lee and Li [12, Remark 6.2(i)].
â
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 NnIâĂ(N2barâ)op.
Hence it remains to show that any identity uâv satisfied by the semigroup NnIâĂ(N2barâ)op is deducible from the identities (6.15).
It is easily shown that if either u or v is a single variable, then the identity uâv is trivial by Lemma 5.1(ii) and so is vacuously deducible from the identities (6.15).
Therefore, assume that u and v are words of length at least two.
By Lemma 6.18, the identities (6.15) can be used to convert u and v into words uⲠand vⲠin canonical form.
By Lemma 6.19, the words uⲠand vⲠshare the same prefix of length two.
Therefore, uⲠand vⲠare both of the form (CF1) or both of the form (CF2).
In any case, it is routinely verified by Lemma 5.1(ii) that uâ˛=vâ˛.
Since
Suppose that S is any finite semigroup that satisfies the identities
[TABLE]
but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨N5Iâ,N2barââŠâŠ that is not ji.
Proof.
By Proposition 6.17 with n=5, the identities satisfied by N5IâĂN2barâ are axiomatized by (6.16).
Hence the inclusion
[TABLE]
holds.
But the two identities in (6.17) are satisfied by N5Iâ and N2barâ, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨N5IââŠâŠ and â¨â¨SâŠâŠââ¨â¨N2barââŠâŠ.
â
6.6. The pseudovariety â¨â¨N2Iâ,(L2barâ)opâŠâŠ
Proposition 6.21**.**
The identities satisfied by the semigroup N2IâĂ(L2barâ)op are axiomatized by
[TABLE]
Proof.
Let T6â={a,b,c,d,e,f} be the semigroup given in Table 9.
The identities satisfied by T6â are axiomatized by (6.18) [16, Proposition 26.1].
It is easily deduced from the proof of this result that any identity violated by T6â is also violated by one of the following subsemigroups of T6â:
[TABLE]
Since L2Iâ,R2âââ¨â¨(L2barâ)opâŠâŠ, any identity violated by T6â is violated by N2Iâ or (L2barâ)op.
Therefore, the semigroup N2IâĂ(L2barâ)op does not generate any proper subpseudovariety of â¨â¨T6ââŠâŠ, whence â¨â¨N2IâĂ(L2barâ)opâŠâŠ=â¨â¨T6ââŠâŠ.
â
Corollary 6.22**.**
Suppose that S is any finite semigroup that satisfies the identities (6.18) but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨N2Iâ,(L2barâ)opâŠâŠ that is not ji.
holds.
But the two identities in (6.19) are satisfied by (L2barâ)op and N2Iâ, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨N2IââŠâŠ and â¨â¨SâŠâŠââ¨â¨(L2barâ)opâŠâŠ.
â
6.7. The pseudovariety â¨â¨L2Iâ,â3â,â3opââŠâŠ
Proposition 6.23** (Zhang and Luo [32, Proposition 3.2(3) and Figure 5]).**
The identities satisfied by the semigroup L2IâĂâ3âĂâ3opâ 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 â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨L2Iâ,â3â,â3opââŠâŠ that is not ji.**
holds.
But the two identities in (6.21) are satisfied by L2IâĂâ3â and â3opâ, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨L2Iâ,â3ââŠâŠ and â¨â¨SâŠâŠââ¨â¨â3opââŠâŠ.
â
6.8. The pseudovariety â¨â¨A0â,â3Iâ,(â3opâ)IâŠâŠ
The identities satisfied by the semigroup A0âĂâ3IâĂ(â3opâ)I are axiomatized by
[TABLE]
Proof.
The identities satisfied by the semigroup A0âĂB0Iâ are axiomatized by the identities (6.22) [9, Proposition 2.8].
Since â¨â¨B0IââŠâŠ=â¨â¨â3Iâ,(â3opâ)IâŠâŠ [9, Figure 4], the result follows.
â
Corollary 6.26**.**
Suppose that S is any finite semigroup that satisfies the identities (6.22) but violates all of the identities
[TABLE]
Then â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨A0â,â3Iâ,(â3opâ)IâŠâŠ that is not ji.**
holds.
But the three identities in (6.23) are satisfied by A0â, (â3opâ)I, and â3Iâ, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨A0ââŠâŠ, â¨â¨SâŠâŠââ¨â¨â3IââŠâŠ, and â¨â¨SâŠâŠââ¨â¨(â3opâ)IâŠâŠ.
â
6.9. The pseudovariety â¨â¨(N2barâ)I,L2barââŠâŠ
The semigroup W={a,b,c,d,e} given in Table 10 is required in this subsection.
Proposition 6.27**.**
(i)
The identities satisfied by the semigroup W are axiomatized by
[TABLE]
2. (ii)
The subpseudovariety of â¨â¨WâŠâŠ defined by the identity
[TABLE]
is the unique maximal proper subpseudovariety of â¨â¨WâŠâŠ.
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 x1â,x2â,âŚ,xmâ are distinct variables and e1â,e2â,âŚ,emââ{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.
(i) It is routinely shown that the semigroup W satisfies the identities (6.24).
Conversely, suppose that uâv is any identity satisfied by W.
By Remark 6.29, the identities (6.24) can be used to convert u and v into some words uⲠand vⲠin canonical form.
Since the subsemigroup {a,c,d} of W and the semigroup R2Iâ are isomorphic, fin(uâ˛)=fin(vâ˛) by Lemma 5.1(iv).
Hence
[TABLE]
for some distinct x1â,x2â,âŚ,xmââA and e1â,e2â,âŚ,emâ,f1â,f2â,âŚ,fmââ{1,2}.
If ekâî =fkâ, then by making the substitutionÂ Ď given by xkââŚb, xiââŚe for any i<k, and xiââŚc for any i>k, the contradiction uâ˛Ďî =vâ˛Ď is obtained.
Therefore, eiâ=fiâ for all i, so that uâ˛=vâ˛.
Consequently, the identity uâv is deducible from the identities (6.24).
(ii) The semigroup W violates the identity (6.25) because e2b2c2î =e2bc2.
Therefore, â¨â¨WâŠâŠâŠ[[\eqrefid:Wmax]] is a proper subpseudovariety of â¨â¨WâŠâŠ.
It remains to verify that every proper subpseudovariety V of â¨â¨WâŠâŠ satisfies the identity (6.25).
Since Vî =â¨â¨WâŠâŠ, there exists an identity uâv of V that is violated by W.
Further, since the identities (6.24) are satisfied by V, it follows from Remark 6.29 that the words u and v can be chosen to be in canonical form.
There are two cases.
Case 1
fin(u)î =fin(v).
Then by Lemma 5.1(iv) and the dual of Theorem 5.17, the pseudovariety V satisfies the pseudoidentity
[TABLE]
Since
[TABLE]
the pseudovariety V satisfies the identity
[TABLE]
Since
[TABLE]
the pseudovariety V satisfies the identity (6.25).
Case 2
fin(u)=fin(v) and uî =v.
Then
[TABLE]
for some distinct x1â,x2â,âŚ,xmââA and e1â,e2â,âŚ,emâ,f1â,f2â,âŚ,fmââ{1,2} such that ekâî =fkâ for some k, say (ekâ,fkâ)=(2,1).
LetÂ Ď denote the substitution given by xkââŚy, xiââŚx2 for any i<k, and xiââŚz2 for any i>k.
Then
[TABLE]
so that the pseudovariety V satisfies the identity (6.25).
â
Proposition 6.30**.**
The pseudovariety â¨â¨WâŠâŠ is sji but not ji.**
Proof.
The pseudovariety â¨â¨WâŠâŠ is sji by Proposition 6.27(ii).
To show that â¨â¨WâŠâŠ is not ji, the semigroups N2barâ={0,a,a,I} and L2barâ={e,f,e,f,I} from Subsections 3.2 and 3.3 are required.
Let T11â denote the subsemigroup of (N2barâ)IĂL2barâ generated by E=(1,e), X=(a,f), and Y=(I,I).
It is routinely checked that
[TABLE]
so that the semigroup T11â consists of the following 11Â elements:
[TABLE]
see Table 11.
Identifying the elements {b,c,e,h,i,j,k} in T11â 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 (N2barâ)I and L2barâ, respectively.
Consequently, â¨â¨WâŠâŠââ¨â¨(N2barâ)IâŠâŠ and â¨â¨WâŠâŠââ¨â¨L2barââŠâŠ.
â
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 â¨â¨SâŠâŠ is a subpseudovariety of â¨â¨(N2barâ)I,L2barââŠâŠ that is not 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 (N2barâ)I and L2barâ, respectively.
Therefore, â¨â¨SâŠâŠââ¨â¨(N2barâ)IâŠâŠ and â¨â¨SâŠâŠââ¨â¨L2barââŠâŠ.
â
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 â¨â¨SâŠâŠ is ji.
Then â¨â¨SâŠâŠ coincides with one of the following 30 pseudovarieties*:**
[TABLE]
Proof.
The 30 pseudovarieties are ji by results in Sec. 5; see Table 12.
Up to isomorphism and anti-isomorphism, there exist 1308Â 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 ji pseudovarieties.
Specifically, by Conditions A1âA23 and their dual conditions, 241 of the 1308 semigroups generate the 30 ji pseudovarieties, while by Conditions B1âB13 and their dual conditions, the remaining 1067 semigroups generate pseudovarieties that are not ji; see Table 13.
The proof of Theorem 7.1 is thus complete.
â
7.1. Conditions sufficient for join irreducibility
The following conditions and their dual conditions are sufficient for a finite semigroup S to generate a ji pseudovariety in Theorem 7.1.
A pseudovariety â¨â¨SâŠâŠ is a non-ji subpseudovariety of â¨â¨(N2barâ)I,L2barââŠâŠ if
â
Sâ¨{x3âx2,xyxây2x},**
2. â
Sî â¨x2âx,* âSî â¨xyx2âyx2.*
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 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 StatesâIsrael Binational Science Foundation #2012080, and NSA MSP #H98230-16-1-0047.
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