On injective endomorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $\omega$

TL;DR

This paper characterizes all injective endomorphisms of a specific semigroup built from inductive subsets of integers, showing they decompose into a composition of a special endomorphism and an automorphism.

Contribution

It provides a complete description of injective endomorphisms of the semigroup  with the family ^2, including their decomposition into endomorphisms and automorphisms.

Findings

Every injective endomorphism is a composition of a specific endomorphism and an automorphism.

All injective ,0,[0) endomorphisms are explicitly described.

The structure of injective endomorphisms is fully characterized.

Abstract

We describe injective endomorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $\omega$. In particular we show that every injective endomorphism $\mathfrak{e}$ of $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^2}$ is presented in the form $\mathfrak{e}=\mathfrak{e}_0\mathfrak{a}$, where $\mathfrak{e}_0$ is an injective $(0,0,[0))$-endomorphism of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$ and $\mathfrak{a}$ is an automorphism $\mathfrak{a}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$. Also we describe all injective $(0,0,[0))$-endomorphisms $\mathfrak{e}_0$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$, i.e., such that $(0,0,[0))\mathfrak{e}_0=(0,0,[0))$.

Equations

$$q^k{p}^l\cdot q^m{p}^n=q^{k+m-\min \{l,m\}}{p}^{l+n-\min \{l,m\}}.$$

$$(i_{1},j_{1})\cdot(i_{2},j_{2})=\left\{\begin{array}[]{ll}(i_{1}-j_{1}+i_{2},j_{2}),&\hbox{if~}j_{1}\leqslant i_{2};\\ (i_{1},j_{1}-i_{2}+j_{2}),&\hbox{if~}j_{1}\geqslant i_{2}.\end{array}\right.$$

$$(i_{1},j_{1},F_{1})\cdot(i_{2},j_{2},F_{2})=\left\{\begin{array}[]{ll}(i_{1}-j_{1}+i_{2},j_{2},(j_{1}-i_{2}+F_{1})\cap F_{2}),&\hbox{if~}j_{1}\leqslant i_{2};\\ (i_{1},j_{1}-i_{2}+j_{2},F_{1}\cap(i_{2}-j_{1}+F_{2})),&\hbox{if~}j_{1}\geqslant i_{2}.\end{array}\right.$$

$$\boldsymbol{B}_{\omega}^{\mathscr{F}}=\left\{\begin{array}[]{ll}(\boldsymbol{B}_{\omega}\times\mathscr{F},\cdot)/\boldsymbol{I},&\hbox{if~}\varnothing\in\mathscr{F};\\ (\boldsymbol{B}_{\omega}\times\mathscr{F},\cdot),&\hbox{if~}\varnothing\notin\mathscr{F}\end{array}\right.$$

$$\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}=\left\{\begin{array}[]{ll}(\boldsymbol{B}_{\mathbb{Z}}\times\mathscr{F},\cdot)/\boldsymbol{I},&\hbox{if~}\varnothing\in\mathscr{F};\\ (\boldsymbol{B}_{\mathbb{Z}}\times\mathscr{F},\cdot),&\hbox{if~}\varnothing\notin\mathscr{F}\end{array}\right.$$

$${\mathbf{B}}_Z^{\{[s)\}}=\{(i,j,[s)):i,j\in \mathbb{Z}\}$$

$$(i,j,[0)){\alpha }_{k,p}$$

$$(i,j,[1)){\alpha }_{k,p}$$

$$(i,j,[0)){\beta }_{k,p}$$

$$(i,j,[1)){\beta }_{k,p}$$

$${\mathbf{B}}_{\mathbb{Z}}^{{\mathcal{F}}^2}(k,k,0)=(k,k,[0))\cdot {\mathbf{B}}_{\mathbb{Z}}^{{\mathcal{F}}^2}\cdot (k,k,[0)).$$

$$(i,j,[0)){\alpha }_{k,p}$$

$$(i,j,[1)){\alpha }_{k,p}$$

$$((i,j,[0))\cdot (s,t,[0))){\alpha }_{k,p}$$

$$((i,j,[1))\cdot (s,t,[1))){\alpha }_{k,p}$$

$$((i,j,[0))\cdot (s,t,[1))){\alpha }_{k,p}$$

$$\displaystyle=\left\{\begin{array}[]{ll}(i+s-j,t,[1))\alpha_{k,p},&\hbox{if~}j<s;\\ (i,t,[1))\alpha_{k,p},&\hbox{if~}j=s;\\ (i,j+t-s,[0))\alpha_{k,p},&\hbox{if~}j>s\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(p+k(i+s-j),p+kt,[1)),&\hbox{if~}j<s;\\ (p+ki,p+kt,[1)),&\hbox{if~}j=s;\\ (ki,k(j+t-s),[0)),&\hbox{if~}j>s,\end{array}\right.$$

$$(i,j,[0))$$

$$\displaystyle=\left\{\begin{array}[]{ll}(ki+p+ks-kj,p+kt,(kj-p-ks+[0))\cap[1)),&\hbox{if~}kj<p+ks;\\ (ki,p+kt,[0)\cap[1)),&\hbox{if~}kj=p+ks;\\ (ki,kj+p+kt-p-ks,[0)\cap(p+ks-kj+[1))),&\hbox{if~}kj>p+ks\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(p+k(i+s-j),p+kt,[1)),&\hbox{if~}kj<p+ks;\\ (ki,p+kt,[1)),&\hbox{if~}kj=p+ks;\\ (ki,k(j+t-s),[0)),&\hbox{if~}kj>p+ks\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(p+k(i+s-j),p+kt,[1)),&\hbox{if~}kj<ks;\\ (p+ki,p+kt,[1)),&\hbox{if~}kj=ks;\\ (ki,kt,[1)),&\hbox{if~}kj=p+ks\hbox{~and~}p=0;\\ \texttt{vagueness},&\hbox{if~}kj=p+ks\hbox{~and~}p\neq 0;\\ (ki,k(j+t-s),[0)),&\hbox{if~}kj>ks\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(p+k(i+s-j),p+kt,[1)),&\hbox{if~}j<s;\\ (p+ki,p+kt,[1)),&\hbox{if~}j=s;\\ (ki,k(j+t-s),[0)),&\hbox{if~}j>s,\end{array}\right.$$

$$((i,j,[1))\cdot (s,t,[0))){\alpha }_{k,p}$$

$$\displaystyle=\left\{\begin{array}[]{ll}(i+s-j,t,[0))\alpha_{k,p},&\hbox{if~}j<s;\\ (i,t,[1))\alpha_{k,p},&\hbox{if~}j=s;\\ (i,j+t-s,[1))\alpha_{k,p},&\hbox{if~}j>s\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(k(i+s-j),kt,[0)),&\hbox{if~}j<s;\\ (p+ki,p+kt,[1)),&\hbox{if~}j=s;\\ (p+ki,p+k(j+t-s),[1)),&\hbox{if~}j>s,\end{array}\right.$$

$$(i,j,[1))$$

$$\displaystyle=\left\{\begin{array}[]{ll}(p+ki+ks-p-kj,kt,(p+kj-ks+[1))\cap[0)),&\hbox{if~}p+kj<ks;\\ (p+ki,kt,[1)\cap[0)),&\hbox{if~}p+kj=ks;\\ (p+ki,p+kj+kt-ks,[1)\cap(ks-p-kj+[0))),&\hbox{if~}p+kj>ks\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(k(i+s-j),kt,[0)),&\hbox{if~}p+kj<ks;\\ (p+ki,kt,[1)),&\hbox{if~}p+kj=ks;\\ (p+ki,p+k(j+t-s),[1)),&\hbox{if~}p+kj>ks\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(k(i+s-j),kt,[0)),&\hbox{if~}kj<ks;\\ (ki,kt,[1)),&\hbox{if~}p+kj=ks\hbox{~and~}p=0;\\ \texttt{vagueness},&\hbox{if~}p+kj=ks\hbox{~and~}p\neq 0;\\ (p+ki,p+kt,[1)),&\hbox{if~}kj=ks;\\ (p+ki,p+k(j+t-s),[1)),&\hbox{if~}kj>ks\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(k(i+s-j),kt,[0)),&\hbox{if~}j<s;\\ (p+ki,p+kt,[1)),&\hbox{if~}j=s;\\ (p+ki,p+k(j+t-s),[1)),&\hbox{if~}j>s,\end{array}\right.$$

$$(i,j,[0)){\beta }_{k,p}$$

$$(i,j,[1)){\beta }_{k,p}$$

$$((i,j,[0))\cdot (s,t,[0))){\beta }_{k,p}$$

$$((i,j,[1))\cdot (s,t,[1))){\beta }_{k,p}$$

$$((i,j,[0))\cdot (s,t,[1))){\beta }_{k,p}$$

$$\displaystyle=\left\{\begin{array}[]{ll}(i+s-j,t,[1))\beta_{k,p},&\hbox{if~}j<s;\\ (i,t,[1))\beta_{k,p},&\hbox{if~}j=s;\\ (i,j+t-s,[0))\beta_{k,p},&\hbox{if~}j>s\end{array}\right.=$$

$$\displaystyle=\left\{\begin{array}[]{ll}(p+k(i+s-j),p+kt,[0)),&\hbox{if~}j<s;\\ (p+ki,p+kt,[0)),&\hbox{if~}j=s;\\ (ki,k(j+t-s),[0)),&\hbox{if~}j>s,\end{array}\right.$$

$$(i,j,[0))$$

Full text

1. Introduction, motivation and main definitions

We shall follow the terminology of [1, 2, 13]. By $\omega$ we denote the set of all non-negative integers and by $\mathbb{Z}$ the set of all integers.

A subset $A$ of $\omega$ is said to be inductive, if $i\in A$ implies $i+1\in A$. Obvious, that $\varnothing$ is an inductive subset of $\omega$.

Remark 1.1** (​[6]).**

1. (1)

By Lemma 6 from [5] a nonempty subset $F\subseteq\omega$ is inductive in $\omega$ if and only $(-1+F)\cap F=F$. 2. (2)

Since the set $\omega$ with the usual order is well-ordered, for any nonempty inductive subset $F$ in $\omega$ there exists nonnegative integer $n_{F}\in\omega$ such that $[n_{F})=F$. 3. (3)

Statement (2) implies that the intersection of an arbitrary finite family of nonempty inductive subsets in $\omega$ is a nonempty inductive subset of $\omega$.

Let $\mathscr{P}(\omega)$ be the family of all subsets of $\omega$. For any $F\in\mathscr{P}(\omega)$ and $n,m\in\omega$ we put $n-m+F=\{n-m+k\colon k\in F\}$ if $F\neq\varnothing$ and $n-m+\varnothing=\varnothing$. A subfamily $\mathscr{F}\subseteq\mathscr{P}(\omega)$ is called ${\omega}$-closed if $F_{1}\cap(-n+F_{2})\in\mathscr{F}$ for all $n\in\omega$ and $F_{1},F_{2}\in\mathscr{F}$. For any $a\in\omega$ we denote $[a)=\{x\in\omega\colon x\geqslant a\}$.

A semigroup $S$ is called inverse if for any element $x\in S$ there exists a unique $x^{-1}\in S$ such that $xx^{-1}x=x$ and $x^{-1}xx^{-1}=x^{-1}$. The element $x^{-1}$ is called the inverse of $x\in S$. If $S$ is an inverse semigroup, then the function $\operatorname{inv}\colon S\to S$ which assigns to every element $x$ of $S$ its inverse element $x^{-1}$ is called the inversion.

If $S$ is a semigroup, then we shall denote the subset of all idempotents in $S$ by $E(S)$. If $S$ is an inverse semigroup, then $E(S)$ is closed under multiplication and we shall refer to $E(S)$ as a band (or the band of $S$). Then the semigroup operation on $S$ determines the following partial order $\preccurlyeq$ on $E(S)$: $e\preccurlyeq f$ if and only if $ef=fe=e$. This order is called the natural partial order on $E(S)$. A semilattice is a commutative semigroup of idempotents.

If $S$ is an inverse semigroup then the semigroup operation on $S$ determines the following partial order $\preccurlyeq$ on $S$: $s\preccurlyeq t$ if and only if there exists $e\in E(S)$ such that $s=te$. This order is called the natural partial order on $S$ [16].

The bicyclic monoid or the bicyclic semigroup ${\mathscr{C}}(p,q)$ is the semigroup with the identity $1$ generated by two elements $p$ and $q$ subjected only to the condition $pq=1$. The semigroup operation on ${\mathscr{C}}(p,q)$ is determined as follows:

[TABLE]

It is well known that the bicyclic monoid ${\mathscr{C}}(p,q)$ is a bisimple (and hence simple) combinatorial $E$-unitary inverse semigroup and every non-trivial congruence on ${\mathscr{C}}(p,q)$ is a group congruence [1].

On the set $\boldsymbol{B}_{\omega}=\omega\times\omega$ we define the semigroup operation “$\cdot$” in the following way

[TABLE]

It is well known that the bicyclic monoid $\mathscr{C}(p,q)$ to the semigroup $\boldsymbol{B}_{\omega}$ is isomorphic by the mapping $\mathfrak{h}\colon\mathscr{C}(p,q)\to\boldsymbol{B}_{\omega}$, $q^{k}p^{l}\mapsto(k,l)$ (see: [1, Section 1.12] or [15, Exercise IV.1.11$(ii)$]).

Next we shall describe the construction which is introduced in [5].

Let $\boldsymbol{B}_{\omega}$ be the bicyclic monoid and $\mathscr{F}$ be an ${\omega}$-closed subfamily of $\mathscr{P}(\omega)$. On the set $\boldsymbol{B}_{\omega}\times\mathscr{F}$ we define the semigroup operation “$\cdot$” in the following way

[TABLE]

In [5] is proved that if the family $\mathscr{F}\subseteq\mathscr{P}(\omega)$ is ${\omega}$-closed then $(\boldsymbol{B}_{\omega}\times\mathscr{F},\cdot)$ is a semigroup. Moreover, if an ${\omega}$-closed family $\mathscr{F}\subseteq\mathscr{P}(\omega)$ contains the empty set $\varnothing$ then the set $\boldsymbol{I}=\{(i,j,\varnothing)\colon i,j\in\omega\}$ is an ideal of the semigroup $(\boldsymbol{B}_{\omega}\times\mathscr{F},\cdot)$. For any ${\omega}$-closed family $\mathscr{F}\subseteq\mathscr{P}(\omega)$ the following semigroup

[TABLE]

is defined in [5]. The semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ generalizes the bicyclic monoid and the countable semigroup of matrix units. It is proven in [5] that $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is a combinatorial inverse semigroup and Green’s relations, the natural partial order on $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its set of idempotents are described. Here, the criteria when the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is simple, [math]-simple, bisimple, [math]-bisimple, or it has the identity, are given. In particularly in [5] it is proved that the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is isomorphic to the semigrpoup of ${\omega}{\times}{\omega}$-matrix units if and only if $\mathscr{F}$ consists of a singleton set and the empty set, and $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is isomorphic to the bicyclic monoid if and only if $\mathscr{F}$ consists of a non-empty inductive subset of $\omega$.

Group congruences on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ and its homomorphic retracts in the case when an ${\omega}$-closed family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$ are studied in [6]. It is proven that a congruence $\mathfrak{C}$ on $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is a group congruence if and only if its restriction on a subsemigroup of $\boldsymbol{B}_{\omega}^{\mathscr{F}}$, which is isomorphic to the bicyclic semigroup, is not the identity relation. Also in [6], all non-trivial homomorphic retracts and isomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ are described.

In [3, 14] the algebraic structure of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is established in the case when ${\omega}$-closed family $\mathscr{F}$ consists of atomic subsets of ${\omega}$.

The set $\boldsymbol{B}_{\mathbb{Z}}=\mathbb{Z}\times\mathbb{Z}$ with the semigroup operation defined by formula (1) is called the extended bicyclic semigroup [17]. On the set $\boldsymbol{B}_{\mathbb{Z}}\times\mathscr{F}$, where $\mathscr{F}$ is an ${\omega}$-closed subfamily of $\mathscr{P}(\omega)$, we define the semigroup operation “$\cdot$” by formula (2). In [7] it is proved that $(\boldsymbol{B}_{\mathbb{Z}}\times\mathscr{F},\cdot)$ is a semigroup. Moreover, if an ${\omega}$-closed family $\mathscr{F}\subseteq\mathscr{P}(\omega)$ contains the empty set $\varnothing$ then the set $\boldsymbol{I}=\{(i,j,\varnothing)\colon i,j\in\mathbb{Z}\}$ is an ideal of the semigroup $(\boldsymbol{B}_{\mathbb{Z}}\times\mathscr{F},\cdot)$. For any ${\omega}$-closed family $\mathscr{F}\subseteq\mathscr{P}(\omega)$ the following semigroup

[TABLE]

is defined in [7] similarly as in [5]. In [7] it is proven that $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ is a combinatorial inverse semigroup. Green’s relations, the natural partial order on the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ and its set of idempotents are described. Here, the criteria when the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ is simple, [math]-simple, bisimple, [math]-bisimple, is isomorphic to the extended bicyclic semigroup, are derived. In particularly in [7] it is proved that the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ is isomorphic to the semigrpoup of ${\omega}{\times}{\omega}$-matrix units if and only if $\mathscr{F}$ consists of a singleton set and the empty set, and $\boldsymbol{B}_{Z}^{\mathscr{F}}$ is isomorphic to the extended bicyclic semigroup if and only if $\mathscr{F}$ consists of a non-empty inductive subset of $\omega$. Also, in [7] it is proved that in the case when the family $\mathscr{F}$ consists of all singletons of $\omega$ and the empty set, the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ is isomorphic to the Brandt $\lambda$-extension of the semilattice $(\omega,\min)$, where $(\omega,\min)$ is the set $\omega$ with the semilattice operation $x\cdot y=\min\{x,y\}$.

It is well-known that every automorphism of the bicyclic monoid $\boldsymbol{B}_{\omega}$ is the identity self-map of $\boldsymbol{B}_{\omega}$ [1], and hence the group $\mathbf{Aut}(\boldsymbol{B}_{\omega})$ of automorphisms of $\boldsymbol{B}_{\omega}$ is trivial. The group $\mathbf{Aut}(\boldsymbol{B}_{\mathbb{Z}})$ of automorphisms of the extended bicyclic semigroup $\boldsymbol{B}_{\mathbb{Z}}$ is established in [4] and there it is proved that $\mathbf{Aut}(\boldsymbol{B}_{\mathbb{Z}})$ is isomorphic to the additive group of integers $\mathbb{Z}(+)$. In the paper [9] we prove that for any family $\mathscr{F}$ of nonempty inductive subsets of $\omega$ the group $\mathbf{Aut}(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}})$ of automorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ is isomorphic to the additive group of integers.

In [12] the semigroups of endomorphisms of the bicyclic semigroup and the extended bicyclic semigroup are described. All types of monoid endomorphisms of the monoid $\boldsymbol{B}_{\omega}^{\mathscr{F}^{2}}$ for two-element family $\mathscr{F}^{2}$ of nonempty inductive subsets of $\omega$ are described in [8, 10, 11].

This paper is a continuation of [7, 9]. We describe injective endomorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^{2}}$ with the two-element family $\mathscr{F}^{2}$ of inductive nonempty subsets of $\omega$. In particular we show that every injective endomorphism $\mathfrak{e}$ of $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^{2}}$ is presented in the form $\mathfrak{e}=\mathfrak{e}_{0}\mathfrak{a}$, where $\mathfrak{e}_{0}$ is an injective $(0,0,[0))$-endomorphism of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ and $\mathfrak{a}$ is an automorphism $\mathfrak{a}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$. Also we describe all injective $(0,0,[0))$-endomorphisms $\mathfrak{e}_{0}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$, i.e., such that $(0,0,[0))\mathfrak{e}_{0}=(0,0,[0))$.

Later we assume that an ${\omega}$-closed family $\mathscr{F}^{2}$ consists of two inductive nonempty subsets of $\omega$.

2. Endomorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^{2}}$ with the fixed point $(0,0,[0))$

If $\mathscr{F}$ is an arbitrary $\omega$-closed family of inductive subsets in $\mathscr{P}(\omega)$ and $[s)\in\mathscr{F}$ for some $s\in\mathbb{Z}$ then

[TABLE]

is a subsemigroup of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ and by Proposition 5 of [7] the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\{[s)\}}$ is isomorphic to the extended bicyclic semigroup.

Remark 2.1**.**

By Proposition 1 of [9] for any $\omega$-closed family $\mathscr{F}$ of inductive subsets in $\mathscr{P}(\omega)$ there exists an $\omega$-closed family $\mathscr{F}^{*}$ of inductive subsets in $\mathscr{P}(\omega)$ such that $[0)\in\mathscr{F}^{*}$ and the semigroups $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ and $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{*}}$ are isomorphic. Hence without loss of generality we may assume that the family $\mathscr{F}^{2}$ contains the set $[0)$.

An endomorphism $\mathfrak{e}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ is called $(0,0,[0))$-endomorphism if $(0,0,[0))\mathfrak{e}=(0,0,[0))$.

Remark 2.2**.**

Theorem 1 of [8] state that for every injective monoid endomorphism $\mathfrak{e}$ of the monoid $\boldsymbol{B}_{\omega}^{\mathscr{F}^{2}}$ only one of the following conditions holds:

1. (1)

there exist a positive integer $k$ and $p\in\{0,\ldots,k-1\}$ such that $\mathfrak{e}=\alpha_{k,p}$, where the mapping $\alpha_{k,p}$ defined by the formula

[TABLE]

$i,j\in\omega$; 2. (2)

there exist a positive integer $k\geqslant 2$ and $p\in\{1,\ldots,k-1\}$ such that $\mathfrak{e}=\beta_{k,p}$, where the mapping $\beta_{k,p}$ defined by the formula

[TABLE]

$i,j\in\omega$.

For any integer $k$ we define

[TABLE]

By Proposition 2 [9], $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(k,k,0)$ is a subsemigroup of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ which is isomorphic to $\boldsymbol{B}_{\omega}^{\mathscr{F}^{2}}$.

Fix an arbitrary positive integer $k$ and any $p\in\{0,\ldots,k-1\}$. For all $i,j\in\mathbb{Z}$ we denote the transformation $\alpha_{k,p}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ in the following way

[TABLE]

Lemma 2.3**.**

For an arbitrary positive integer $k$ and any $p\in\{0,\ldots,k-1\}$ the map $\alpha_{k,p}$ is an injective endomorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$.

Proof.

By by Proposition 5 of [7] the subsemigroups $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$ and $\boldsymbol{B}_{\mathbb{Z}}^{\{[1)\}}$ are isomorphic to the extended bicyclic semigroup. By Proposition of [12] we have that the restrictions of $\alpha_{k,p}$ onto the subsemigroups $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$ and $\boldsymbol{B}_{\mathbb{Z}}^{\{[1)\}}$ are endomorphisms of $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$ and $\boldsymbol{B}_{\mathbb{Z}}^{\{[1)\}}$, respectively. This implies that for all $i,j,s,t\in\mathbb{Z}$ the following equalities hold

[TABLE]

For any $i,j,p,q\in\mathbb{Z}$ we have that

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

because $p\in\{0,\ldots,k-1\}$. Thus, $\alpha_{k,p}$ is an endomorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$. ∎

Fix an arbitrary positive integer $k\geqslant 2$ and any $p\in\{1,\ldots,k-1\}$. For all $i,j\in\mathbb{Z}$ we define the transformation $\beta_{k,p}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ in the following way

[TABLE]

It is obvious that $\beta_{k,p}$ is an injective transformation of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$.

Lemma 2.4**.**

For an arbitrary positive integer $k\geqslant 2$ and any $p\in\{1,\ldots,k-1\}$ the map $\beta_{k,p}$ is an injective endomorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$.

Proof.

By by Proposition 5 of [7] the subsemigroups $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$ and $\boldsymbol{B}_{\mathbb{Z}}^{\{[1)\}}$ are isomorphic to the extended bicyclic semigroup. By Lemma 3 of [12], the restriction of $\beta_{k,p}$ onto the subsemigroup $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$ is an endomorphisms of $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$, and the restriction of $\beta_{k,p}$ onto the subsemigroup $\boldsymbol{B}_{\mathbb{Z}}^{\{[1)\}}$ is a homomorphisms of $\boldsymbol{B}_{\mathbb{Z}}^{\{[1)\}}$ into $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$. This implies that for all $i,j,s,t\in\mathbb{Z}$ the following equalities hold

[TABLE]

For any $i,j,p,q\in\mathbb{Z}$ we have that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

because $p\in\{1,\ldots,k-1\}$. Thus, $\beta_{k,p}$ is an endomorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$. ∎

Lemma 2.5**.**

Let $\mathfrak{e}$ be a $(0,0,[0))$-endomorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$. Then there exists a non-negative integer $n$ such that $(i,j,[0))\mathfrak{e}=(ni,nj,[0))$ for all $i,j\in\mathbb{Z}$.

Proof.

By by Proposition 5 of [7] the subsemigroup $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$ is isomorphic to the extended bicyclic semigroup. By Lemma 3 of [12], the restriction of the transformation $\mathfrak{e}$ onto the subsemigroup $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$ is an endomorphisms of $\boldsymbol{B}_{\mathbb{Z}}^{\{[0)\}}$. Then Lemma 5 of [12] implies the statement of the lemma. ∎

Theorem 2.6**.**

Let $\mathfrak{e}$ be an injective $(0,0,[0))$-endomorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$. Then one of the following conditions holds:

1. (1)

there exist a positive integer $k$ and $p\in\{0,\ldots,k-1\}$ such that $\mathfrak{e}=\alpha_{k,p}$; 2. (2)

there exist a positive integer $k\geqslant 2$ and $p\in\{1,\ldots,k-1\}$ such that $\mathfrak{e}=\beta_{k,p}$.

Proof.

It is obvious that for any $(i,j,[l))\in\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$, $l=0,1$, there exists a non-negative integer $n$ such that $(i,j,[l))\in\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(-n,-n,0)$. This implies the equality

[TABLE]

Also by the semigroup operation of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ for $m,n\in\omega$ we have that $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(-n,-n,0)\subseteq\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(-m,-m,0)$ if and only if $m\geqslant n$.

Since $\mathfrak{e}$ is an injective $(0,0,[0))$-endomorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$, Lemma 2.5 implies that there exists a positive integer $k$ such that $(i,j,[0))\mathfrak{e}=(ki,kj,[0))$ for all $i,j\in\mathbb{Z}$.

Fix an arbitrary positive integer $n$. By Proposition 2 [9], $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(-n,-n,0)$ is a subsemigroup of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ which is isomorphic to $\boldsymbol{B}_{\omega}^{\mathscr{F}^{2}}$. This implies that the semigroups $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(-n,-n,0)$ and $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(0,0,0)$ are isomorphic. By Corollary 2 from [6] every automorphism of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^{2}}$ is the identity map, and hence every automorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(0,0,0)$ is the identity map, too.

We define the isomorphism $\mathfrak{I}_{0}^{-n}\colon\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(-n,-n,0)\to\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(0,0,0)$ by the formula

[TABLE]

for any positive integers $i,j$ and $s\in\{0,1\}$. The above arguments imply that so defined isomorphism is unique. Hence we have that for any injective endomorphism $\mathfrak{e}_{-n}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(-n,-n,0)$ there exists an injective endomorphism $\mathfrak{e}_{0}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(0,0,0)$ such that the following diagram

[TABLE]

is commutative. Hence, by Remark 2.2 we have that for any injective endomorphism $\mathfrak{e}_{-n}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}(-n,-n,0)$ one of the following conditions holds:

1. (1)

there exist a positive integer $k$ and $p\in\{0,\ldots,k-1\}$ such that $\mathfrak{e}_{0}=\alpha_{k,p}$; 2. (2)

there exist a positive integer $k\geqslant 2$ and $p\in\{1,\ldots,k-1\}$ such that $\mathfrak{e}_{0}=\beta_{k,p}$.

If $\mathfrak{e}_{0}=\alpha_{k,p}$ then

[TABLE]

and

[TABLE]

for any positive integers $i,j$.

If $\mathfrak{e}_{0}=\beta_{k,p}$ then

[TABLE]

and

[TABLE]

for any positive integers $i,j$.

This completes the proof of the theorem. ∎

3. On injective endomorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$

Remark 3.1**.**

1. (1)

By Theorem 1 of [9] every $(0,0,[0))$-automorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ is the identity map. 2. (2)

For every integer $s$ the map $\mathfrak{h}_{s}\colon\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}\to\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$, $(i,j,[p))\mapsto(i+s,j+s,[q))$, $i,j\in\mathbb{Z}$, $q\in\{0,1\}$, is an automorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ (Proposition 6 of [9]). 3. (3)

The map $\widetilde{\mathfrak{a}}\colon\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}\to\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$, $(i,j,[p))\mapsto(i+q,j+q,[1-q))$, $i,j\in\mathbb{Z}$, $q\in\{0,1\}$, is an automorphism of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ (Lemma 2 of [9]), and moreover $\widetilde{\mathfrak{a}}^{2}=\mathfrak{h}_{1}$.

Lemma 3.2**.**

For any endomorphism $\mathfrak{e}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ there exists an automorphism $\mathfrak{a}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ such that $(0,0,[0))\mathfrak{e}=(0,0,[0))\mathfrak{a}$. Moreover, $\mathfrak{a}=\widetilde{\mathfrak{a}}^{2s}=\mathfrak{h}_{s}$ in the case when $(0,0,[0))\mathfrak{e}=(s,s,[0))$, and $\mathfrak{a}=\widetilde{\mathfrak{a}}^{2s+1}=\mathfrak{h}_{s}\widetilde{\mathfrak{a}}$ in the case when $(0,0,[0))\mathfrak{e}=(s,s,[1))$ for some integer $s$.

Proof.

Since any homomorphic image of an idempotent is again an idempotent, by Lemma 1 of [7] there exist an integer $s$ and $q\in\{0,1\}$ such that $(0,0,[0))\mathfrak{e}=(s,s,[q))$. Simple verifications and Remark 3.1 imply that $(0,0,[0))\widetilde{\mathfrak{a}}^{2s}=0,0,[0))=(s,s,[0))$ and $(0,0,[0))\widetilde{\mathfrak{a}}^{2s+1}=(0,0,[0))(\mathfrak{h}_{s}\widetilde{\mathfrak{a}})=(s,s,[1))$. ∎

Theorem 3.3**.**

For any endomorphism $\mathfrak{e}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ there exist a $(0,0,[0))$-endomorphism $\mathfrak{e}_{0}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ and an automorphism $\mathfrak{a}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ such that $\mathfrak{e}=\mathfrak{e}_{0}\mathfrak{a}$. Moreover, $\mathfrak{e}=\mathfrak{e}_{0}\widetilde{\mathfrak{a}}^{2s}=\mathfrak{e}_{0}\mathfrak{h}_{s}$ in the case when $(0,0,[0))\mathfrak{e}=(s,s,[0))$, and $\mathfrak{e}=\mathfrak{e}_{0}\widetilde{\mathfrak{a}}^{2s+1}=\mathfrak{e}_{0}\mathfrak{h}_{s}\widetilde{\mathfrak{a}}$ in the case when $(0,0,[0))\mathfrak{e}=(s,s,[1))$ for some integer $s$.

Proof.

By Lemma 3.2 there exists an automorphism $\mathfrak{a}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ such that $(0,0,[0))\mathfrak{e}=(0,0,[0))\mathfrak{a}$. Then the product $\mathfrak{e}\mathfrak{a}^{-1}$ is a $(0,0,[0))$-endomorphism of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$. Let be $\mathfrak{e}_{0}=\mathfrak{e}\mathfrak{a}^{-1}$. Since for an arbitrary monoid $S$ every right translation $\rho_{u}\colon S\to S$, $s\mapsto su$ on an element of the group of units of $S$ is a bijective map, we conclude that the equality $\mathfrak{e}_{0}=\mathfrak{e}\mathfrak{a}^{-1}$ implies that $\mathfrak{e}=\mathfrak{e}_{0}\mathfrak{a}$. The last statement follows from the second statemnet of Lemma 3.2. ∎

Since the composition of two injective maps is an injective map, Theorems 2.6 and 3.3 imply the following theorem, which describes the structure of all injective endomorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$.

Theorem 3.4**.**

For any injective endomorphism $\mathfrak{e}$ of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ there exist an injective $(0,0,[0))$-endomorphism $\mathfrak{e}_{0}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ and an automorphism $\mathfrak{a}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^{2}}$ such that $\mathfrak{e}=\mathfrak{e}_{0}\mathfrak{a}$. Moreover, $\mathfrak{e}=\mathfrak{e}_{0}\widetilde{\mathfrak{a}}^{2s}=\mathfrak{e}_{0}\mathfrak{h}_{s}$ in the case when $(0,0,[0))\mathfrak{e}=(s,s,[0))$, $\mathfrak{e}=\mathfrak{e}_{0}\widetilde{\mathfrak{a}}^{2s+1}=\mathfrak{e}_{0}\mathfrak{h}_{s}\widetilde{\mathfrak{a}}$ in the case when $(0,0,[0))\mathfrak{e}=(s,s,[1))$ for some integer $s$, and one of the following conditions holds:

1. (1)

there exist a positive integer $k$ and $p\in\{0,\ldots,k-1\}$ such that $\mathfrak{e}_{0}=\alpha_{k,p}$; 2. (2)

there exist a positive integer $k\geqslant 2$ and $p\in\{1,\ldots,k-1\}$ such that $\mathfrak{e}_{0}=\beta_{k,p}$.