On the monoid of monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images, II

TL;DR

This paper investigates the algebraic structure of a semigroup of monotone injective partial selfmaps of the ordered set of natural number pairs with cofinite domains and images, revealing its decomposition, order relations, and quotient structures.

Contribution

It characterizes the semigroup's structure, describes its natural partial order, and establishes isomorphisms with semidirect products and free commutative monoids, extending understanding of such algebraic objects.

Findings

Semigroup is isomorphic to a semidirect product involving orientation-preserving maps and Z_2.

The natural partial order coincides with the order induced from the symmetric inverse monoid.

Quotients of the semigroup are isomorphic to free commutative monoids and their semidirect products.

Abstract

Let $\mathbb{N}^{2}_{\leqslant}$ be the set $\mathbb{N}^{2}$ with the partial order defined as the product of usual order $\leq$ on the set of positive integers $\mathbb{N}$. We study the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ of monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ having cofinite domain and image. We describe the natural partial order on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ and show that it coincides with the natural partial order which is induced from symmetric inverse monoid $\mathscr{I}_{\mathbb{N}\times\mathbb{N}}$ onto $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. We proved that the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ is isomorphic to the semidirect product $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})\rtimes \mathbb{Z}_2$ of the monoid $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})$ of orientation-preserving monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images by the cyclic group $\mathbb{Z}_2$ of the order two. Also we describe the congruence $\sigma$ on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ which is generated by the natural order $\preccurlyeq$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. We prove that the quotient semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})/\sigma$ is isomorphic to the free commutative monoid $\mathfrak{AM}_\omega$ over an infinite countable set and show that the quotient semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})/\sigma$ is isomorphic to the semidirect product of the free commutative monoid $\mathfrak{AM}_\omega$ by $\mathbb{Z}_2$.

Equations

$$\uparrow x=\left\{y\in X:x⩽y\right\}.$$

$$(i,m)⩽(j,n)if\; and\; only\; if(i\le j)and(m\le n).$$

$$\begin{array}{rl}{\text{V}}^n=\{(n,j):j\in \mathbb{N}\};& {\text{H}}^n=\{(j,n):j\in \mathbb{N}\};\\ {\text{V}}_{\mathrm{dom}\alpha }^n={\text{V}}^n\cap \mathrm{dom}\alpha ;& {\text{V}}_{\mathrm{ran}\alpha }^n={\text{V}}^n\cap \mathrm{ran}\alpha ;\\ {\text{H}}_{\mathrm{dom}\alpha }^n={\text{H}}^n\cap \mathrm{dom}\alpha ;& {\text{H}}_{\mathrm{ran}\alpha }^n={\text{H}}^n\cap \mathrm{ran}\alpha ,\end{array}$$

$$(i_{\alpha [\ast ,j]},j_{\alpha [i,\ast ]})=(i,j)\alpha ,for\; every(i,j)\in \mathrm{dom}\alpha .$$

$$\left\{\chi \in \mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)\mid \alpha \cdot \chi =\beta \right\}\text{mbox}and\left\{\chi \in \mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)\mid \chi \cdot \alpha =\beta \right\}$$

$$\alpha \preccurlyeq \beta if\; and\; only\; if\alpha =\beta \epsilon for\; some\epsilon \in E(\mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2)).$$

$$\alpha \epsilon =(\beta \epsilon )\epsilon =\beta (\epsilon \epsilon )=\beta \epsilon =\alpha .$$

$$\alpha \sigma \beta if\; and\; only\; if\; there\; exists\epsilon \in E(\mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2))such\; that\alpha \epsilon =\beta \epsilon ,$$

$$\mathcal{P}\mathcal{O}{}_{\infty }^{+}({\mathbb{N}}_{⩽}^2)=\left\{\alpha \in \mathcal{P}\mathcal{O}{}_{\infty }({\mathbb{N}}_{⩽}^2):({\text{H}}_{\mathrm{dom}\alpha }^1)\alpha \subseteq {\text{H}}^1\right\}.$$

$$(\alpha \beta )\mathfrak{h}=\varpi (\alpha \beta )\varpi =\varpi (\alpha \mathbb{I}\beta )\varpi =\varpi (\alpha \varpi \varpi \beta )\varpi =(\varpi \alpha \varpi )(\varpi \beta \varpi )=(\alpha )\mathfrak{h}(\beta )\mathfrak{h},$$

$$(\varpi \alpha \varpi )\mathfrak{h}=\varpi \varpi \alpha \varpi \varpi =\mathbb{I}\alpha \mathbb{I}=\alpha ,$$

$$\alpha =\mathbb{I}\alpha \mathbb{I}=\varpi \varpi \alpha \varpi \varpi =((\alpha )\mathfrak{h})\mathfrak{h}=((\beta )\mathfrak{h})\mathfrak{h}=\varpi \varpi \beta \varpi \varpi =\mathbb{I}\beta \mathbb{I}=\beta ,$$

$$\left(s_1,t_1\right)\cdot \left(s_2,t_2\right)=\left(s_1\cdot (s_2){\mathfrak{h}}_{t_1},t_1\cdot t_2\right),s_1,s_2\in S,t_1,t_2\in T,$$

$$\begin{array}{rl}\left(({\alpha }_1,g_1)\cdot ({\alpha }_2,g_2)\right)\mathfrak{I}& =\left({\alpha }_1\cdot ({\alpha }_2)(g_1)\mathfrak{Q},g_1\cdot g_2\right)\mathfrak{I}=\left({\alpha }_1\cdot g_1\cdot {\alpha }_2\cdot g_1,g_1\cdot g_2\right)\mathfrak{I}=\\ & ={\alpha }_1\cdot g_1\cdot {\alpha }_2\cdot g_1\cdot g_1\cdot g_2={\alpha }_1\cdot g_1\cdot {\alpha }_2\cdot g_2=\\ & =({\alpha }_1,g_1)\mathfrak{I}\cdot ({\alpha }_2,g_2)\mathfrak{I},\end{array}$$

$$\begin{array}{rl}& {\text{H}}_{\mathrm{ran}\alpha }^i\cap \left\{(p,i):p⩾h_{\alpha }^i\right\}=\left\{(p,i):p⩾h_{\alpha }^i\right\};\\ & {\text{V}}_{\mathrm{ran}\alpha }^j\cap \left\{(j,q):q⩾v_{\alpha }^j\right\}=\left\{(j,q):q⩾v_{\alpha }^j\right\},\end{array}$$

$$(k,i),(j,l)\in \mathrm{dom}\alpha ,(k,i)\alpha \in {\text{H}}^i,(j,l)\alpha \in {\text{V}}^j,$$

$${\bar{h}}_{\alpha }=\max \left\{h_{\alpha }^i:i=1,\dots ,n_{\alpha }-1\right\}and{\bar{v}}_{\alpha }=\max \left\{v_{\alpha }^j:j=1,\dots ,n_{\alpha }-1\right\}.$$

$${\text{H}}_{\mathrm{ran}\alpha }^i\cap \left\{(p,i):p⩾{\bar{h}}_{\alpha }\right\}=\left\{(p,i):p⩾{\bar{h}}_{\alpha }\right\};$$

$${\text{V}}_{\mathrm{ran}\alpha }^j\cap \left\{(j,q):q⩾{\bar{v}}_{\alpha }\right\}=\left\{(j,q):q⩾{\bar{v}}_{\alpha }\right\},$$

$$(k,i),(j,l)\in \mathrm{dom}\alpha ,(k,i)\alpha \in {\text{H}}^i,(j,l)\alpha \in {\text{V}}^j,$$

$$D_{\alpha }=\left(\mathbb{N}\times \mathbb{N}\right)\setminus \left(\left\{(i,j):i⩽{\bar{h}}_{\alpha }andj⩽n_{\alpha }\right\}\cup \left\{(i,j):i⩽n_{\alpha }andj⩽{\bar{v}}_{\alpha }\right\}\right).$$

$$\mathrm{dom}{\alpha }_{\text{f}}=D_{\alpha },\mathrm{ran}{\alpha }_{\text{f}}=(D_{\alpha })\alpha and(i,j){\alpha }_{\text{f}}=(i,j)\alpha for\; all(i,j)\in \mathrm{dom}{\alpha }_{\text{f}}.$$

$$\big{(}\bar{h}_{\alpha}+2\big{)}_{\alpha_{\!\texttt{f}}[*,1]}=\big{(}\bar{h}_{\alpha}+1\big{)}_{\alpha_{\!\texttt{f}}[*,1]}+1,$$

$$\begin{array}[]{l}\operatorname{dom}\gamma_{n}=\mathbb{N}\times\mathbb{N}\setminus\left\{(1,i)\colon i=1,\ldots,n\right\},\\ \operatorname{dom}\upsilon_{n}=\mathbb{N}\times\mathbb{N}\setminus\left\{(i,1)\colon i=1,\ldots,n\right\},\\ \operatorname{ran}\gamma_{n}=\operatorname{ran}\upsilon_{n}=\mathbb{N}\times\mathbb{N}\end{array}$$

$$\begin{array}[]{l}(i,j)\gamma_{n}=\left\{\begin{array}[]{ll}(i-1,j),&\hbox{if}~{}j\leqslant n;\\ (i,j),&\hbox{if}~{}j>n\end{array}\right.\qquad\hbox{for}\quad(i,j)\in\operatorname{dom}\gamma_{n},\\ \\ (i,j)\upsilon_{n}=\left\{\begin{array}[]{ll}(i,j-1),&\hbox{if}~{}i\leqslant n;\\ (i,j),&\hbox{if}~{}i>n\end{array}\right.\qquad\hbox{for}\quad(i,j)\in\operatorname{dom}\upsilon_{n}.\end{array}$$

$$\alpha\sigma\big{(}\gamma_{k_{1}}^{p_{1}}\ldots\gamma_{k_{i}}^{p_{i}}\upsilon_{l_{1}}^{q_{1}}\ldots\upsilon_{l_{j}}^{q_{j}}\big{)},$$

$$\alpha\sigma\big{(}\gamma_{k_{1}}^{p_{1}}\ldots\gamma_{k_{i}}^{p_{i}}\upsilon_{l_{1}}^{q_{1}}\ldots\upsilon_{l_{j}}^{q_{j}}\big{)}\qquad\hbox{and}\qquad\beta\sigma\big{(}\gamma_{a_{1}}^{b_{1}}\ldots\gamma_{a_{i}}^{b_{i}}\upsilon_{c_{1}}^{d_{1}}\ldots\upsilon_{c_{j}}^{d_{j}}\big{)}$$

$$\iota {\gamma }_{k_1}^{p_1}\dots {\gamma }_{k_i}^{p_i}{\upsilon }_{l_1}^{q_1}\dots {\upsilon }_{l_j}^{q_j}\ne \iota {\gamma }_{a_1}^{b_1}\dots {\gamma }_{a_i}^{b_i}{\upsilon }_{c_1}^{d_1}\dots {\upsilon }_{c_j}^{d_j}$$

$$\big{(}j,\widehat{h}_{\alpha}-1\big{)}\alpha_{\!\texttt{f}}=\big{(}j_{\alpha_{\!\texttt{f}}[*,\widehat{h}_{\alpha}-1]},\widehat{h}_{\alpha}-1\big{)}<(j,\widehat{h}_{\alpha}-1)\quad\hbox{and}\quad(j+1)_{\alpha_{\!\texttt{f}}[*,\widehat{h}_{\alpha}-1]}-j_{\alpha_{\!\texttt{f}}[*,\widehat{h}_{\alpha}-1]}=1,$$

$$p_{{\hat{h}}_{\alpha }-s}=j-j_{{\alpha }_{\text{f}}[\ast ,{\hat{h}}_{\alpha }-s]}-\left(p_{{\hat{h}}_{\alpha }-1}+\dots +p_{{\hat{h}}_{\alpha }-s+1}\right),$$

Full text

We shall follow the terminology of [2] and [10].

In this paper we shall denote the first infinite cardinal by $\omega$ and the cardinality of the set $A$ by $|A|$. We shall identify every set $X$ with its cardinality $|X|$. By $\mathbb{Z}_{2}$ we shall denote the cyclic group of order two. Also, for infinite subsets $A$ and $B$ of an infinite set $X$ we shall write $A{\subseteq^{*}}B$ if and only if there exists a finite subset $A_{0}$ of $A$ such that $A\setminus A_{0}\subseteq B$.

An algebraic 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 a semigroup, then we shall denote the subset of idempotents in $S$ by $E(S)$. If $S$ is an inverse semigroup, then $E(S)$ is closed under multiplication and we shall refer to $E(S)$ a band (or the band of $S$). If the band $E(S)$ is a non-empty subset of $S$, then the semigroup operation on $S$ determines the following partial order $\leqslant$ on $E(S)$: $e\leqslant f$ 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 $\alpha\colon X\rightharpoonup Y$ is a partial map, then by $\operatorname{dom}\alpha$ and $\operatorname{ran}\alpha$ we denote the domain and the range of $\alpha$, respectively.

Let $\mathscr{I}_{\lambda}$ denote the set of all partial one-to-one transformations of an infinite set $X$ of cardinality $\lambda$ together with the following semigroup operation: $x(\alpha\beta)=(x\alpha)\beta$ if $x\in\operatorname{dom}(\alpha\beta)=\{y\in\operatorname{dom}\alpha\mid y\alpha\in\operatorname{dom}\beta\}$, for $\alpha,\beta\in\mathscr{I}_{\lambda}$. The semigroup $\mathscr{I}_{\lambda}$ is called the symmetric inverse semigroup over the set $X$ (see [2, Section 1.9]). The symmetric inverse semigroup was introduced by Vagner [18] and it plays a major role in the theory of semigroups. An element $\alpha\in\mathscr{I}_{\lambda}$ is called cofinite, if the sets $\lambda\setminus\operatorname{dom}\alpha$ and $\lambda\setminus\operatorname{ran}\alpha$ are finite.

Let $(X,\leqslant)$ be a partially ordered set (a poset). For an arbitrary $x\in X$ we denote

[TABLE]

We shall say that a partial map $\alpha\colon X\rightharpoonup X$ is monotone if $x\leqslant y$ implies $(x)\alpha\leqslant(y)\alpha$ for $x,y\in\operatorname{dom}\alpha$.

Let $\mathbb{N}$ be the set of positive integers with the usual linear order $\leq$. On the Cartesian product $\mathbb{N}\times\mathbb{N}$ we define the product partial order, i.e.,

[TABLE]

Later the set $\mathbb{N}\times\mathbb{N}$ with so defined partial order will be denoted by $\mathbb{N}^{2}_{\leqslant}$.

By $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ we denote the semigroup of injective partial monotone selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images. Obviously, $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is a submonoid of the symmetric inverse semigroup $\mathscr{I}_{\omega}$ and $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is a countable semigroup.

Furthermore, we shall denote the identity of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ by $\mathbb{I}$ and the group of units of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ by $H(\mathbb{I})$.

For any positive integer $n$ and an arbitrary $\alpha\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ we denote:

[TABLE]

and

[TABLE]

It well known that each partial injective cofinite selfmap $f$ of $\lambda$ induces a homeomorphism $f^{*}\colon\lambda^{*}\rightarrow\lambda^{*}$ of the remainder $\lambda^{*}=\beta\lambda\setminus\lambda$ of the Stone-Čech compactification of the discrete space $\lambda$. Moreover, under some set theoretic axioms (like PFA or OCA), each homeomorphism of $\omega^{*}$ is induced by some partial injective cofinite selfmap of $\omega$ (see [12]–[17]). So, the inverse semigroup $\mathscr{I}^{\mathrm{cf}}_{\lambda}$ of injective partial selfmaps of an infinite cardinal $\lambda$ with cofinite domains and images admits a natural homomorphism $\mathfrak{h}\colon\mathscr{I}^{\mathrm{cf}}_{\lambda}\rightarrow\mathscr{H}(\lambda^{*})$ to the homeomorphism group $\mathscr{H}(\lambda^{*})$ of $\lambda^{*}$ and this homomorphism is surjective under certain set theoretic assumptions.

In the paper [9] algebraic properties of the semigroup $\mathscr{I}^{\mathrm{cf}}_{\lambda}$ are studied. It is showed that $\mathscr{I}^{\mathrm{cf}}_{\lambda}$ is a bisimple inverse semigroup and that for every non-empty chain $L$ in $E(\mathscr{I}^{\mathrm{cf}}_{\lambda})$ there exists an inverse subsemigroup $S$ of $\mathscr{I}^{\mathrm{cf}}_{\lambda}$ such that $S$ is isomorphic to the bicyclic semigroup and $L\subseteq E(S)$, the Green relations on $\mathscr{I}^{\mathrm{cf}}_{\lambda}$ are described and it is proved that every non-trivial congruence on $\mathscr{I}^{\mathrm{cf}}_{\lambda}$ is a group congruence. Also, the structure of the quotient semigroup $\mathscr{I}^{\mathrm{cf}}_{\lambda}/\sigma$ is described, where $\sigma$ is the least group congruence on $\mathscr{I}^{\mathrm{cf}}_{\lambda}$.

The semigroups $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ and $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{Z})$ of injective isotone partial selfmaps with cofinite domains and images of positive integers and integers are studied in [7] and [8], respectively. It was proved that the semigroups $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ and $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{Z})$ have similar properties to the bicyclic semigroup: they are bisimple and every non-trivial homomorphic image $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ and $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{Z})$ is a group, and moreover the semigroup $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ has $\mathbb{Z}(+)$ as a maximal group image and $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{Z})$ has $\mathbb{Z}(+)\times\mathbb{Z}(+)$, respectively.

In the paper [6] we studied the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^{n}_{\operatorname{lex}})$ of monotone injective partial selfmaps of the set of $L_{n}\times_{\operatorname{lex}}\mathbb{Z}$ having cofinite domain and image, where $L_{n}\times_{\operatorname{lex}}\mathbb{Z}$ is the lexicographic product of $n$-elements chain and the set of integers with the usual linear order. In this paper we described Green’s relations on $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^{n}_{\operatorname{lex}})$, showed that the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^{n}_{\operatorname{lex}})$ is bisimple and established its projective congruences. Also, we proved that $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^{n}_{\operatorname{lex}})$ is finitely generated, every automorphism of $\mathscr{I\!O}\!_{\infty}(\mathbb{Z})$ is inner and showed that in the case $n\geqslant 2$ the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^{n}_{\operatorname{lex}})$ has non-inner automorphisms. In [6] we also proved that for every positive integer $n$ the quotient semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^{n}_{\operatorname{lex}})/\sigma$, where $\sigma$ is a least group congruence on $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^{n}_{\operatorname{lex}})$, is isomorphic to the direct power $\left(\mathbb{Z}(+)\right)^{2n}$. The structure of the sublattice of congruences on $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^{n}_{\operatorname{lex}})$ that are contained in the least group congruence is described in [4].

In the paper [5] we studied algebraic properties of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. We described properties of elements of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ as monotone partial bijection of $\mathbb{N}^{2}_{\leqslant}$ and showed that the group of units of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is isomorphic to the cyclic group of order two. Also in [5] the subsemigroup of idempotents of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ and the Green relations on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ are described. In particular, here we proved that $\mathscr{D}=\mathscr{J}$ in $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$.

The present paper is a continuation of [5]. We describe the natural partial order $\preccurlyeq$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ and show that it coincides with the natural partial order which is induced from symmetric inverse monoid $\mathscr{I}_{\mathbb{N}\times\mathbb{N}}$ over the set $\mathbb{N}\times\mathbb{N}$ onto the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. We proved that the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is isomorphic to the semidirect product $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})\rtimes\mathbb{Z}_{2}$ of the monoid $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ of orientation-preserving monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images by the cyclic group $\mathbb{Z}_{2}$ of the order two. Also we describe the congruence $\sigma$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$, which is generated by the natural order $\preccurlyeq$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$: $\alpha\sigma\beta$ if and only if $\alpha$ and $\beta$ are comparable in $\left(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}),\preccurlyeq\right)$. We prove that the quotient semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})/\sigma$ is isomorphic to the free commutative monoid $\mathfrak{AM}_{\omega}$ over an infinite countable set and show that quotient semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma$ is isomorphic to the semidirect product of the free commutative monoid $\mathfrak{AM}_{\omega}$ by the group $\mathbb{Z}_{2}$.

The following proposition implies that the equations of the form $a\cdot x=b$ and $x\cdot c=d$ in the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ have finitely many solutions. This property holds for the bicyclic monoid, many its generalizations and other semigroups (see corresponding results in [1, 3, 6, 7, 8, 9]).

[TABLE]

are finite. Consequently, every right translation and every left translation by an element of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is a finite-to-one map.

The definition of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ and the equality $\alpha\cdot\chi=\beta$ imply that $\operatorname{dom}\beta\subseteq\operatorname{dom}\alpha$ and $\operatorname{ran}\chi\subseteq\operatorname{ran}\alpha$. Since any element of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ has a cofinite domain and a cofinite image in $\mathbb{N}\times\mathbb{N}$, we conclude that if an element $\chi_{0}$ satisfies the equality $\alpha\cdot\chi=\beta$ then for every other root $\chi$ of the equation $\alpha\cdot\chi=\beta$ there exist finitely many $(i,j)\in(\mathbb{N}\times\mathbb{N})\setminus\operatorname{ran}\beta$ such that one of the following conditions holds:

• $(1)$

$(i,j)\chi\neq(i,j)\chi_{0}$;

• $(2)$

$(i,j)\chi$ is determined and $(i,j)\chi_{0}$ is undetermined;

• $(3)$

$(i,j)\chi_{0}$ is determined and $(i,j)\chi$ is undetermined.

This implies that the equation $\alpha\cdot\chi=\beta$ has finitely many solutions, which completes the proof of the proposition. ∎

Later we shall describe the natural partial order “$\preccurlyeq$” on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. For $\alpha,\beta\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ we put

[TABLE]

We need the following proposition from [11].

• $(i)$

$a\preccurlyeq b$;

• $(ii)$

$a=wb=bz$, $az=a$ for some $w,z\in S^{1}$;

• $(iii)$

$a=xb=by$, $xa=ay=a$ for some $x,y\in S^{1}$.

[TABLE]

Let $\iota\colon\operatorname{dom}(\beta\varepsilon)\to\operatorname{dom}(\beta\varepsilon)$ be the identity map of the set $\operatorname{dom}(\beta\varepsilon)$. Then $\iota\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ and the definition of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ implies that $\operatorname{dom}(\beta\varepsilon)=\operatorname{dom}(\iota\beta)$, because $\varepsilon$ is an idempotent of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. This implies that $(i,j)\iota\beta=(i,j)\beta\varepsilon$ for each $(i,j)\in\operatorname{dom}(\iota\beta)$ and hence we get that $\alpha=\beta\varepsilon=\iota\beta$. Next we apply Proposition 2. ∎

We define a relation $\sigma$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ in the following way:

[TABLE]

for $\alpha,\beta\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$.

• $(i)$

$\alpha\sigma\beta$;

• $(ii)$

there exist $\varsigma,\upsilon\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ such that $\alpha\varsigma=\beta\upsilon$;

• $(iii)$

there exist $\varsigma,\upsilon\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ such that $\alpha\varsigma=\upsilon\beta$;

• $(iv)$

there exists $\iota\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ such that $\iota\alpha=\iota\beta$;

• $(v)$

there exist $\varsigma,\upsilon\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ such that $\varsigma\alpha=\upsilon\beta$.

Thus $\sigma$ is a congruence on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$.

$(ii)\Rightarrow(i)$ If we have that $\alpha\varsigma=\beta\upsilon$ for some $\varsigma,\upsilon\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ then $\alpha\varsigma(\varsigma\upsilon)=\beta\upsilon(\varsigma\upsilon)$. Since $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is a subsemigroup of the symmetric inverse monoid $\mathscr{I}_{|\mathbb{N}\times\mathbb{N}|}$, the idempotents in the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ commute and hence $\alpha(\varsigma\upsilon)=\beta(\varsigma\upsilon)$. This implies that $\alpha\sigma\beta$.

$(ii)\Rightarrow(iii)$ Suppose that $\alpha\varsigma=\beta\upsilon$ for some $\varsigma,\upsilon\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$. Let $\iota\colon\operatorname{dom}(\beta\upsilon)\to\operatorname{dom}(\beta\upsilon)$ be the identity map of the set $\operatorname{dom}(\beta\upsilon)$. Then $\iota\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ and the definition of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ implies that $\operatorname{dom}(\beta\upsilon)=\operatorname{dom}(\iota\beta)$, because $\upsilon$ is an idempotent of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. This implies that $(i,j)\iota\beta=(i,j)\beta\upsilon$ for each $(i,j)\in\operatorname{dom}(\iota\beta)$ and hence we get that $\alpha\varsigma=\beta\upsilon=\iota\beta$.

$(iii)\Rightarrow(ii)$ Suppose that $\alpha\varsigma=\upsilon\beta$ for some $\varsigma,\upsilon\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$. Let $\iota\colon\operatorname{ran}(\upsilon\beta)\to\operatorname{ran}(\upsilon\beta)$ be the identity map of the set $\operatorname{ran}(\upsilon\beta)$. Then $\iota\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ and the definition of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ implies that $\operatorname{ran}(\upsilon\beta)=\operatorname{ran}(\beta\iota)$, because $\upsilon$ is an idempotent of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. Since all elements of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ are partial bijections of $\mathbb{N}\times\mathbb{N}$ we get that $\operatorname{dom}(\upsilon\beta)=\operatorname{dom}(\beta\iota)$. This implies that $(i,j)\beta\iota=(i,j)\upsilon\beta$ for each $(i,j)\in\operatorname{dom}(\beta\iota)$ and hence we get that $\alpha\varsigma=\upsilon\beta=\beta\iota$.

The proofs of equivalences $(iii)\Leftrightarrow(iv)$ and $(iv)\Leftrightarrow(v)$ are similar.

It is obvious that $\sigma$ is a reflexive and symmetric relation on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. Suppose that $\alpha\sigma\beta$ and $\beta\sigma\gamma$ in $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. Then there exist $\varsigma,\upsilon\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ such that $\alpha\varsigma=\beta\varsigma$ and $\beta\upsilon=\gamma\upsilon$. This implies that $\alpha\varsigma\upsilon=\beta\varsigma\upsilon$ and $\beta\upsilon\varsigma=\gamma\upsilon\varsigma$, and since the idempotents in $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ commute we get that $\alpha\varsigma\upsilon=\beta\varsigma\upsilon=\beta\upsilon\varsigma=\gamma\upsilon\varsigma$, and hence $\alpha\sigma\gamma$.

Suppose that $\alpha\sigma\beta$ for some $\alpha,\beta\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. Then by $(iv)$ there exists $\iota\in E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$ such that $\iota\alpha=\iota\beta$. This implies that $\iota\alpha\gamma=\iota\beta\gamma$ for each $\gamma\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ and hence by item $(iv)$ we get that $(\alpha\gamma)\sigma(\beta\gamma)$. The proof of the statement that $(\gamma\alpha)\sigma(\gamma\beta)$ for each $\gamma\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is similar, and hence $\sigma$ is a congruence on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. ∎

• $(i)$

$\alpha\sigma\beta$;

• $(ii)$

$\alpha\varpi\sigma\beta\varpi$;

• $(iii)$

$\varpi\alpha\sigma\varpi\beta$.

The proof of $(i)\Leftrightarrow(ii)$ is similar. ∎

Also the definition of the congruence $\sigma$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ implies the following simple property of $\sigma$-equivalent elements of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$:

• $(i)$

$(\textsf{H}^{1}_{\operatorname{dom}\alpha})\alpha{\subseteq}\textsf{H}^{1}$* if and only if $(\textsf{H}^{1}_{\operatorname{dom}\beta})\beta{\subseteq}\textsf{H}^{1}$;*

• $(ii)$

$(\textsf{H}^{1}_{\operatorname{dom}\alpha})\alpha{\subseteq}\textsf{V}^{1}$* if and only if $(\textsf{H}^{1}_{\operatorname{dom}\beta})\beta{\subseteq}\textsf{V}^{1}$.*

We define

[TABLE]

Then Lemma 3 and Theorem 1 from [5] imply that $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ is a subsemigroup of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. The subsemigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ is called the monoid of orientation-preserving monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images. Moreover it is obvious that $E(\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})=E(\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant}))$. Also, later by $\preccurlyeq$ and $\sigma$ we denote the corresponding induced relations of the relations $\preccurlyeq$ and $\sigma$ from the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ onto its subsemigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$.

The proofs of the following propositions are similar to those of Propositions 3 and 5, respectively.

By $\varpi$ we denote the bijective transformation of $\mathbb{N}\times\mathbb{N}$ defined by the formula $(i,j)\varpi=(j,i)$, for any $(i,j)\in\mathbb{N}\times\mathbb{N}$. It is obvious that $\varpi$ is an element of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ and $\varpi\varpi=\mathbb{I}$.

• $(i)$

$\alpha\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ if and only if $\alpha\varpi,\varpi\alpha\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})\setminus\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$;

• $(ii)$

$\alpha\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ if and only if $\varpi\alpha\varpi\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$.

We define a map $\mathfrak{h}\colon\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})\to\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ by the formula $(\alpha)\mathfrak{h}=\varpi\alpha\varpi$, for $\alpha\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$.

[TABLE]

and hence $\mathfrak{h}\colon\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})\to\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is a homomorphism.

Fix an arbitrary $\alpha\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. Then the definition of $\mathfrak{h}$ implies that

[TABLE]

and hence the map $\mathfrak{h}\colon\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})\to\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is surjective. Suppose that $(\alpha)\mathfrak{h}=(\beta)\mathfrak{h}$ for some $\alpha,\beta\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$. Then

[TABLE]

and hence the map $\mathfrak{h}\colon\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})\to\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is injective. Thus the map $\mathfrak{h}$ is an automorphism of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$.

Now, Remark 10 implies that the restriction $\mathfrak{h}|_{\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})}\colon\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})\to\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ is an automorphism of the semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$, too. ∎

For the automorphism $\mathfrak{h}\colon\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})\to\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ of the semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ we have that $\mathfrak{h}^{2}=\operatorname{\textsf{Id}}_{\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})}$ is the identity automorphism of $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$. This implies that the element $\mathfrak{h}$ generates the group which is isomorphic to the cyclic group of order two $\mathbb{Z}_{2}$. By Proposition 4 from [5] the group of units $H(\mathbb{I})$ of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is isomorphic to $\mathbb{Z}_{2}$. We define a map $\mathfrak{Q}$ from $H(\mathbb{I})$ into the group $\operatorname{Aut}(\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant}))$ of automorphisms of the semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ in the following way $(\mathbb{I})\mathfrak{Q}=\operatorname{\textsf{Id}}_{\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})}$ and $(\varpi)\mathfrak{Q}=\mathfrak{h}$. It is obvious that so defined map $\mathfrak{Q}\colon H(\mathbb{I})\to\operatorname{Aut}(\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant}))$ is an injective homomorphism.

Let $S$ and $T$ be semigroups and let $\mathfrak{H}$ be a homomorphism from $T$ into the semigroup of endomorphisms $\operatorname{\textsf{End}}(S)$ of $S$, $\mathfrak{H}\colon t\mapsto\mathfrak{h}_{t}$. Then the Cartesian product $S\times T$ with the following semigroup operation

[TABLE]

is called a semidirect product of the semigroup $S$ by $T$ and is denoted by $S\rtimes_{\mathfrak{H}}T$. We remark that if $1_{T}$ is the unit of the semigroup $T$ then $(1_{T})\mathfrak{H}=\mathfrak{h}_{1_{T}}$ is the identity homomorphism of $S$ and in the case when $T$ is a group then $(t)\mathfrak{H}=\mathfrak{h}_{t}$ is an automorphism of $S$ for any $t\in T$.

[TABLE]

because $g^{2}=\mathbb{I}$ for any $g\in H(\mathbb{I})$, and hence the map $\mathfrak{I}\colon\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})\rtimes_{\mathfrak{Q}}H(\mathbb{I})\to\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is a homomorphism.

By Lemma 3 from [5] for every $\alpha\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ there exist $\alpha^{+}\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ and $g_{\alpha}\in H(\mathbb{I})$ such that $\alpha=\alpha^{+}g_{\alpha}$. Indeed,

• $(a)$

in the case when $(\textsf{H}^{1}_{\operatorname{dom}\alpha})\alpha\subseteq\textsf{H}^{1}$ we put $\alpha^{+}=\alpha$ and $g_{\alpha}=\mathbb{I}$;

• $(b)$

in the case when $(\textsf{H}^{1}_{\operatorname{dom}\alpha})\alpha\subseteq\textsf{V}^{1}$ we put $\alpha^{+}=\alpha\omega$ and $g_{\alpha}=\omega$.

Let $\alpha^{+},\beta^{+}\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ and $g_{\alpha},g_{\beta}\in H(\mathbb{I})$ be such that $\alpha^{+}g_{\alpha}=(\alpha^{+},g_{\alpha})\mathfrak{I}=(\beta^{+},g_{\beta})\mathfrak{I}=\beta^{+}g_{\beta}$. Since $(\textsf{H}^{1}_{\operatorname{dom}\alpha^{+}})\alpha^{+}\subseteq\textsf{H}^{1}$ and $(\textsf{H}^{1}_{\operatorname{dom}\beta^{+}})\beta^{+}\subseteq\textsf{H}^{1}$, Lemma 3 from [5] implies that $g_{\alpha}=g_{\beta}$. By Proposition 4 from [5] the group of units $H(\mathbb{I})$ of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is isomorphic to $\mathbb{Z}_{2}$ and hence $\alpha^{+}=\alpha^{+}g_{\alpha}^{2}=\alpha^{+}g_{\alpha}g_{\beta}=\beta^{+}g_{\beta}^{2}=\beta^{+}$. Therefore, we get that so defined map $\mathfrak{I}\colon\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})\rtimes_{\mathfrak{Q}}H(\mathbb{I})\to\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ is an isomorphism. ∎

By Theorem 2($ii_{1}$) from [5] for every $\alpha\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ there exists a smallest positive integer $n_{\alpha}$ such that $(i,j)\alpha=(i,j)$ for each $(i,j)\in\operatorname{dom}\alpha\cap{\uparrow}(n_{\alpha},n_{\alpha})$.

• $(i)$

$\alpha\sigma\alpha_{\texttt{f}}$**;

• $(ii)$

$(i+1)_{\alpha_{\texttt{f}}[*,j]}-(i+1)=i_{\alpha_{\texttt{f}}[*,j]}-i$* for arbitrary $(i,j)\in\operatorname{dom}\alpha_{\texttt{f}}$ with $j<n_{\alpha}$, (i,j)\alpha_{\texttt{f}}=\big{(}i_{\alpha_{\texttt{f}}[*,j]},j_{\alpha_{\texttt{f}}[i,*]}\big{)} and (i+1,j)\alpha_{\texttt{f}}=\big{(}(i+1)_{\alpha_{\texttt{f}}[*,j]},j_{\alpha_{\texttt{f}}[i+1,*]}\big{)}, i.e., $\alpha_{\texttt{f}}$ acts as a partial shift on the set $\textsf{H}^{j}$;*

• $(iii)$

$(j+1)_{\alpha_{\texttt{f}}[i,*]}-(j+1)=j_{\alpha_{\texttt{f}}[i,*]}-j$* for arbitrary $(i,j)\in\operatorname{dom}\alpha_{\texttt{f}}$ with $i<n_{\alpha}$, (i,j)\alpha_{\texttt{f}}=\big{(}i_{\alpha_{\texttt{f}}[*,j]},j_{\alpha_{\texttt{f}}[i,*]}\big{)} and (i,j+1)\alpha_{\texttt{f}}=\big{(}i_{\alpha_{\texttt{f}}[*,j+1]},(j+1)_{\alpha_{\texttt{f}}[i,*]}\big{)}, i.e., $\alpha_{\texttt{f}}$ acts as a partial shift on the set $\textsf{V}^{i}$.*

Moreover, there exist smallest positive integers $\widehat{h}_{\alpha},\widehat{v}_{\alpha}\leqslant n_{\alpha}$ such that $(i,j)\alpha_{\texttt{f}}=(i,j)$ for arbitrary $(i,j)\in\operatorname{dom}\alpha_{\texttt{f}}$ with $i\geqslant\widehat{h}_{\alpha}$ and $(k,l)\alpha_{\texttt{f}}=(k,l)$ for arbitrary $(k,l)\in\operatorname{dom}\alpha_{\texttt{f}}$ with $l\geqslant\widehat{v}_{\alpha}$.

[TABLE]

and

[TABLE]

for all positive integers $k\geqslant h^{i}_{\alpha}$ and $l\geqslant v^{j}_{\alpha}$.

We put

[TABLE]

The above arguments imply that

[TABLE]

[TABLE]

and

[TABLE]

for all positive integers $k\geqslant\bar{h}_{\alpha}$ and $l\geqslant\bar{v}_{\alpha}$.

Next we put

[TABLE]

We define $\alpha_{\!\texttt{f}}=\alpha|_{D_{\alpha}}$, i.e.,

[TABLE]

Since $\alpha_{\!\texttt{f}}=\varepsilon_{\alpha}\alpha_{\!\texttt{f}}=\varepsilon_{\alpha}\alpha$ for the identity partial map $\varepsilon_{\alpha}\colon\mathbb{N}\times\mathbb{N}\rightharpoonup\mathbb{N}\times\mathbb{N}$ with $\operatorname{dom}\varepsilon_{\alpha}=\operatorname{ran}\varepsilon_{\alpha}=D_{\alpha}$, Proposition 5 implies that $\alpha\sigma\alpha_{\!\texttt{f}}$.

Then condition (1) and the definition of the positive integer $\bar{h}_{\alpha}$ imply that

[TABLE]

and by similar arguments and induction we have that $(i+1)_{\alpha_{\!\texttt{f}}[*,1]}=(i,1)_{\alpha_{\!\texttt{f}}[*,1]}+1$ for arbitrary $i\geqslant\bar{h}_{\alpha}+1$. Next, if we apply condition (1) and induction for arbitrary $j<n_{\alpha}$ then we get that $(i+1)_{\alpha_{\!\texttt{f}}[*,j]}=(i)_{\alpha_{\!\texttt{f}}[*,j]}+1$ for arbitrary $i\geqslant\bar{h}_{\alpha}+1$. This implies assertion $(ii)$.

The proof of item $(iii)$ is similar to $(ii)$.

The last statement of the lemma follows from the above arguments and Theorem 2($1$) from [5]. ∎

For every positive integer $n$ we define partial maps $\gamma_{n}\colon\mathbb{N}\times\mathbb{N}\rightharpoonup\mathbb{N}\times\mathbb{N}$ and $\upsilon_{n}\colon\mathbb{N}\times\mathbb{N}\rightharpoonup\mathbb{N}\times\mathbb{N}$ in the following way:

[TABLE]

and

[TABLE]

Simple verifications show that $\gamma_{n},\upsilon_{n}\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ for every positive integer $n$, and moreover the subsemigroups $\left\langle\gamma_{k}\mid k\in\mathbb{N}\right\rangle$ and $\left\langle\upsilon_{k}\mid k\in\mathbb{N}\right\rangle$ of the semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$, generated by the sets $\left\{\gamma_{k}\colon k\in\mathbb{N}\right\}$ and $\left\{\upsilon_{k}\colon k\in\mathbb{N}\right\}$, respectively, are isomorphic to the free Abelian semigroup over an infinite countable set.

[TABLE]

for some positive integers $p_{1},\ldots,p_{i},q_{1},\ldots,q_{j}$. Moreover if

[TABLE]

for some $\alpha,\beta\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ then $(\alpha,\beta)\notin\sigma$ if and only if

[TABLE]

for any idempotent $\iota\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$.

By Lemma 13 and Theorem 1(1) of [5] we have that

[TABLE]

for arbitrary \big{(}j,\widehat{h}_{\alpha}-1\big{)},\big{(}j+1,\widehat{h}_{\alpha}-1\big{)}\in\operatorname{dom}\alpha_{\!\texttt{f}}. Then we put $p_{\widehat{h}_{\alpha}-1}=j-j_{\alpha_{\!\texttt{f}}[*,\widehat{h}_{\alpha}-1]}$. Next, for $s=2,\ldots,\widehat{h}_{\alpha}-2$ we define integers $p_{\widehat{h}_{\alpha}-s},\ldots,p_{1}$ by induction,

[TABLE]

where \big{(}j,\widehat{h}_{\alpha}-s\big{)}\alpha_{\!\texttt{f}}=\big{(}j_{\alpha_{\!\texttt{f}}[*,\widehat{h}_{\alpha}-s]},\widehat{h}_{\alpha}-s\big{)}\leqslant(j,\widehat{h}_{\alpha}-s) for arbitrary \big{(}j,\widehat{h}_{\alpha}-s\big{)}\in\operatorname{dom}\alpha_{\!\texttt{f}}.

Similarly, by Lemma 13 and Theorem 1(1) of [5] we have that

[TABLE]

for arbitrary \big{(}\widehat{v}_{\alpha}-1,i\big{)},\big{(}\widehat{v}_{\alpha}-1,i+1\big{)}\in\operatorname{dom}\alpha_{\!\texttt{f}}. Then we put $q_{\widehat{v}_{\alpha}-1}=i-i_{\alpha_{\!\texttt{f}}[\widehat{v}_{\alpha}-1,*]}$. Next, for $t=2,\ldots,\widehat{v}_{\alpha}-2$ we define integers $q_{\widehat{v}_{\alpha}-t},\ldots,q_{1}$ by induction

[TABLE]

where \big{(}\widehat{v}_{\alpha}-t,i\big{)}\alpha_{\!\texttt{f}}=\big{(}\widehat{v}_{\alpha}-t,i_{\alpha_{\!\texttt{f}}[\widehat{v}_{\alpha}-t,*]}\big{)}\leqslant\big{(}\widehat{v}_{\alpha}-t,i\big{)} for arbitrary \big{(}\widehat{v}_{\alpha}-t,i\big{)}\in\operatorname{dom}\alpha_{\!\texttt{f}}.

For any $\alpha\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ put $\varepsilon_{\alpha}\colon\mathbb{N}\times\mathbb{N}$ be the identity partial map with $\operatorname{dom}\varepsilon_{\alpha}=\operatorname{ran}\varepsilon_{\alpha}=D_{\alpha}$, where the set $D_{\alpha}$ is defined by formula (3). Simple verification shows that \varepsilon_{\alpha}\alpha=\varepsilon_{\alpha}\big{(}\gamma_{1}^{p_{1}}\ldots\gamma_{\widehat{h}_{\alpha}-1}^{p_{\widehat{h}_{\alpha}-1}}\upsilon_{1}^{q_{1}}\ldots\upsilon_{l_{j}}^{q_{\widehat{v}_{\alpha}-1}}\big{)} and hence

[TABLE]

which implies that relation (4) holds.

Since $\gamma^{0}_{m}=\upsilon^{0}_{m}=\mathbb{I}$ for any positive integer $m$, without loss of generality we may assume that $p_{1},\ldots,{p_{i}},q_{1},\ldots,q_{j}$ are positive integers in formula (4).

Also, the last statement of the lemma follows from the definition of the congruence $\sigma$ on the semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$. ∎

[TABLE]

[TABLE]

where $\bar{h}_{\alpha}$ and $\bar{v}_{\alpha}$ are the positive integers defined in the proof of Lemma 13. We define the identity partial map $\widehat{\varepsilon}_{\alpha}\colon\mathbb{N}\times\mathbb{N}\rightharpoonup\mathbb{N}\times\mathbb{N}$ with $\operatorname{dom}\widehat{\varepsilon}_{\alpha}=\operatorname{ran}\widehat{\varepsilon}_{\alpha}=M_{\alpha}$, where

[TABLE]

Then $\widehat{\varepsilon}_{\alpha}\preccurlyeq\varepsilon_{\alpha}$ where $\varepsilon_{\alpha}$ is the idempotent of the semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ defined in the proof of Lemma 13. This implies that

[TABLE]

and the equlity

[TABLE]

follows from the definition of the idempotent $\widehat{\varepsilon}_{\alpha}\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$. ∎

The following theorem describes the quotient semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})/\sigma$.

We define the map $\mathfrak{H}_{\sigma}\colon\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})\rightarrow\mathfrak{AM}_{X}$ in the following way:

• $(a)$

if \alpha\sigma\big{(}\gamma_{k_{1}}^{p_{1}}\ldots\gamma_{k_{i}}^{p_{i}}\upsilon_{l_{1}}^{q_{1}}\ldots\upsilon_{l_{j}}^{q_{j}}\big{)} for some positive integers $p_{1},\ldots,p_{i}$, $q_{1},\ldots,q_{j}$, $k_{1}<\ldots<k_{i}$, $l_{1}<\ldots<l_{j}$, then

[TABLE]

• $(b)$

$(\mathbb{I})\mathfrak{H}_{\sigma}=e$, where $e$ is the unit of the free commutative monoid $\mathfrak{AM}_{X}$.

Then Lemmas 14 and 15 imply that $(\alpha)\mathfrak{H}_{\sigma}=(\beta)\mathfrak{H}_{\sigma}$ if and only if $\alpha\sigma\beta$ in $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ and hence the quotient semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})/\sigma$ is isomorphic to the free commutative monoid $\mathfrak{AM}_{X}$. ∎

The following corollary of Theorem 16 shows that the semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ has infinitely many congruences similar as the free commutative monoid $\mathfrak{AM}_{\omega}$ over an infinite countable set.

Its obvious that every non-unit element $u$ of the free commutative monoid $\mathfrak{AM}_{\omega}$ over the infinite countable set $\left\{a_{i}\colon i\in\omega\right\}\cup\left\{b_{j}\colon j\in\omega\right\}$ can be represented in the form $u=a_{1}^{i_{1}}\ldots a_{k}^{i_{k}}b_{1}^{j_{1}}\ldots b_{l}^{j_{l}}$, where $i_{1},\ldots,i_{k},j_{1},\ldots,l_{l}$ are positive integers. We define a map $\mathfrak{f}\colon\mathfrak{AM}_{\omega}\to\mathfrak{AM}_{\omega}$ by the formula

[TABLE]

for $u=a_{1}^{i_{1}}\ldots a_{k}^{i_{k}}b_{1}^{j_{1}}\ldots b_{l}^{j_{l}}\in\mathfrak{AM}_{\omega}$ and $(e)\mathfrak{f}=e$, for unit element $e$ of $\mathfrak{AM}_{\omega}$.

[TABLE]

for some non-negative integers $p,i_{1},\ldots,i_{p},j_{1},\ldots,j_{p},s_{1},\ldots,s_{p},t_{1},\ldots,t_{p}$, where $a^{i}=b^{i}=e$ for $i=0$.

Then we have that

[TABLE]

It is obvious that $\mathfrak{f}\colon\mathfrak{AM}_{\omega}\to\mathfrak{AM}_{\omega}$ is a bijective map and hence $\mathfrak{f}\colon\mathfrak{AM}_{\omega}\to\mathfrak{AM}_{\omega}$ is an automorphism. ∎

The relationships between elements of the subsemigroup $\left\langle\gamma_{k}\mid k\in\mathbb{N}\right\rangle$ and of the subsemigroup $\left\langle\upsilon_{k}\mid k\in\mathbb{N}\right\rangle$ in $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ is described by the following proposition.

We observe that the cyclic group $\mathbb{Z}_{2}$ acts on the free commutative monoid $\mathfrak{AM}_{\omega}$ over the infinite countable set $\left\{a_{i}\colon i\in\omega\right\}\cup\left\{b_{j}\colon j\in\omega\right\}$ in the following way

[TABLE]

where the map $\mathfrak{f}\colon\mathfrak{AM}_{\omega}\to\mathfrak{AM}_{\omega}$ is defined by formula(5). By Proposition 18 the map $\mathfrak{f}$ is an automorphism of the free commutative monoid $\mathfrak{AM}_{\omega}$.

• $(i)$

$\varpi\gamma_{k_{1}}^{p_{1}}\ldots\gamma_{k_{i}}^{p_{i}}\varpi=\upsilon_{k_{1}}^{p_{1}}\ldots\upsilon_{k_{i}}^{p_{i}}$;

• $(ii)$

$\gamma_{k_{1}}^{p_{1}}\ldots\gamma_{k_{i}}^{p_{i}}\varpi=\varpi\upsilon_{k_{1}}^{p_{1}}\ldots\upsilon_{k_{i}}^{p_{i}}$;

• $(iii)$

$\varpi\gamma_{k_{1}}^{p_{1}}\ldots\gamma_{k_{i}}^{p_{i}}=\upsilon_{k_{1}}^{p_{1}}\ldots\upsilon_{k_{i}}^{p_{i}}\varpi$;

• $(iv)$

$\varpi\upsilon_{k_{1}}^{p_{1}}\ldots\upsilon_{k_{i}}^{p_{i}}\varpi=\gamma_{k_{1}}^{p_{1}}\ldots\gamma_{k_{i}}^{p_{i}}$.

Later we assume that $\mathbb{Z}_{2}=\left\{\bar{0},\bar{1}\right\}$.

The following theorem describes the quotient semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma$.

• (1)

$(\textsf{H}^{1}_{\operatorname{dom}\alpha_{x}})\alpha_{x}{\subseteq}\textsf{H}^{1}$;

• (2)

$(\textsf{H}^{1}_{\operatorname{dom}\alpha_{x}})\alpha_{x}{\subseteq}\textsf{V}^{1}$.

We put

[TABLE]

for all $\alpha_{x}\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ with $(\alpha_{x})\mathfrak{P}_{\sigma}=x$. Then the definition of the congruence $\sigma$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ and Corollary 7 imply that the map $\mathfrak{I}\colon\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma\to\mathfrak{AM}_{\omega}\times\mathbb{Z}_{2}$ is well defined.

We observe that formula (6) implies that $(x_{\mathbb{I}})\mathfrak{I}=\left(e,\bar{0}\right)$ for $x_{\mathbb{I}}=(\mathbb{I})\mathfrak{P}_{\sigma}$ and $(x_{\varpi})\mathfrak{I}=\left(e,\bar{1}\right)$ for $x_{\varpi}=(\varpi)\mathfrak{P}_{\sigma}$. Hence we have that

[TABLE]

and

[TABLE]

Also, since $\sigma$ is congruence on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$, we get

[TABLE]

and

• $(i)$

in the case when $(\textsf{H}^{1}_{\operatorname{dom}\alpha_{x}})\alpha_{x}{\subseteq}\textsf{H}^{1}$ for $\alpha_{x}=\gamma_{1}^{i_{1}}\ldots\gamma_{p}^{i_{p}}\upsilon_{1}^{j_{1}}\ldots\upsilon_{p}^{j_{p}}$, for some non-negative integers $p,i_{1},\ldots,i_{p},j_{1},\ldots,j_{p}$, where $\gamma^{0}=\upsilon^{0}=\mathbb{I}$, we get that $(\textsf{H}^{1}_{\operatorname{dom}(\varpi\alpha_{x})})\varpi\alpha_{x}{\subseteq}\textsf{V}^{1}$,

[TABLE]

and by Proposition 19,

[TABLE]

• $(ii)$

in the case when $(\textsf{H}^{1}_{\operatorname{dom}\alpha_{x}})\alpha_{x}{\subseteq}\textsf{V}^{1}$ we get for $\alpha_{x}=\gamma_{1}^{i_{1}}\ldots\gamma_{p}^{i_{p}}\upsilon_{1}^{j_{1}}\ldots\upsilon_{p}^{j_{p}}\varpi$, for some non-negative integers $p,i_{1},\ldots,i_{p},j_{1},\ldots,j_{p}$, where $\gamma^{0}=\upsilon^{0}=\mathbb{I}$, we get that $(\textsf{H}^{1}_{\operatorname{dom}(\varpi\alpha_{x}\varpi)})\varpi\alpha_{x}\varpi{\subseteq}\textsf{H}^{1}$,

[TABLE]

[TABLE]

and by Proposition 19,

[TABLE]

which implies that $(x_{\varpi}\cdot x)\mathfrak{I}=(x_{\varpi})\mathfrak{I}\cdot(x)\mathfrak{I}$.

Therefore we have showed that $(x_{\mathbb{I}})\mathfrak{I}$ is the identity element of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma$ and $(x_{\varpi})\mathfrak{I}\cdot(x_{\varpi})\mathfrak{I}=(x_{\mathbb{I}})\mathfrak{I}$.

Next we shall show that so defined map $\mathfrak{I}$ is a homomorphism from $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma$ into the semigroup $\mathfrak{AM}_{\omega}\rtimes_{\mathfrak{Q}}\mathbb{Z}_{2}$. Fix arbitrary elements $x$ and $y$ of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma$. We consider the following four possible cases:

• $(i)$

$(\textsf{H}^{1}_{\operatorname{dom}\alpha_{x}})\alpha_{x}{\subseteq}\textsf{H}^{1}$ and $(\textsf{H}^{1}_{\operatorname{dom}\alpha_{y}})\alpha_{y}{\subseteq}\textsf{H}^{1}$ for any $\alpha_{x},\alpha_{y}\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ such that $(\alpha_{x})\mathfrak{P}_{\sigma}=x$ and $(\alpha_{y})\mathfrak{P}_{\sigma}=y$;

• $(ii)$

$(\textsf{H}^{1}_{\operatorname{dom}\alpha_{x}})\alpha_{x}{\subseteq}\textsf{V}^{1}$ and $(\textsf{H}^{1}_{\operatorname{dom}\alpha_{y}})\alpha_{y}{\subseteq}\textsf{H}^{1}$ for any $\alpha_{x},\alpha_{y}\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ such that $(\alpha_{x})\mathfrak{P}_{\sigma}=x$ and $(\alpha_{y})\mathfrak{P}_{\sigma}=y$;

• $(iii)$

$(\textsf{H}^{1}_{\operatorname{dom}\alpha_{x}})\alpha_{x}{\subseteq}\textsf{H}^{1}$ and $(\textsf{H}^{1}_{\operatorname{dom}\alpha_{y}})\alpha_{y}{\subseteq}\textsf{V}^{1}$ for any $\alpha_{x},\alpha_{y}\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ such that $(\alpha_{x})\mathfrak{P}_{\sigma}=x$ and $(\alpha_{y})\mathfrak{P}_{\sigma}=y$;

• $(iv)$

$(\textsf{H}^{1}_{\operatorname{dom}\alpha_{x}})\alpha_{x}{\subseteq}\textsf{V}^{1}$ and $(\textsf{H}^{1}_{\operatorname{dom}\alpha_{y}})\alpha_{y}{\subseteq}\textsf{V}^{1}$ for any $\alpha_{x},\alpha_{y}\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ such that $(\alpha_{x})\mathfrak{P}_{\sigma}=x$ and $(\alpha_{y})\mathfrak{P}_{\sigma}=y$.

Assume that $(i)$ hods. Then we have that $\alpha_{x},\alpha_{y},\alpha_{x}\alpha_{y}\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$. Since $\sigma$ is a congruence on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$, we may choose an element $\alpha_{xy}=\alpha_{x}\alpha_{y}\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$. Then $(\alpha_{xy})\mathfrak{P}_{\sigma}=xy$ Also, since $\mathfrak{P}_{\sigma}\colon\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})\rightarrow\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma$ is the natural homomorphism generated by the congruence $\sigma$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$ we get that

[TABLE]

If $(ii)$ hods then by Propositions 1 and 3 from [5], $\alpha_{x}\varpi,\alpha_{y},\alpha_{x}\alpha_{y}\varpi\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ and by Lemma 14 without loss of generality we may assume that

[TABLE]

for some non-negative integers $p,i_{1},\ldots,i_{p},j_{1},\ldots,j_{p},s_{1},\ldots,s_{p},t_{1},\ldots,t_{p}$, where $\gamma^{0}=\upsilon^{0}=\mathbb{I}$. This and the fact that $\sigma$ is a congruence on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$, Proposition 19 imply that

[TABLE]

If $(iii)$ hods then by Propositions 1 and 3 from [5], $\alpha_{x},\alpha_{y}\varpi,\alpha_{x}\alpha_{y}\varpi\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ and by Lemma 14 without loss of generality we may assume that

[TABLE]

for some non-negative integers $p,i_{1},\ldots,i_{p},j_{1},\ldots,j_{p},s_{1},\ldots,s_{p},t_{1},\ldots,t_{p}$, where $\gamma^{0}=\upsilon^{0}=\mathbb{I}$. Since $\sigma$ is a congruence on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$, this and Proposition 19 imply that

[TABLE]

Assume that $(iv)$ hods. Then by Propositions 1 and 3 from [5] we have that $\alpha_{x}\varpi,\alpha_{y}\varpi,\alpha_{x}\alpha_{y}$, $\alpha_{x}\varpi\alpha_{y}\varpi\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ and by Lemma 14 without loss of generality we may assume that

[TABLE]

for some non-negative integers $p,i_{1},\ldots,i_{p},j_{1},\ldots,j_{p},s_{1},\ldots,s_{p},t_{1},\ldots,t_{p}$, where $\gamma^{0}=\upsilon^{0}=\mathbb{I}$. Since $\sigma$ is a congruence on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})$, this and Proposition 19 imply that

[TABLE]

Thus the map $\mathfrak{I}\colon\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma\to\mathfrak{AM}_{\omega}\rtimes_{\mathfrak{Q}}\mathbb{Z}_{2}$ is a homomorphism. Also, since $(x_{\mathbb{I}})\mathfrak{I}=\left(e,\bar{0}\right)$, $(x_{\varpi})\mathfrak{I}=\left(e,\bar{1}\right)$ and for any $\alpha_{x}=\gamma_{1}^{i_{1}}\ldots\gamma_{p}^{i_{p}}\upsilon_{1}^{j_{1}}\ldots\upsilon_{p}^{j_{p}}$, where $p,i_{1},\ldots,i_{p},j_{1},\ldots,j_{p}$ are some positive integers, our above arguments imply that

[TABLE]

where $x=(\alpha_{x})\mathfrak{P}_{\sigma}$ and $y=(\alpha_{x}\varpi)\mathfrak{P}_{\sigma}$. This implies that the homomorphism $\mathfrak{I}$ is surjective.

Now suppose that $(x)\mathfrak{I}=(y)\mathfrak{I}=(u,g)$ for some $x,y\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma$. Then there exist $\alpha_{x},\alpha_{y}\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ such that $(\alpha_{x})\mathfrak{P}_{\sigma}=x$ and $(\alpha_{y})\mathfrak{P}_{\sigma}=y$ in the case when $g=\bar{0}$, and $(\alpha_{x}\varpi)\mathfrak{P}_{\sigma}=x$ and $(\alpha_{y}\varpi)\mathfrak{P}_{\sigma}=y$ in the case when $g=\bar{1}$. If $g=\bar{0}$ then $x,y\in\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ and the condition $\alpha_{x}\sigma\alpha_{y}$ in $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ implies the equality $x=y$. Similarly, if $g=\bar{1}$ then $x,y\in\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})\setminus\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ and the condition $\alpha_{x}\varpi\sigma\alpha_{y}\varpi$ in $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^{2}_{\leqslant})$ implies the equality $x=y$. Hence $\mathfrak{I}\colon\mathscr{P\!O}\!_{\infty}(\mathbb{N}^{2}_{\leqslant})/\sigma\to\mathfrak{AM}_{\omega}\rtimes_{\mathfrak{Q}}\mathbb{Z}_{2}$ is an isomorphism. ∎

Acknowledgements

The authors acknowledge Taras Banakh and Alex Ravsky for their comments and suggestions.