Ranks and presentations of some normally ordered inverse semigroups

TL;DR

This paper determines the rank and provides presentations for certain inverse semigroups related to $P$-stable and $P$-order preserving partial permutations, contributing to the understanding of normally ordered inverse semigroups.

Contribution

It computes the rank and exhibits presentations for monoids of $P$-stable and $P$-order preserving partial permutations, advancing the structural knowledge of normally ordered inverse semigroups.

Findings

Computed the rank of the monoids.

Provided explicit presentations for these semigroups.

Enhanced understanding of the generators of the pseudovariety ${ m NO}$.

Abstract

In this paper we compute the rank and exhibit a presentation for the monoids of all $P$-stable and $P$-order preserving partial permutations on a finite set $\Omega$, with $P$ an ordered uniform partition of $\Omega$. These (inverse) semigroups constitute a natural class of generators of the pseudovariety of inverse semigroups ${\sf NO}$ of all normally ordered (finite) inverse semigroups.

Equations

$$\begin{array}[]{rccccccc}\phi:&M&\rightarrow&\mathcal{I}(E)&&&\\ &s&\mapsto&\phi_{s}:&Ess^{-1}&\rightarrow&Es^{-1}s\\ &&&&e&\mapsto&s^{-1}es\leavevmode\nobreak\ ,\end{array}$$

$$I_{1}=\{1,\dots,n_{1}\}\quad\text{and}\quad I_{i}=\{n_{1}+\cdots+n_{i-1}+1,\ldots,n_{1}+\cdots+n_{i}\},\leavevmode\nobreak\ \mbox{for $2\leq i\leq k$.}$$

$$\begin{array}[]{rccc}\Psi:&\mathcal{POI}_{\Omega,P}&\longrightarrow&\mathcal{I}_{n}\\ &\alpha&\longmapsto&\sigma^{-1}\alpha\sigma\end{array}$$

$$i_{a(\alpha \Psi )}=i_{a{\sigma }^{-1}\alpha \sigma }=i_{a{\sigma }^{-1}\alpha }\le i_{b{\sigma }^{-1}\alpha }=i_{b{\sigma }^{-1}\alpha \sigma }=i_{b(\alpha \Psi )},$$

$${\mathcal{P}\mathcal{O}\mathcal{I}}_{k\times m}/\mathcal{J}=\{J_0{<}_{\mathcal{J}}{J}_1{<}_{\mathcal{J}}\cdots {<}_{\mathcal{J}}{J}_k\},$$

$$\alpha=\left(\begin{array}[]{c|c|c|c}I_{i_{1}}&I_{i_{2}}&\cdots&I_{i_{t}}\\ I_{j_{1}}&I_{j_{2}}&\cdots&I_{j_{t}}\end{array}\right),$$

$$\left(\begin{array}[]{c|c|c|c}I_{i_{1}}&I_{i_{2}}&\cdots&I_{i_{t}}\\ I_{i_{1}}&I_{i_{2}}&\cdots&I_{i_{t}}\end{array}\right)$$

$$\mathrm{Dom}(\theta \psi )=\cup \{I_i\mid i\in \mathrm{Dom}(\theta )\}\text{and}\mathrm{Im}(\theta \psi )=\cup \{I_i\mid i\in \mathrm{Im}(\theta )\},$$

$$\theta=\left(\begin{array}[]{cccc}i_{1}&i_{2}&\cdots&i_{t}\\ j_{1}&j_{2}&\cdots&j_{t}\end{array}\right)\in\mathcal{POI}_{k},$$

$$\theta\psi=\left(\begin{array}[]{c|c|c|c}I_{i_{1}}&I_{i_{2}}&\cdots&I_{i_{t}}\\ I_{j_{1}}&I_{j_{2}}&\cdots&I_{j_{t}}\end{array}\right)\in\mathcal{POI}_{n}\cap\mathcal{POI}_{k\times m}.$$

$${x}_{0}=\left(\begin{array}[]{cccc}2&\cdots&k-1&k\\ 1&\cdots&k-2&k-1\end{array}\right),\leavevmode\nobreak\ {x}_{i}=\left(\begin{array}[]{ccccccc}1&\cdots&k-i-1&k-i&k-i+2&\cdots&k\\ 1&\cdots&k-i-1&k-i+1&k-i+2&\cdots&k\end{array}\right),\leavevmode\nobreak\ 1\leq i\leq k-1,$$

$$\bar{x}_{0}=\left(\begin{array}[]{c|c|c|c}I_{2}&\cdots&I_{k-1}&I_{k}\\ I_{1}&\cdots&I_{k-2}&I_{k-1}\end{array}\right),\leavevmode\nobreak\ \bar{x}_{i}=\left(\begin{array}[]{c|c|c|c|c|c|c}I_{1}&\cdots&I_{k-i-1}&I_{k-i}&I_{k-i+2}&\cdots&I_{k}\\ I_{1}&\cdots&I_{k-i-1}&I_{k-i+1}&I_{k-i+2}&\cdots&I_{k}\end{array}\right),\leavevmode\nobreak\ 1\leq i\leq k-1.$$

$$a_i=(1,\dots ,1,a,1,\dots ,1),\text{leavevmode}{b}_i=(1,\dots ,1,b,1,\dots ,1)\text{leavevmode}\text{and}\text{leavevmode}{c}_i=(1,\dots ,1,c,1,\dots ,1),$$

$$\{a_1,a_2,\dots ,a_k,b_1,b_2,\dots ,b_k\}\text{leavevmode}\text{and}\text{leavevmode}\{b_1,b_2,\dots ,b_k,c_1,c_2,\dots ,c_k\}$$

$$\{d_1,d_2,\dots ,d_k\}$$

$$\begin{array}[]{ccc}\mathcal{S}_{m}^{k}&\longrightarrow&\mathcal{G}_{k\times m}\\ z&\longmapsto&\bar{z},\end{array}$$

$$p=\max \{i\in \{1,\dots ,t\}\mid {\alpha }_i\text{ has rank equal to }m(k-1)\}.$$

$$⟨a,b\mid a^2=b^m=(ba{)}^{m-1}=(a{b}^{m-1}ab{)}^3=(a{b}^{m-j}a{b}^j{)}^2=1,2\le j\le m-2⟩$$

$$⟨b,c\mid (c{b}^{m-1}{)}^2=b^m=(bc{b}^{m-1}{)}^{m-1}=(c{b}^{m-2}c{)}^3=(c{b}^{m-j-1}c{b}^{j-1}{)}^2=1,2\le j\le m-2⟩$$

Full text

2010 *Mathematics subject classification: *20M20, 20M05, 20M07.

Keywords: transformations, normally ordered inverse semigroups, ranks, presentations.

1 Introduction and preliminaries

Let $\Omega$ be a set. We denote by $\mathcal{PT}(\Omega)$ the monoid (under composition) of all partial transformations on $\Omega$, by $\mathcal{T}(\Omega)$ the submonoid of $\mathcal{PT}(\Omega)$ of all full transformations on $\Omega$, by $\mathcal{I}(\Omega)$ the symmetric inverse semigroup on $\Omega$, i.e. the inverse submonoid of $\mathcal{PT}(\Omega)$ of all partial permutations on $\Omega$, and by $\mathcal{S}(\Omega)$ the symmetric group on $\Omega$, i.e. the subgroup of $\mathcal{PT}(\Omega)$ of all permutations on $\Omega$. If $\Omega$ is a finite set with $n$ elements ($n\in\mathbb{N}$), say $\Omega=\Omega_{n}=\{1,\ldots,n\}$, we denote $\mathcal{PT}(\Omega)$, $\mathcal{T}(\Omega)$, $\mathcal{I}(\Omega)$ and $\mathcal{S}(\Omega)$ simply by $\mathcal{PT}_{n}$, $\mathcal{T}_{n}$, $\mathcal{I}_{n}$ and $\mathcal{S}_{n}$, respectively.

Now, consider a linear order $\leq$ on $\Omega_{n}$, e.g the usual order. We say that a transformation $\alpha\in\mathcal{PT}_{n}$ is order preserving if $x\leq y$ implies $x\alpha\leq y\alpha$, for all $x,y\in\mathop{\mathrm{Dom}}\nolimits(\alpha)$. Denote by $\mathcal{PO}_{n}$ the submonoid of $\mathcal{PT}_{n}$ of all order preserving partial transformations, by $\mathcal{O}_{n}$ the submonoid of $\mathcal{T}_{n}$ of all order preserving full transformations of $\Omega_{n}$ and by $\mathcal{POI}_{n}$ the inverse submonoid of $\mathcal{I}_{n}$ of all order preserving partial permutations of $\Omega_{n}$.

A pseudovariety of [inverse] semigroups is a class of finite [inverse] semigroups closed under homomorphic images of [inverse] subsemigroups and finitary direct products.

In the “Szeged International Semigroup Colloquium” (1987) J.-E. Pin asked for an effective description of the pseudovariety (i.e. an algorithm to decide whether or not a finite semigroup belongs to the pseudovariety) of semigroups ${\sf O}$ generated by the semigroups $\mathcal{O}_{n}$, with $n\in\mathbb{N}$. Despite, as far as we know, this question is still open, some progresses have been made. First, Higgins [13] proved that ${\sf O}$ is self-dual and does not contain all $\mathscr{R}$-trivial semigroups (and so ${\sf O}$ is properly contained in ${\sf A}$, the pseudovariety of all finite aperiodic semigroups), although every finite band belongs to ${\sf O}$. Next, Vernitskii and Volkov [19] generalized Higgins’s result by showing that every finite semigroup whose idempotents form an ideal is in ${\sf O}$ and, in [5], Fernandes proved that the pseudovariety of semigroups ${\sf POI}$ generated by the semigroups $\mathcal{POI}_{n}$, with $n\in\mathbb{N}$, is a (proper) subpseudovariety of ${\sf O}$. On the other hand, Almeida and Volkov [3] showed that the interval $[{\sf O},{\sf A}]$ of the lattice of all pseudovarieties of semigroups has the cardinality of the continuum and Repnitskiĭ and Volkov [17] proved that ${\sf O}$ is not finitely based. Another contribution to the resolution of Pin’s problem was given by Fernandes [9] who showed that ${\sf O}$ contains all semidirect products of a chain (considered as a semilattice) by a semigroup of injective order preserving partial transformations on a finite chain. This result was later generalized by Fernandes and Volkov [12] for semidirect products of a chain by any semigroup from ${\sf O}$.

The inverse counterpart of Pin’s problem can be formulated by asking for an effective description of the pseudovariety of inverse semigroups ${\sf PCS}$ generated by $\{\mathcal{POI}_{n}\mid n\in\mathbb{N}\}$. In [4] Cowan and Reilly proved that ${\sf PCS}$ is properly contained in ${\sf A}\cap{\sf Inv}$ (being ${\sf Inv}$ the class of all inverse semigroups) and also that the interval $[{\sf PCS},{\sf A}\cap{\sf Inv}]$ of the lattice of all pseudovarieties of inverse semigroups has the cardinality of the continuum. From Cowan and Reilly’s results it can be deduced that a finite inverse semigroup with $n$ elements belongs to ${\sf PCS}$ if and only if it can be embedded into the semigroup $\mathcal{POI}_{n}$. This is in fact an effective description of ${\sf PCS}$. On the other hand, in [6] Fernandes introduced the class ${\sf NO}$ of all normally ordered inverse semigroups. This notion is deeply related with the Munn representation of an inverse semigroup $M$, an idempotent-separating homomorphism that may be defined by

[TABLE]

with $E$ the semilattice of all idempotents of $M$. A finite inverse semigroup $M$ is said to be normally ordered if there exists a linear order $\sqsubseteq$ in the semilattice $E$ of the idempotents of $M$ preserved by all partial permutations $\phi_{s}$ (i.e. for $e,f\in Ess^{-1}$, $e\sqsubseteq f$ implies $e\phi_{s}\sqsubseteq f\phi_{s}$), with $s\in M$. It was proved in [6] that ${\sf NO}$ is a pseudovariety of inverse semigroups and also that the class of all fundamental normally ordered inverse semigroups consists of all aperiodic normally ordered inverse semigroups. Moreover, Fernandes showed that ${\sf PCS}={\sf NO}\cap{\sf A}$, giving in this way a Cowan and Reilly alternative (effective) description of ${\sf PCS}$. In fact, this also led Fernandes [6] to the following refinement of Cowan and Reilly’s description of ${\sf PCS}$: a finite inverse semigroup with $n$ idempotents belongs to ${\sf PCS}$ if and only if it can be embedded into $\mathcal{POI}_{n}$. Another refinement (in fact, the best possible) was also given by Fernandes [11], by considering only join irreducible idempotents. Notice that, in [6] it was also proved that ${\sf NO}={\sf PCS}\vee{\sf G}$ (the join of ${\sf PCS}$ and ${\sf G}$, the pseudovariety of all groups).

Now, let $\Omega$ be a finite set.

An ordered partition of $\Omega$ is a partition of $\Omega$ endowed with a linear order. By convention, whenever we take an ordered partition $P=\{X_{i}\}_{i=1,\dots,k}$ of $\Omega$, we will assume that $P$ is the chain $\{X_{1}<X_{2}<\cdots<X_{k}\}$. We say that $P=\{X_{i}\}_{i=1,\dots,k}$ is an uniform partition of $\Omega$ if $|X_{i}|=|X_{j}|$ for all $i,j\in\{1,\ldots,k\}$.

Let $P=\{X_{i}\}_{i=1,\dots,k}$ be an ordered partition of $\Omega$ and, for each $x\in\Omega$, denote by $i_{x}$ the integer $i\in\{1,\ldots,k\}$ such that $x\in X_{i}$. Let $\alpha$ be a partial transformation on $\Omega$. We say that $\alpha$ is:

$P$-stable if $X_{i_{x}}\subseteq\mathop{\mathrm{Dom}}\nolimits(\alpha)$ and $X_{i_{x}}\alpha=X_{i_{x\alpha}}$, for all $x\in\mathop{\mathrm{Dom}}\nolimits(\alpha)$; and

$P$-order preserving if $i_{x}\leq i_{y}$ implies $i_{x\alpha}\leq i_{y\alpha}$, for all $x,y\in\mathop{\mathrm{Dom}}\nolimits(\alpha)$

(where $\leq$ denotes the usual order on $\{1,\ldots,k\}$).

Denote by $\mathcal{POI}_{\Omega,P}$ the set of all $P$-stable and $P$-order preserving partial permutations on $\Omega$.

Notice that, the identity mapping belongs to $\mathcal{POI}_{\Omega,P}$ and it is easy to check that $\mathcal{POI}_{\Omega,P}$ is an inverse submonoid of $\mathcal{I}(\Omega)$. Observe also that if $P$ is the trivial partition of $\Omega_{n}$, i.e. $P=\{\{i\}\}_{i=1,\ldots,n}$, then $\mathcal{POI}_{\Omega_{n},P}$ coincides with $\mathcal{POI}_{n}$ and, on the other hand, if $P=\{\Omega_{n}\}$ (the universal partition of $\Omega_{n}$) then $\mathcal{POI}_{\Omega_{n},P}$ is exactly the symmetric group $\mathcal{S}_{n}$, for $n\in\mathbb{N}$.

These monoids, considered for the first time by Fernandes in [6], were inspired by the work of Almeida and Higgins [2], despite they are quite different from the ones considered by these two last authors. The main relevance of the monoids $\mathcal{POI}_{\Omega,P}$ lies in the fact that they constitute a family of generators of the pseudovariety ${\sf NO}$ of inverse semigroups. More precisely, Fernandes proved in [6, Theorem 4.4] that ${\sf NO}$ is the class of all inverse subsemigroups (up to an isomorphism) of semigroups of the form $\mathcal{POI}_{\Omega,P}$. In fact, by the proof of [6, Theorem 4.4], it is clear that it suffices to consider semigroups of the form $\mathcal{POI}_{\Omega,P}$, with $P$ a uniform partition of $\Omega$, i.e. we also have the following result:

Theorem 1.1

The class ${\sf NO}$ is the pseudovariety of inverse semigroups generated by all semigroups of the form $\mathcal{POI}_{\Omega,P}$, where $\Omega$ is a finite set and $P$ is an ordered uniform partition of $\Omega$.

Let $n\in\mathbb{N}$. An ordered partition of $n$ is a non-empty sequence of positive integers whose elements sum to $n$.

Let $\pi=(n_{1},\ldots,n_{k})$ be an ordered partition of $n$. Define the ordered partition $P_{\pi}$ of $\Omega_{n}$ as being the partition into intervals $P_{\pi}=\{I_{i}\}_{i=1,\dots,k}$ of $\Omega_{n}$ (endowed with the usual order), where

[TABLE]

Notice that $\pi=(|I_{1}|,\ldots,|I_{k}|)$.

Next, we show that $\Omega_{n}$ and its partitions into intervals allow us to construct, up to an isomorphism, all monoids of type $\mathcal{POI}_{\Omega,P}$, with $\Omega$ a set with $n$ elements and $P$ and ordered partition of $\Omega$.

Theorem 1.2

Let $\Omega$ be a set with $n$ elements and let $P=\{X_{i}\}_{i=1,\dots,k}$ be an ordered partition of $\Omega$. Then, being $\pi$ the ordered partition $(|X_{1}|,\ldots,|X_{k}|)$ of $n$, the monoids $\mathcal{POI}_{\Omega,P}$ and $\mathcal{POI}_{\Omega_{n},P_{\pi}}$ are isomorphic.

• Proof.

Let $\pi=(n_{1},\ldots,n_{k})$ and $P_{\pi}=\{I_{i}\}_{i=1,\dots,k}$. Then $|X_{i}|=|I_{i}|$, for all $i\in\{1,\ldots,k\}$, and so we may consider a bijection $\sigma:\Omega\longrightarrow\Omega_{n}$ such that $X_{i}\sigma=I_{i}$, for all $i\in\{1,\ldots,k\}$. Hence, it is clear that the mapping

[TABLE]

is a homomorphism of monoids such that $\alpha\Psi$ is $P_{\pi}$-stable, for all $\alpha\in\mathcal{POI}_{\Omega,P}$.

For each $x\in\Omega$, denote by $i_{x}$ the integer $i\in\{1,\ldots,k\}$ such that $x\in X_{i}$ and, for each $a\in\Omega_{n}$, denote by $i_{a}$ the integer $i\in\{1,\ldots,k\}$ such that $a\in I_{i}$. Clearly, $i_{x}=i_{x\sigma}$ and $i_{a}=i_{a\sigma^{-1}}$, for all $x\in\Omega$ and $a\in\Omega_{n}$.

Let $\alpha\in\mathcal{POI}_{\Omega,P}$. Notice that $\mathop{\mathrm{Dom}}\nolimits(\alpha\Psi)=(\mathop{\mathrm{Dom}}\nolimits(\alpha))\sigma$ (and $\mathop{\mathrm{Im}}\nolimits(\alpha\Psi)=(\mathop{\mathrm{Im}}\nolimits(\alpha))\sigma$). Take $a,b\in\mathop{\mathrm{Dom}}\nolimits(\alpha\Psi)$ such that $i_{a}\leq i_{b}$. Then $i_{a\sigma^{-1}}\leq i_{b\sigma^{-1}}$ and, since $a\sigma^{-1},b\sigma^{-1}\in\mathop{\mathrm{Dom}}\nolimits(\alpha)$, we have $i_{(a\sigma^{-1})\alpha}\leq i_{(b\sigma^{-1})\alpha}$. Hence

[TABLE]

which proves that $\alpha\Psi\in\mathcal{POI}_{\Omega_{n},P_{\pi}}$.

Thus, we may consider $\Psi$ as a homomorphism of monoids from $\mathcal{POI}_{\Omega,P}$ into $\mathcal{POI}_{\Omega_{n},P_{\pi}}$. Analogously, we build a homomorphism of monoids $\Phi:\mathcal{POI}_{\Omega_{n},P_{\pi}}\longrightarrow\mathcal{POI}_{\Omega,P}$ by defining $\beta\Phi=\sigma\beta\sigma^{-1}$, for each $\beta\in\mathcal{POI}_{\Omega_{n},P_{\pi}}$. Clearly, $\Psi$ and $\Phi$ are mutually inverse mappings and so they are isomorphisms of monoids, as required.

Now, let $k,m\in\mathbb{N}$ be such that $n=km$. Let $\pi=(m,\ldots,m)\in\Omega_{n}^{k}$. Denote the uniform partition into intervals $P_{\pi}$ of $\Omega_{n}$ by $P_{k\times m}$ (i.e. we have $P_{k\times m}=\{I_{i}\}_{i=1,\dots,k}$, with $I_{i}=\{(i-1)m+1,\ldots,im\},$ for $i\in\{1,\ldots,k\}$) and denote the monoid $\mathcal{POI}_{\Omega_{n},P_{k\times m}}$ by $\mathcal{POI}_{k\times m}$. Therefore, combining Theorems 1.1 and 1.2, we immediately obtain the following result:

Corollary 1.3

The pseudovariety of inverse semigroups ${\sf NO}$ is generated by the class $\{\mathcal{POI}_{k\times m}\mid k,m\in\mathbb{N}\}$.

This fact gave us the main motivation for the work presented in this paper, which is about the monoids $\mathcal{POI}_{k\times m}$, with $k,m\in\mathbb{N}$. The remaining of this paper is organized as follows. In Section 2 we calculate their sizes and ranks and in Section 3 we construct presentations for them.

For general background on Semigroup Theory and standard notation, we refer the reader to Howie’s book [15]. For general background on pseudovarieties and finite semigroups, we refer the reader to Almeida’s book [1]. All semigroups considered in this paper are finite.

2 Size and rank of $\mathcal{POI}_{k\times m}$

Let $M$ be a monoid. Recall that the quasi-order $\leq_{\mathscr{J}}$ is defined on $M$ as follows: for all $u,v\in M$, $u\leq_{\mathscr{J}}v$ if and only if $MuM\subseteq MvM$. As usual, the $\mathscr{J}$-class of an element $u\in M$ is denoted by $J_{u}$ and a partial order relation $\leq_{\mathscr{J}}$ is defined on the set $M/\mathscr{J}$ by $J_{u}\leq_{\mathscr{J}}J_{v}$ if and only if $u\leq_{\mathscr{J}}v$. Given $u,s\in M$, we write $u<_{\mathscr{J}}v$ or $J_{u}<_{\mathscr{J}}J_{v}$ if and only if $u\leq_{\mathscr{J}}v$ and $(u,v)\notin\mathscr{J}$.

Recall also that the rank of a (finite) monoid $M$ is the minimum size of a generating set of $M$.

Let $P$ be an ordered partition of $\Omega$. Let $\alpha,\beta\in\mathcal{POI}_{\Omega,P}$. Since $\mathcal{POI}_{\Omega,P}$ is an inverse submonoid of $\mathcal{I}(\Omega)$, we immediately have that $\alpha\mathscr{R}\beta$ if and only if $\mathop{\mathrm{Dom}}\nolimits(\alpha)=\mathop{\mathrm{Dom}}\nolimits(\beta)$ and that $\alpha\mathscr{L}\beta$ if and only if $\mathop{\mathrm{Im}}\nolimits(\alpha)=\mathop{\mathrm{Im}}\nolimits(\beta)$. If $P$ is uniform, it is easy to check also that $\alpha\mathscr{J}\beta$ if and only if $|\mathop{\mathrm{Im}}\nolimits(\alpha)|=|\mathop{\mathrm{Im}}\nolimits(\beta)|$ (see [7, Proposition 5.2.2]). In fact, more specifically, we have that $J_{\alpha}\leq_{\mathscr{J}}J_{\beta}$ if and only if $|\mathop{\mathrm{Im}}\nolimits(\alpha)|\leq|\mathop{\mathrm{Im}}\nolimits(\beta)|$.

Notice that $\mathcal{POI}_{n\times 1}$ is isomorphic to $\mathcal{POI}_{n}$, whose size is $\binom{2n}{n}$ and rank is $n$ (see [5, Proposition 2.2] and [8, Proposition 2.8]), and $\mathcal{POI}_{1\times n}$ is isomorphic to $\mathcal{S}_{n}$, whose size is well known to be $n!$ and rank is well known to be $2$, for $n\geq 3$, and $1$, for $n\in\{1,2\}$.

From now on let $k,m\in\mathbb{N}$ be such that $k,m\geq 2$ and let $n=km$.

Now, we turn our attention to the $\mathscr{J}$-classes of $\mathcal{POI}_{k\times m}$. Let $\alpha\in\mathcal{POI}_{k\times m}$. Then $|\mathop{\mathrm{Im}}\nolimits(\alpha)|=im$, for some $0\leq i\leq k$. Hence

[TABLE]

where $J_{i}=\{\alpha\in\mathcal{POI}_{k\times m}\mid|\mathop{\mathrm{Im}}\nolimits(\alpha)|=im\}$, for $0\leq i\leq k$.

Let $t\in\{1,\ldots,k\}$. We write

[TABLE]

for all transformations $\alpha\in\mathcal{POI}_{k\times m}$ such that $\mathop{\mathrm{Dom}}\nolimits(\alpha)=I_{i_{1}}\cup I_{i_{2}}\cup\cdots\cup I_{i_{t}}$, $\mathop{\mathrm{Im}}\nolimits(\alpha)=I_{j_{1}}\cup I_{j_{2}}\cup\cdots\cup I_{j_{t}}$ and $I_{i_{r}}\alpha=I_{j_{r}}$, for $1\leq r\leq t$. In this case, we assume always that $1\leq i_{1}<i_{2}<\cdots<i_{t}\leq k$ and $1\leq j_{1}<j_{2}<\cdots<j_{t}\leq k$. Clearly, the set of all such transformations forms an $\mathscr{H}$-class of $\mathcal{POI}_{k\times m}$ contained in $J_{t}$.

In particular, it is easy to check that the $\mathscr{H}$-class of the transformations of the form

[TABLE]

constitutes a group isomorphic to $\mathcal{S}_{m}^{t}$ and so it has ${(m!)}^{t}$ elements.

On the other hand, since there are $\binom{k}{t}$ distinct possibilities for domains (and images) of the transformations of $J_{t}$, we deduce that $|J_{t}|=\binom{k}{t}^{2}{(m!)}^{t}$.

Thus, we have:

Proposition 2.1

For $k,m\geq 2$, the monoid $\mathcal{POI}_{k\times m}$ has $\sum_{t=0}^{k}\binom{k}{t}^{2}{(m!)}^{t}$ elements.

Next, let $\psi:\mathcal{POI}_{k}\longrightarrow\mathcal{POI}_{n}$ be the mapping defined by

[TABLE]

for all $\theta\in\mathcal{POI}_{k}$. Notice that, if

[TABLE]

with $1\leq t\leq k$, $1\leq i_{1}<i_{2}<\cdots<i_{t}\leq k$ and $1\leq j_{1}<j_{2}<\cdots<j_{t}\leq k$, then

[TABLE]

Moreover, it is a routine matter to show that $\mathop{\mathrm{Im}}\nolimits(\psi)=\mathcal{POI}_{n}\cap\mathcal{POI}_{k\times m}$ and $\psi$ is an injective homomorphism of monoids.

Let

[TABLE]

and take $\bar{x}_{i}={x}_{i}\psi$, for $0\leq i\leq k-1$. Observe that $\bar{x}_{0},\bar{x}_{1},\ldots,\bar{x}_{k-1}$ are order preserving and $P$-order preserving transformations such that

[TABLE]

Since $\mathcal{POI}_{k}$ is generated by $\{x_{0},x_{1},\ldots,x_{k-1}\}$ (see [8]) and $\psi$ is a homomorphism, then $\bar{X}=\{\bar{x}_{0},\bar{x}_{1},\ldots,\bar{x}_{k-1}\}$ is a generating set for $\mathop{\mathrm{Im}}\nolimits(\psi)=\mathcal{POI}_{n}\cap\mathcal{POI}_{k\times m}$.

Next, recall that is well known that $\mathcal{S}_{m}$ is generated by the permutations $a=(1\ 2)$ and $b=(1\ 2\cdots m)$. Take $c=ab=(1\ 3\ 4\cdots m)$. Thus, since $a=cb^{m-1}$, it is clear that $\mathcal{S}_{m}$ is also generated by the permutations $b$ and $c$. Let

[TABLE]

where $a$, $b$ and $c$ are in the position $i$, for $1\leq i\leq k$ (and $1$ denoting the identity of $\mathcal{S}_{m}$). Clearly,

[TABLE]

are generating sets of the direct product $\mathcal{S}_{m}^{k}$.

Let $d_{i}=b_{i}c_{i+1}$, for $1\leq i\leq k-1$, and $d_{k}=b_{k}c_{1}$. For $1\leq i\leq k$, we have $b_{i}^{m}=c_{i}^{m-1}=1$, whence $b_{i}^{{(m-1)}^{2}}=b_{i}$ and $c_{i}^{m}=c_{i}$. Moreover, since $b_{i}c_{j}=c_{j}b_{i}$, for $1\leq i,j\leq k$ and $i\neq j$, it is easy to check that $c_{1}=d_{k}^{m}$, $c_{i+1}=d_{i}^{m}$, for $1\leq i\leq k-1$, and $b_{i}=d_{i}^{{(m-1)}^{2}}$, for $1\leq i\leq k$. Therefore

[TABLE]

is also a generating set of $\mathcal{S}_{m}^{k}$. Observe that, as $m,k\geq 2$, the rank of $\mathcal{S}_{m}^{k}$ is $k$ (for instance, see [20]).

Let $\mathcal{G}_{k\times m}$ be the group of units of $\mathcal{POI}_{k\times m}$, i.e. $\mathcal{G}_{k\times m}=\{\alpha\in\mathcal{POI}_{k\times m}\mid|\mathop{\mathrm{Im}}\nolimits(\alpha)|=n\}=\mathcal{S}_{n}\cap\mathcal{POI}_{k\times m}$. We have a natural isomorphism

[TABLE]

defined by $x\bar{z}=(x-(i-1)m)z_{i}+(i-1)m$, for $x\in I_{i}$, $1\leq i\leq k$, and $z=(z_{1},z_{2},\ldots,z_{k})\in\mathcal{S}_{m}^{k}$.

Let $\bar{A}=\{\bar{a}_{1},\bar{a}_{2},\ldots,\bar{a}_{k}\}$, $\bar{B}=\{\bar{b}_{1},\bar{b}_{2},\ldots,\bar{b}_{k}\}$, $\bar{C}=\{\bar{c}_{1},\bar{c}_{2},\ldots,\bar{c}_{k}\}$ and $\bar{D}=\{\bar{d}_{1},\bar{d}_{2},\ldots,\bar{d}_{k}\}$. By the above observations, $\bar{A}\cup\bar{B}$, $\bar{B}\cup\bar{C}$ and $\bar{D}$ are three generating sets of $\mathcal{G}_{k\times m}$. Moreover, the number of elements of $\bar{D}$ is $k$, which is precisely the rank of $\mathcal{G}_{k\times m}$.

Proposition 2.2

For $k,m\geq 2$, the monoid $\mathcal{POI}_{k\times m}$ is generated by $\bar{X}\cup\mathcal{G}_{k\times m}$.

• Proof.

Let $\alpha\in\mathcal{POI}_{k\times m}$ be a nonempty transformation (notice that, clearly, $\langle{\bar{X}}\rangle$ contains the empty transformation) and suppose that \alpha=\left(\begin{array}[]{c|c|c|c}I_{i_{1}}&I_{i_{2}}&\cdots&I_{i_{t}}\\ I_{j_{1}}&I_{j_{2}}&\cdots&I_{j_{t}}\end{array}\right), with $1\leq t\leq k$.

Take \theta=\left(\begin{array}[]{cccc}i_{1}&i_{2}&\cdots&i_{t}\\ j_{1}&j_{2}&\cdots&j_{t}\end{array}\right)\in\mathcal{POI}_{k} and let $\bar{\alpha}=\theta\psi$. Then $\bar{\alpha}\in\langle{\bar{X}}\rangle$.

On the other hand, define $\gamma\in\mathcal{T}_{n}$ by $x\gamma=(x+(i_{r}-j_{r})m)\alpha$, for $x\in I_{j_{r}}$, $1\leq r\leq t$, and $x\gamma=x$, for $x\in\Omega_{n}\setminus\mathop{\mathrm{Im}}\nolimits(\alpha)$. Then $\gamma\in\mathcal{G}_{k\times m}$ and it is a routine matter to check that $\alpha=\bar{\alpha}\gamma$, which proves the result.

It follows immediately:

Corollary 2.3

For $k,m\geq 2$, $\bar{A}\cup\bar{B}\cup\bar{X}$, $\bar{B}\cup\bar{C}\cup\bar{X}$ and $\bar{D}\cup\bar{X}$ are generating sets of $\mathcal{POI}_{k\times m}$.

Notice that, in particular, $\bar{D}\cup\bar{X}$ is a generating set of $\mathcal{POI}_{k\times m}$ with $2k$ elements.

Proposition 2.4

For $k,m\geq 2$, the rank of $\mathcal{POI}_{k\times m}$ is $2k$.

• Proof.

Let $L$ be a generating set of $\mathcal{POI}_{k\times m}$.

Take a transformation $\alpha\in\mathcal{POI}_{k\times m}$ of rank $m(k-1)$. Then, $\mathop{\mathrm{Im}}\nolimits(\alpha)=\Omega_{n}\setminus I_{j}$, for some $1\leq j\leq k$. Let $\alpha_{1},\alpha_{2},\ldots,\alpha_{t}\in L$ be such that $\alpha=\alpha_{1}\alpha_{2}\cdots\alpha_{t}$. Hence, for $1\leq i\leq t$, the rank of $\alpha_{i}$ is either $mk$ or $m(k-1)$. Moreover, at least one of the transformations $\alpha_{1},\alpha_{2},\ldots,\alpha_{t}$ must have rank equal to $m(k-1)$. Let

[TABLE]

Then $\xi=\alpha_{p+1}\cdots\alpha_{t}$ has rank $km$ (here $\xi$ denotes the identity, if $p=t$) and so \xi=\left(\begin{array}[]{c|c|c|c}I_{1}&I_{2}&\cdots&I_{k}\\ I_{1}&I_{2}&\cdots&I_{k}\end{array}\right)\in\mathcal{G}_{k\times m}. Thus $\mathop{\mathrm{Im}}\nolimits(\alpha)=(\mathop{\mathrm{Im}}\nolimits(\alpha_{1}\alpha_{2}\cdots\alpha_{p}))\xi=\mathop{\mathrm{Im}}\nolimits(\alpha_{1}\alpha_{2}\cdots\alpha_{p})\subseteq\mathop{\mathrm{Im}}\nolimits(\alpha_{p})$. As $|\mathop{\mathrm{Im}}\nolimits(\alpha)|=|\mathop{\mathrm{Im}}\nolimits(\alpha_{p})|$, we deduce that $\mathop{\mathrm{Im}}\nolimits(\alpha_{p})=\mathop{\mathrm{Im}}\nolimits(\alpha)$. Therefore $L$ contains at least one element whose image is $\Omega_{n}\setminus I_{j}$, for each $1\leq j\leq k$, and so $L$ must contain at least $k$ elements of rank $m(k-1)$.

Next, we consider the elements of $\mathcal{POI}_{k\times m}$ of rank $km$. They constitute the group of units $\mathcal{G}_{k\times m}$ of $\mathcal{POI}_{k\times m}$, which has rank $k$ (as observed above). Therefore, we must also have at least $k$ elements of rank $km$ in the generating set $L$.

Thus $|L|\geq 2k$. Since $\bar{D}\cup\bar{X}$ is a generating set of $\mathcal{POI}_{k\times m}$ with $2k$ elements, the result follows.

3 Presentations for $\mathcal{POI}_{k\times m}$

We begin this section by recalling some notions and facts on presentations. Let $A$ be a set and denote by $A^{*}$ the free monoid generated by $A$. Usually, the set $A$ is called alphabet and the elements of $A$ and $A^{*}$ are called letters and words, respectively. A monoid presentation is an ordered pair $\langle A\mid R\rangle$, where $A$ is an alphabet and $R$ is a subset of $A^{*}\times A^{*}$. An element $(u,v)$ of $A^{*}\times A^{*}$ is called a relation and it is usually represented by $u=v$. To avoid confusion, given $u,v\in A^{*}$, we will write $u\equiv v$, instead of $u=v$, whenever we want to state precisely that $u$ and $v$ are identical words of $A^{*}$. A monoid $M$ is said to be defined by a presentation $\langle A\mid R\rangle$ if $M$ is isomorphic to $A^{*}/\rho_{R}$, where $\rho_{R}$ denotes the smallest congruence on $A^{*}$ containing $R$. For more details see [16] or [18].

A direct method to find a presentation for a monoid is described by the following well-known result (e.g. see [18, Proposition 1.2.3]).

Proposition 3.1

Let $M$ be a monoid generated by a set $A$ (also considered as an alphabet) and let $R\subseteq A^{*}\times A^{*}$. Then $\langle A\mid R\rangle$ is a presentation for $M$ if and only if the following two conditions are satisfied:

The generating set $A$ of $M$ satisfies all the relations from $R$; 2. 2.

If $u,v\in A^{*}$ are any two words such that the generating set $A$ of $M$ satisfies the relation $u=v$, then $u=v$ is a consequence of $R$.

Given a presentation for a monoid, a method to find a new presentation consists in applying Tietze transformations. For a monoid presentation $\langle A\mid R\rangle$, the (four) elementary Tietze transformations are:

Adding a new relation $u=v$ to $\langle A\mid R\rangle$, provided that $u=v$ is a consequence of $R$;

Deleting a relation $u=v$ from $\langle A\mid R\rangle$, provided that $u=v$ is a consequence of $R\backslash\{u=v\}$;

Adding a new generating symbol $b$ and a new relation $b=w$, where $w\in A^{*}$;

If $\langle A\mid R\rangle$ possesses a relation of the form $b=w$, where $b\in A$, and $w\in(A\backslash\{b\})^{*}$, then deleting $b$ from the list of generating symbols, deleting the relation $b=w$, and replacing all remaining appearances of $b$ by $w$.

The next result is also well-known (e.g. see [18, Proposition 3.2.5]):

Proposition 3.2

Two finite presentations define the same monoid if and only if one can be obtained from the other by applying a finite number of elementary Tietze transformations.

Another tool that we will use is given by the following proposition (e.g. see [14]):

Proposition 3.3

Let $M$ and $N$ be two monoids defined by the monoid presentations $\langle A\mid R\rangle$ and $\langle B\mid S\rangle$, respectively. Then the monoid presentation $\langle A,B\mid R,S,ab=ba,\,a\in A,b\in B\rangle$ defines the direct product $M\times T$.

Our strategy for obtainning a presentation for $\mathcal{POI}_{k\times m}$ will use well-known presentations of $\mathcal{S}_{m}$ and $\mathcal{POI}_{k}$.

First, we consider the following (monoid) presentation of $\mathcal{S}_{m}$, with $m+1$ relations in terms of the generators $a$ and $b$ defined in the previous section:

[TABLE]

(for instance, see [10]). From this presentation, applying Tietze transformations, we can easily deduce the following presentation for $\mathcal{S}_{m}$, also with $m+1$ relations, in terms of the generators $b$ and $c$ (also defined in the previous section):

[TABLE]

(recall that $c=ab$ and $a=cb^{m-1}$). Notice that $c^{m-1}=1$.

Next, we use these presentations of $\mathcal{S}_{m}$ for getting two presentations of $\mathcal{S}_{m}^{k}$.

Consider the alphabets $A=\{a_{i}\mid 1\leq i\leq k\}$ and $B=\{b_{i}\mid 1\leq i\leq k\}$ (with $k$ letters each) and the set $R$ formed by the following $2k^{2}+(m-1)k$ monoid relations:

($R_{1}$)

$a_{i}^{2}=1$, $1\leq i\leq k$;

($R_{2}$)

$b_{i}^{m}=1$, $1\leq i\leq k$ ;

($R_{3}$)

$(b_{i}a_{i})^{m-1}=1$, $1\leq i\leq k$;

($R_{4}$)

$(a_{i}b_{i}^{m-1}a_{i}b_{i})^{3}=1$, $1\leq i\leq k$;

($R_{5}$)

$(a_{i}b_{i}^{m-j}a_{i}b_{i}^{j})^{2}=1$, $2\leq j\leq m-2$, $1\leq i\leq k$;

($R_{6}$)

$a_{i}a_{j}=a_{j}a_{i}$, $b_{i}b_{j}=b_{j}b_{i}$, $1\leq i<j\leq k$;  $a_{i}b_{j}=b_{j}a_{i}$, $1\leq i,j\leq k$, $i\neq j$.

Then, by Proposition 3.3, the monoid $\mathcal{S}_{m}^{k}$ is defined by the presentation $\langle A,B\mid R\rangle$.

Now, consider the alphabet $C=\{c_{i}\mid 1\leq i\leq k\}$ (with $k$ letters) and the set $U$ formed by the following $2k^{2}+(m-1)k$ monoid relations:

($U_{1}$)

$(c_{i}b_{i}^{m-1})^{2}=1$, $1\leq i\leq k$;

($U_{2}$)

$b_{i}^{m}=1$, $1\leq i\leq k$ ;

($U_{3}$)

$(b_{i}c_{i}b_{i}^{m-1})^{m-1}=1$, $1\leq i\leq k$;

($U_{4}$)

$(c_{i}b_{i}^{m-2}c_{i})^{3}=1$, $1\leq i\leq k$;

($U_{5}$)

$(c_{i}b_{i}^{m-j-1}c_{i}b_{i}^{j-1})^{2}=1$, $2\leq j\leq m-2$, $1\leq i\leq k$;

($U_{6}$)

$b_{i}b_{j}=b_{j}b_{i}$, $c_{i}c_{j}=c_{j}c_{i}$, $1\leq i<j\leq k$;  $b_{i}c_{j}=c_{j}b_{i}$, $1\leq i,j\leq k$, $i\neq j$.

By Proposition 3.3, the monoid $\mathcal{S}_{m}^{k}$ is also defined by the presentation $\langle B,C\mid U\rangle$.

Let us also consider the $k$-letters alphabet $D=\{d_{i}\mid 1\leq i\leq k\}$. Recall that, as elements of $\mathcal{S}_{m}^{k}$, we have $d_{i}=b_{i}c_{i+1}$, for $1\leq i\leq k-1$, and $d_{k}=b_{k}c_{1}$. Moreover, $c_{1}=d_{k}^{m}$, $c_{i+1}=d_{i}^{m}$, for $1\leq i\leq k-1$, and $b_{i}=d_{i}^{(m-1)^{2}}$, for $1\leq i\leq k$. Also, notice that $d_{i}^{m(m-1)}=1$, whence $b_{i}^{m-1}=d_{i}^{(m-1)^{3}}=d_{i}^{m-1}$, for $1\leq i\leq k$. By applying Tietze transformations to the previous presentation, it is easy to check that $\mathcal{S}_{m}^{k}$ is also defined by the presentation $\langle D\mid V\rangle$, where $V$ is formed by the following $2k^{2}+(m-2)k$ monoid relations:

($V_{1}$)

$(d_{k}^{m}d_{1}^{m-1})^{2}=1$; $(d_{i}^{m}d_{i+1}^{m-1})^{2}=1$, $1\leq i\leq k-1$;

($V_{2}$)

$d_{i}^{m(m-1)}=1$, $1\leq i\leq k$;

($V_{3}$)

$(d_{1}^{(m-1)^{2}}d_{k}^{m}d_{1}^{m-1})^{m-1}=1$; $(d_{i+1}^{(m-1)^{2}}d_{i}^{m}d_{i+1}^{m-1})^{m-1}=1$, $1\leq i\leq k-1$;

($V_{4}$)

$(d_{k}^{m}d_{1}^{(m-1)^{2}(m-2)}d_{k}^{m})^{3}=1$; $(d_{i}^{m}d_{i+1}^{(m-1)^{2}(m-2)}d_{i}^{m})^{3}=1$, $1\leq i\leq k-1$;

($V_{5}$)

$(d_{k}^{m}d_{1}^{(m-1)^{2}(m-j-1)}d_{k}^{m}d_{1}^{(m-1)^{2}(j-1)})^{2}=1$, $2\leq j\leq m-2$;

$(d_{i}^{m}d_{i+1}^{(m-1)^{2}(m-j-1)}d_{i}^{m}d_{i+1}^{(m-1)^{2}(j-1)})^{2}=1$, $2\leq j\leq m-2$, $1\leq i\leq k-1$;

($V_{6}$)

$d_{i}^{m}d_{j}^{m}=d_{j}^{m}d_{i}^{m},d_{i}^{(m-1)^{2}}d_{j}^{(m-1)^{2}}=d_{j}^{(m-1)^{2}}d_{i}^{(m-1)^{2}}$, $1\leq i<j\leq k$;

$d_{i}^{(m-1)^{2}}d_{k}^{m}=d_{k}^{m}d_{i}^{(m-1)^{2}}$, $2\leq i\leq k-1$;  $d_{i}^{(m-1)^{2}}d_{j}^{m}=d_{j}^{m}d_{i}^{(m-1)^{2}}$, $1\leq i\leq k$, $1\leq j\leq k-1$, $i\not\in\{j,j+1\}$.

We move on to the monoid $\mathcal{POI}_{k}$. Let $X=\{x_{i}\mid 0\leq i\leq k-1\}$ be an alphabet (with $k$ letters). For $k\geq 2$, let $W$ be the set formed by the following $\frac{1}{2}(k^{2}+5k-4)$ monoid relations:

($W_{1}$)

$x_{i}x_{0}=x_{0}x_{i+1}$, $1\leq i\leq k-2$;

($W_{2}$)

$x_{j}x_{i}=x_{i}x_{j}$, $2\leq i+1<j\leq k-1$;

($W_{3}$)

$x_{0}^{2}x_{1}=x_{0}^{2}=x_{k-1}x_{0}^{2}$;

($W_{4}$)

$x_{i+1}x_{i}x_{i+1}=x_{i+1}x_{i}=x_{i}x_{i+1}x_{i}$, $1\leq i\leq k-2$;

($W_{5}$)

$x_{i}x_{i+1}\ldots x_{k-1}x_{0}x_{1}\ldots x_{i-1}x_{i}=x_{i}$, $0\leq i\leq k-1$;

($W_{6}$)

$x_{i+1}\ldots x_{k-1}x_{0}x_{1}\ldots x_{i-1}x_{i}^{2}=x_{i}^{2}$, $1\leq i\leq k-1$.

The presentation $\langle X\mid W\rangle$ defines the monoid $\mathcal{POI}_{k}$ (see [8] or [10]).

Finally, we define three sets of relations that envolve the letters from $X$ together with the previous alphabets considered. Foremost, let $R^{\prime}$ be the set formed by the following $2k^{2}+2k$ monoid relations over the alphabet $A\cup B\cup X$:

($R^{\prime}_{1}$)

$a_{1}x_{0}=x_{0}$, $b_{1}x_{0}=x_{0}$;

($R^{\prime}_{2}$)

$x_{0}a_{i}=a_{i+1}x_{0}$, $x_{0}b_{i}=b_{i+1}x_{0}$, $1\leq i\leq k-1$;

($R^{\prime}_{3}$)

$x_{i}a_{k-i}=x_{i}$, $x_{i}b_{k-i}=x_{i}$, $0\leq i\leq k-1$;

($R^{\prime}_{4}$)

$a_{i}x_{k-i+1}=x_{k-i+1}$, $b_{i}x_{k-i+1}=x_{k-i+1}$, $2\leq i\leq k$;

($R^{\prime}_{5}$)

$x_{i}a_{k-i+1}=a_{k-i}x_{i}$, $x_{i}b_{k-i+1}=b_{k-i}x_{i}$, $1\leq i\leq k-1$;

($R^{\prime}_{6}$)

$x_{i}a_{j}=a_{j}x_{i}$, $x_{i}b_{j}=b_{j}x_{i}$, $1\leq i\leq k-1$, $1\leq j\leq k$, $j\notin\{k-i,k-i+1\}$.

Secondly, consider the set $U^{\prime}$ formed by the following $2k^{2}+2k$ monoid relations over the alphabet $B\cup C\cup X$:

($U^{\prime}_{1}$)

$c_{1}b_{1}^{m-1}x_{0}=x_{0}$, $b_{1}x_{0}=x_{0}$;

($U^{\prime}_{2}$)

$x_{0}c_{i}b_{i}^{m-1}=c_{i+1}b_{i+1}^{m-1}x_{0}$, $x_{0}b_{i}=b_{i+1}x_{0}$, $1\leq i\leq k-1$;

($U^{\prime}_{3}$)

$x_{i}c_{k-i}b_{k-i}^{m-1}=x_{i}$, $x_{i}b_{k-i}=x_{i}$, $0\leq i\leq k-1$;

($U^{\prime}_{4}$)

$c_{i}b_{i}^{m-1}x_{k-i+1}=x_{k-i+1}$, $b_{i}x_{k-i+1}=x_{k-i+1}$, $2\leq i\leq k$;

($U^{\prime}_{5}$)

$x_{i}c_{k-i+1}b_{k-i+1}^{m-1}=c_{k-i}b_{k-i}^{m-1}x_{i}$, $x_{i}b_{k-i+1}=b_{k-i}x_{i}$, $1\leq i\leq k-1$;

($U^{\prime}_{6}$)

$x_{i}c_{j}b_{j}^{m-1}=c_{j}b_{j}^{m-1}x_{i}$, $x_{i}b_{j}=b_{j}x_{i}$, $1\leq i\leq k-1$, $1\leq j\leq k$, $j\notin\{k-i,k-i+1\}$.

Lastly, let $V^{\prime}$ be the set formed by the following $2k^{2}+2k$ monoid relations over the alphabet $D\cup X$:

($V^{\prime}_{1}$)

$d_{k}^{m}d_{1}^{m-1}x_{0}=x_{0}$, $d_{1}^{(m-1)^{2}}x_{0}=x_{0}$;

($V^{\prime}_{2}$)

$x_{0}d_{k}^{m}d_{1}^{m-1}=d_{1}^{m}d_{2}^{m-1}x_{0}$;  $x_{0}d_{i}^{m}d_{i+1}^{m-1}=d_{i+1}^{m}d_{i+2}^{m-1}x_{0}$, $1\leq i\leq k-2$;

$x_{0}d_{i}^{(m-1)^{2}}=d_{i+1}^{(m-1)^{2}}x_{0}$, $1\leq i\leq k-1$;

($V^{\prime}_{3}$)

$x_{i}d_{k-i-1}^{m}d_{k-i}^{m-1}=x_{i}$, $0\leq i\leq k-2$;  $x_{k-1}d_{k}^{m}d_{1}^{m-1}=x_{k-1}$; $x_{i}d_{k-i}^{{(m-1)}^{2}}=x_{i}$, $0\leq i\leq k-1$;

($V^{\prime}_{4}$)

$d_{i}^{m}d_{i+1}^{m-1}x_{k-i}=x_{k-i}$, $d_{i+1}^{(m-1)^{2}}x_{k-i}=x_{k-i}$, $1\leq i\leq k-1$;

($V^{\prime}_{5}$)

$x_{i}d_{k-i}^{m}d_{k-i+1}^{m-1}=d_{k-i-1}^{m}d_{k-i}^{m-1}x_{i}$, $1\leq i\leq k-2$;  $x_{k-1}d_{1}^{m}d_{2}^{m-1}=d_{k}^{m}d_{1}^{m-1}x_{k-1}$;

$x_{i}d_{k-i+1}^{(m-1)^{2}}=d_{k-i}^{(m-1)^{2}}x_{i}$, $1\leq i\leq k-1$;

($V^{\prime}_{6}$)

$x_{i}d_{k}^{m}d_{1}^{m-1}=d_{k}^{m}d_{1}^{m-1}x_{i}$, $1\leq i\leq k-2$;  $x_{i}d_{j}^{m}d_{j+1}^{m-1}=d_{j}^{m}d_{j+1}^{m-1}x_{i}$, $1\leq i,j\leq k-1$, $j\notin\{k-i-1,k-i\}$;

$x_{i}d_{j}^{(m-1)^{2}}=d_{j}^{(m-1)^{2}}x_{i}$, $1\leq i\leq k-1$, $1\leq j\leq k$, $j\notin\{k-i,k-i+1\}$.

Clearly, the presentations $\langle A\cup B\cup X\mid R\cup W\cup R^{\prime}\rangle$, $\langle B\cup C\cup X\mid U\cup W\cup U^{\prime}\rangle$ and $\langle D\cup X\mid V\cup W\cup V^{\prime}\rangle$ can be obtained from each other by applying a finite number of Tietze transformations. Therefore, by Proposition 3.2, they define the same monoid. We will prove that they define the monoid $\mathcal{POI}_{k\times m}$ by showing that $\langle A\cup B\cup X\mid R\cup W\cup R^{\prime}\rangle$ is a presentation for $\mathcal{POI}_{k\times m}$ in terms of its generators $\bar{A}\cup\bar{B}\cup\bar{X}$ defined in the previous section. The method described in Proposition 3.1 will be used.

Let $f:A\cup B\cup X\longrightarrow\mathcal{POI}_{k\times m}$ be the mapping defined by $a_{i}f=\bar{a}_{i}$, $b_{i}f=\bar{b}_{i}$ and $x_{j}f=\bar{x}_{j}$, $i\in\{1,2,\ldots,k\}$, $j\in\{0,1,\ldots,k-1\}$. Let $\varphi:{(A\cup B\cup X)}^{*}\longrightarrow\mathcal{POI}_{k\times m}$ be the natural homomorphism that extends $f$ to ${(A\cup B\cup X)}^{*}$.

It is a routine matter to prove the following lemma:

Lemma 3.4

The generating set $\bar{A}\cup\bar{B}\cup\bar{X}$ of $\mathcal{POI}_{k\times m}$ satisfies (via $\varphi$) all the relations from $R\cup W\cup R^{\prime}$.

Observe that, it follows from the previous lemma that $w_{1}\varphi=w_{2}\varphi$, for all $w_{1},w_{2}\in(A\cup B\cup X)^{*}$ such that $w_{1}=w_{2}$ is a consequence of $R\cup W\cup R^{\prime}$.

Lemma 3.5

Let $e\in A\cup B$ and $x\in X$. Then, there exists $f\in A\cup B\cup\{1\}$ such that the relation $ex=xf$ is a consequence of $R^{\prime}$.

• Proof.

The result follows immediately from relations $(R^{\prime}_{1})$ and $(R^{\prime}_{2})$, for $x=x_{0}$, and from relations $(R^{\prime}_{4})$, $(R^{\prime}_{5})$ and $(R^{\prime}_{6})$, for $x\in X\setminus\{x_{0}\}$.

Let us denote by $|w|$ the length of a word $w\in(A\cup B\cup X)^{*}$.

Lemma 3.6

For each $w\in{(A\cup B\cup X)}^{*}$ there exist $u\in X^{*}$ and $s\in(A\cup B)^{*}$ such that the relation $w=us$ is a consequence of $R^{\prime}$.

• Proof.

We will prove the lemma by induction on the length of $w\in{(A\cup B\cup X)}^{*}$.

Clearly, the result is trivial for any $w\in{(A\cup B\cup X)}^{*}$ such that $|w|\leq 1$.

Let $t\geq 2$ and, by induction hypothesis, admit the result for all $w\in{(A\cup B\cup X)}^{*}$ such that $|w|<t$.

Let $w\in{(A\cup B\cup X)}^{*}$ be such that $|w|=t$. Then, there exist $v\in{(A\cup B\cup X)}^{*}$ and $x\in A\cup B\cup X$ such that $w\equiv vx$ and $|v|=t-1$.

Since $|v|<t$, by induction hypothesis, there exist $u_{1}\in X^{*}$ and $s_{1}\in(A\cup B)^{*}$ such that the relation $v=u_{1}s_{1}$ is a consequence of $R^{\prime}$. Hence the relation $w=u_{1}s_{1}x$ is also a consequence of $R^{\prime}$.

If $|s_{1}|=0$ or $x\in A\cup B$ then the result is proved.

So, suppose that $|s_{1}|\geq 1$ and $x\in X$. Let $s^{\prime}\in(A\cup B)^{*}$ and $e\in A\cup B$ be such that $s_{1}\equiv s^{\prime}e$. By Lemma 3.5, there exists $f\in A\cup B\cup\{1\}$ such that the relation $ex=xf$ is a consequence of $R^{\prime}$. On the other hand, since $|s^{\prime}|<|s_{1}|\leq|v|<t$, we have $|s^{\prime}x|<t$ and so, by induction hypothesis, there exist $u_{2}\in X^{*}$ and $s_{2}\in(A\cup B)^{*}$ such that the relation $s^{\prime}x=u_{2}s_{2}$ is a consequence of $R^{\prime}$. Thus, the relation $w=u_{1}u_{2}s_{2}f$, where $u_{1}u_{2}\in X^{*}$ and $s_{2}f\in(A\cup B)^{*}$, is also a consequence of $R^{\prime}$, as required.

Lemma 3.7

For all $j\in\{1,2,\ldots,k\}$ and $u\in X^{*}$ such that $I_{j}\nsubseteq\mathop{\mathrm{Im}}\nolimits(u\varphi)$, the relations $ua_{j}=u$ and $ub_{j}=u$ are consequences of $R^{\prime}$.

• Proof.

We will prove the lemma for relations of the form $ua_{j}=u$, with $1\leq j\leq k$, by induction on the length of $u\in X^{*}$. For relations of the form $ub_{j}=u$, with $1\leq j\leq k$, the proof is analogous.

Let $j\in\{1,2,\ldots,k\}$ and let $u\in X^{*}$ be such that $I_{j}\nsubseteq\mathop{\mathrm{Im}}\nolimits(u\varphi)$ and $|u|=1$. Then $u\equiv x_{i}$, for some $0\leq i\leq k-1$. As $I_{j}\nsubseteq\mathop{\mathrm{Im}}\nolimits(x_{i}\varphi)=\mathop{\mathrm{Im}}\nolimits(\bar{x}_{i})$, by definition of $\bar{x}_{i}$, we have $j=k-i$. Hence, the relation $ua_{j}=u$, i.e. $x_{i}a_{k-i}=x_{i}$, is one of the relations $(R^{\prime}_{3})$.

Next, let $t\geq 1$ and, by induction hypothesis, admit that for all $j\in\{1,2,\ldots,k\}$ and $u\in X^{*}$ such that $I_{j}\nsubseteq\mathop{\mathrm{Im}}\nolimits(u\varphi)$ and $|u|=t$, the relation $ua_{j}=u$ is a consequence of $R^{\prime}$.

Let $j\in\{1,2,\ldots,k\}$ and let $u\in X^{*}$ be such that $I_{j}\nsubseteq\mathop{\mathrm{Im}}\nolimits(u\varphi)$ and $|u|=t+1$. Let $\bar{u}=u\varphi$.

Take $v\in X^{*}$ and $i\in\{0,1,\ldots,k-1\}$ such that $u\equiv vx_{i}$. Observe that $|v|=t$. Let $\bar{v}=v\varphi$. Hence $\bar{u}=\bar{v}\bar{x}_{i}$.

First, consider $i=0$. If $j=k$ then $x_{0}a_{k}=x_{0}$ is a relation from $(R^{\prime}_{3})$ and so $vx_{0}a_{k}=vx_{0}$, i.e. $ua_{j}=u$, is a consequence of $R^{\prime}$. On the other hand, suppose that $1\leq j\leq k-1$. If $I_{j+1}\subseteq\mathop{\mathrm{Im}}\nolimits(\bar{v})$ then, as $I_{j+1}\bar{x}_{0}=I_{j}$, we obtain $I_{j}\subseteq\mathop{\mathrm{Im}}\nolimits(\bar{v}\bar{x}_{0})=\mathop{\mathrm{Im}}\nolimits(\bar{u})$, which is a contradiction. Hence, $I_{j+1}\nsubseteq\mathop{\mathrm{Im}}\nolimits(\bar{v})$ and so, by induction hypothesis, the relation $va_{j+1}=v$ is a consequence of $R^{\prime}$. Since $x_{0}a_{j}=a_{j+1}x_{0}$ is a relation from $(R^{\prime}_{2})$, we deduce $ua_{j}\equiv vx_{0}a_{j}=va_{j+1}x_{0}=vx_{0}\equiv u$, as a consequence of $R^{\prime}$.

Now, suppose that $1\leq i\leq k-1$.

If $j=k-i$ then $x_{i}a_{j}=x_{i}$ is a relation from $(R^{\prime}_{3})$ and so $vx_{i}a_{j}=vx_{i}$, i.e. $ua_{j}=u$, is a consequence of $R^{\prime}$.

Next, let $j=k-i+1$. If $I_{j-1}\subseteq\mathop{\mathrm{Im}}\nolimits(\bar{v})$ then, as $I_{j-1}\bar{x}_{i}=I_{k-i}\bar{x}_{i}=I_{j}$, we have $I_{j}\subseteq\mathop{\mathrm{Im}}\nolimits(\bar{v}\bar{x}_{i})=\mathop{\mathrm{Im}}\nolimits(\bar{u})$, which is a contradiction. Hence, $I_{j-1}\nsubseteq\mathop{\mathrm{Im}}\nolimits(\bar{v})$ and so, by induction hypothesis, the relation $va_{j-1}=v$ is a consequence of $R^{\prime}$. Since $x_{i}a_{j}=a_{j-1}x_{i}$ is a relation from $(R^{\prime}_{5})$, we obtain $ua_{j}\equiv vx_{i}a_{j}=va_{j-1}x_{i}=vx_{i}\equiv u$, as a consequence of $R^{\prime}$.

Finally, admit that $j\not\in\{k-i,k-i+1\}$. If $I_{j}\subseteq\mathop{\mathrm{Im}}\nolimits(\bar{v})$ then, as $I_{j}\bar{x}_{i}=I_{j}$, we get $I_{j}\subseteq\mathop{\mathrm{Im}}\nolimits(\bar{v}\bar{x}_{i})=\mathop{\mathrm{Im}}\nolimits(\bar{u})$, which is a contradiction. Hence, $I_{j}\nsubseteq\mathop{\mathrm{Im}}\nolimits(\bar{v})$ and so, by induction hypothesis, the relation $va_{j}=v$ is a consequence of $R^{\prime}$ and so $ua_{j}\equiv vx_{i}a_{j}=va_{j}x_{i}=vx_{i}\equiv u$ are also consequences of $R^{\prime}$, as required.

Lemma 3.8

For each $w\in{(A\cup B\cup X)}^{*}$ there exist $u\in X^{*}$ and $s\in(A\cup B)^{*}$ such that the relation $w=us$ is a consequence of $R\cup R^{\prime}$ and $\ell(s\varphi)=\ell$, for all $\ell\in\Omega_{n}\setminus\mathop{\mathrm{Im}}\nolimits(u\varphi)$.

• Proof.

Let $u\in X^{*}$ and $s_{0}\in(A\cup B)^{*}$ be such that the relation $w=us_{0}$ is a consequence of $R^{\prime}$ (by applying Lemma 3.6). Hence, we may take $s\in(A\cup B)^{*}$ such that $w=us$ is a consequence of $R\cup R^{\prime}$ and $s$ has minimum length among all $s^{\prime}\in(A\cup B)^{*}$ such that $w=us^{\prime}$ is a consequence of $R\cup R^{\prime}$.

Let $\ell\in\Omega_{n}\setminus\mathop{\mathrm{Im}}\nolimits(u\varphi)$. Then $\ell\in I_{j}$, for some $1\leq j\leq k$.

Suppose that $a_{j}$ or $b_{j}$ occur in $s$. Let $s_{1}\in((A\cup B)\setminus\{a_{j},b_{j}\})^{*}$, $y\in\{a_{j},b_{j}\}$ and $s_{2}\in(A\cup B)^{*}$ be such that $s\equiv s_{1}ys_{2}$. Then, clearly, the relation $s_{1}y=ys_{1}$ is a consequence of relations from $(R_{6})$ and so the relation $s=ys_{1}s_{2}$ is a consequence of $R$. On the other hand, as $I_{j}\nsubseteq\mathop{\mathrm{Im}}\nolimits(u\varphi)$, by Lemma 3.7, the relation $uy=u$ is a consequence of $R^{\prime}$, whence the relation $w=us_{1}s_{2}$ is a consequence of $R\cup R^{\prime}$, $s_{1}s_{2}\in(A\cup B)^{*}$ and $|s_{1}s_{2}|=|s|-1$, which is a contradiction.

Therefore, $a_{j}$ and $b_{j}$ do not occur in $s$ and so the restriction of $s\varphi$ to $I_{j}$ is the identity of $I_{j}$. In particular, $\ell(s\varphi)=\ell$, as required.

Let $u\in X^{*}$ and $s\in(A\cup B)^{*}$. Notice that, as $s\varphi\in\mathcal{G}_{k\times m}$, then $I_{j}(s\varphi)=I_{j}$, for all $1\leq j\leq k$, and so $\mathop{\mathrm{Dom}}\nolimits((us)\varphi)=\mathop{\mathrm{Dom}}\nolimits(u\varphi)$ and $\mathop{\mathrm{Im}}\nolimits((us)\varphi)=\mathop{\mathrm{Im}}\nolimits(u\varphi)$.

We are now in a position to prove our last lemma.

Lemma 3.9

Let $w_{1},w_{2}\in(A\cup B\cup X)^{*}$. If $w_{1}\varphi=w_{2}\varphi$ then $w_{1}=w_{2}$ is a consequence of $R\cup W\cup R^{\prime}$.

• Proof.

By Lemma 3.8 we can consider $u_{1},u_{2}\in X^{*}$ and $s_{1},s_{2}\in(A\cup B)^{*}$ such that the relations $w_{1}=u_{1}s_{1}$ and $w_{2}=u_{2}s_{2}$ are consequences of $R\cup R^{\prime}$, $\ell(s_{1}\varphi)=\ell$, for all $\ell\in\Omega_{n}\setminus\mathop{\mathrm{Im}}\nolimits(u_{1}\varphi)$, and $\ell(s_{2}\varphi)=\ell$, for all $\ell\in\Omega_{n}\setminus\mathop{\mathrm{Im}}\nolimits(u_{2}\varphi)$.

Observe that $\mathop{\mathrm{Dom}}\nolimits(u_{1}\varphi)=\mathop{\mathrm{Dom}}\nolimits((u_{1}s_{1})\varphi)=\mathop{\mathrm{Dom}}\nolimits(w_{1}\varphi)=\mathop{\mathrm{Dom}}\nolimits(w_{2}\varphi)=\mathop{\mathrm{Dom}}\nolimits((u_{2}s_{2})\varphi)=\mathop{\mathrm{Dom}}\nolimits(u_{2}\varphi)$ and $\mathop{\mathrm{Im}}\nolimits(u_{1}\varphi)=\mathop{\mathrm{Im}}\nolimits((u_{1}s_{1})\varphi)=\mathop{\mathrm{Im}}\nolimits(w_{1}\varphi)=\mathop{\mathrm{Im}}\nolimits(w_{2}\varphi)=\mathop{\mathrm{Im}}\nolimits((u_{2}s_{2})\varphi)=\mathop{\mathrm{Im}}\nolimits(u_{2}\varphi).$ Since $u_{1}\varphi,u_{2}\varphi\in\mathcal{POI}_{k\times m}\cap\mathcal{POI}_{n}$, it follows that $u_{1}\varphi=u_{2}\varphi$. On the other hand, $\mathcal{POI}_{k\times m}\cap\mathcal{POI}_{n}\simeq\mathcal{POI}_{k}$ and the monoid $\mathcal{POI}_{k}$ is defined by the presentation $\langle B\mid W\rangle$. Therefore, the relation $u_{1}=u_{2}$ is a consequence of $W$.

Next, we turn our attention to $s_{1}\varphi,s_{2}\varphi\in\mathcal{G}_{k\times m}$. Let $\ell\in\Omega_{n}\setminus\mathop{\mathrm{Im}}\nolimits(u_{1}\varphi)=\Omega_{n}\setminus\mathop{\mathrm{Im}}\nolimits(u_{2}\varphi)$. Then $\ell(s_{1}\varphi)=\ell=\ell(s_{2}\varphi)$. On the other hand, let $\ell\in\mathop{\mathrm{Im}}\nolimits(u_{1}\varphi)=\mathop{\mathrm{Im}}\nolimits(u_{2}\varphi)$. Take $t\in\Omega_{n}$ such that $t(u_{1}\varphi)=\ell$. Then $\ell(s_{1}\varphi)=(t(u_{1}\varphi))(s_{1}\varphi)=t((u_{1}\varphi)(s_{1}\varphi))=t((u_{1}s_{1})\varphi)=t(w_{1}\varphi)=t(w_{2}\varphi)=t((u_{2}s_{2})\varphi)=t((u_{2}\varphi)(s_{2}\varphi))=(t(u_{2}\varphi))(s_{2}\varphi)=\ell(s_{2}\varphi)$. Hence $s_{1}\varphi=s_{2}\varphi$. Since $\mathcal{G}_{k\times m}\simeq\mathcal{S}_{m}^{k}$ and $\mathcal{S}_{m}^{k}$ is defined by the presentation $\langle A,B\mid R\rangle$, it follows that the relation $s_{1}=s_{2}$ is a consequence of $R$.

Thus, the relation $u_{1}s_{1}=u_{2}s_{2}$ is a consequence of $R\cup W$ and so the relation $w_{1}=w_{2}$ is a consequence of $R\cup W\cup R^{\prime}$, as required.

In view of Proposition 3.1, it follows immediately from Lemmas 3.4 and 3.9:

Theorem 3.10

For $k,m\geq 2$, the monoid $\mathcal{POI}_{k\times m}$ is defined by the presentation $\langle A\cup B\cup X\mid R\cup W\cup R^{\prime}\rangle$ on $3k$ generators and $\frac{1}{2}(9k^{2}+(2m+7)k-4)$ relations.

In view of Proposition 3.2, as corollaries of the previous theorem, we also have:

Theorem 3.11

For $k,m\geq 2$, the monoid $\mathcal{POI}_{k\times m}$ is defined by the presentation $\langle B\cup C\cup X\mid U\cup W\cup U^{\prime}\rangle$ on $3k$ generators and $\frac{1}{2}(9k^{2}+(2m+7)k-4)$ relations.

Theorem 3.12

For $k,m\geq 2$, the monoid $\mathcal{POI}_{k\times m}$ is defined by the presentation $\langle D\cup X\mid V\cup W\cup V^{\prime}\rangle$ on $2k$ generators and $\frac{1}{2}\left(9k^{2}+(2m+5)k-4\right)$ relations.