# Product decompositions of semigroups induced by action pairs

###### Abstract

This paper concerns a class of semigroups that arise as products $US$,
associated to what we call ‘action pairs’. Here $U$ and $S$ are subsemigroups
of a common monoid and, roughly speaking, $S$ has an action on the monoid
completion $U^{1}$ that is suitably compatible with the product in the over-
monoid.

The semigroups encapsulated by the action pair construction include many
natural classes such as inverse semigroups and (left) restriction semigroups,
as well as many important concrete examples such as transformational wreath
products, linear monoids, (partial) endomorphism monoids of independence
algebras, and the singular ideals of many of these. Action pairs provide a
unified framework for systematically studying such semigroups, within which we
build a suite of tools to ensure a comprehensive understanding of them. We
then apply our abstract results to many special cases of interest.

The first part of the paper constitutes a detailed structural analysis of
semigroups arising from action pairs. We show that any such semigroup $US$ is
a quotient of a semidirect product $U\rtimes S$, and we classify all
congruences on semidirect products that correspond to action pairs. We also
prove several covering and embedding theorems, each of which naturally extends
celebrated results of McAlister on proper (a.k.a. $E$-unitary) inverse
semigroups.

The second part of the paper concerns presentations by generators and
relations for semigroups arising from action pairs. We develop a substantial
body of general results and techniques that allow us to build presentations
for $US$ out of presentations for the constituents $U$ and $S$ in many cases,
and then apply these to several examples, including those listed above. Due to
the broad applicability of the action pair construction, many results in the
literature are special cases of our more general ones.

_Keywords_ : Actions; Semidirect products; Covers; Embeddings; Presentations;
Inverse semigroups; Left restriction semigroups; Independence algebras;
Endomorphism monoids; Singular endomorphism semigroups; Transformation
semigroups; Wreath products.

MSC: 20M10, 20M05, 20M20, 20M30, 08A05, 08A30, 08A35.

Scott Carson,111Department of Mathematics, University of York, York YO10 5DD,
U.K. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>Igor
Dolinka,222Department of Mathematics and Informatics, University of Novi Sad,
Trg Dositeja Obradovića 4, 21101 Novi Sad, Serbia. Email<EMAIL_ADDRESS>James East,333Centre for Research in Mathematics and Data Science, Western
Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia. Email:
<EMAIL_ADDRESS>Victoria Gould,1 Rida-e Zenab444Department of
Mathematics, Sukkur I.B.A. University, Pakistan. Email: ridaezenab@iba-
suk.edu.pk

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Semigroups
    2. 2.2 Transformation semigroups
    3. 2.3 Left restriction semigroups
    4. 2.4 Presentations
  3. I Structure
    1. 3 Actions and semidirect products
      1. 3.1 Actions
      2. 3.2 Semidirect products
    2. 4 Action pairs
      1. 4.1 Basic definitions and properties
      2. 4.2 Examples
      3. 4.3 Left restriction monoids
      4. 4.4 The first structure theorem
      5. 4.5 Congruence conditions
    3. 5 Proper action pairs
      1. 5.1 Background
      2. 5.2 From actions to action pairs
      3. 5.3 Proper action pairs and a covering theorem
      4. 5.4 Embedding theorems for proper monoids I
      5. 5.5 Embedding theorems for proper monoids II
      6. 5.6 Embedding theorems for proper monoids III
      7. 5.7 Embedding theorems for proper monoids IV
  4. II Presentations
    1. 6 Presentations
      1. 6.1 Two submonoids I
      2. 6.2 Two submonoids II
      3. 6.3 Two submonoids III
      4. 6.4 One submonoid I
      5. 6.5 One submonoid II
    2. 7 Free left restriction monoids
      1. 7.1 Preliminaries
      2. 7.2 Presentations
    3. 8 Independence algebras
      1. 8.1 Preliminaries
      2. 8.2 (Partial) endomorphisms
      3. 8.3 Maximal subalgebras
      4. 8.4 Finite-codimensional subalgebras
      5. 8.5 Generating (finitary) automorphisms
      6. 8.6 Action pairs and subsemigroups
      7. 8.7 Congruence conditions
      8. 8.8 Presentations
    4. 9 Wreath products and free acts
      1. 9.1 Preliminaries
      2. 9.2 Action pairs and subsemigroups I
      3. 9.3 Congruence conditions I
      4. 9.4 Presentations I
      5. 9.5 Action pairs and subsemigroups II
      6. 9.6 Congruence conditions II
      7. 9.7 Presentations II

###  1 Introduction

The motivation for the current work comes from many different directions. The
basic philosophical underpinning is the desire to understand mathematical
structures by sets of data that are sufficient to determine the entire
structure, yet are intrinsically simpler. In particular, we draw inspiration
from classical techniques and constructions in group theory, and seek to
understand the (much) more complicated and wider context of these techniques
and constructions within the realm of semigroups and monoids.

One approach is to use _presentations_. Here the idea is to define an
algebraic structure—often, and originally, a group—in terms of generators and
relations between them. It is possible in this way to capture some infinite
groups using a finite amount of information: i.e., via a finite number of
generators and a finite number of relations, resulting in a finitely presented
group. Important examples include the Artin braid groups [5, 4] and Thompson’s
groups $F$, $T$ and $V$ [13]. On the other hand, it is also useful to
determine finite groups via (finite) presentations. For example, the well-
known Coxeter presentation of a finite symmetric group [92] tells us how to
calculate all products by dint of the products between certain transpositions;
cf. [66]. Group presentations were introduced as far back as the mid-
nineteenth century [59, 23], and have since been central in approaches to
algebraic and differential topology, geometric group theory, algebraic
combinatorics, representation theory and many other branches of mathematics.
Semigroup presentations go back to the mid-twentieth century, and again have
proven a crucial tool in the above fields. For some early studies, see for
example [2, 1, 100, 101], and for early papers with connections to logic see
[116, 84, 102]; some historical information can be found in [97].

A second approach to the above philosophy involves the concept of _products_ ,
which covers a wide range of mathematical constructions prominently featuring
in the areas of algebra, topology, category theory, mathematical logic and
graph theory, just to name a few. The idea behind the introduction and use of
such constructions is two-fold. Firstly, one of the most basic tasks of almost
any coherent mathematical field or theory is to describe the many ways in
which the objects of study are constructed; in many cases, the philosophy is
to go from smaller, simpler structures towards larger, more complicated ones.
This is exactly how various types of products permeate mathematics: think of
direct products in general algebra, semidirect, wreath, and free (amalgamated)
products in group theory, product topologies, tensor products in linear
algebra, ultraproducts in model theory, cap and cup products in algebraic
topology, various types of graph products, pullbacks in category theory, and
so on. However, there is also the reverse task: to understand the structure of
a given mathematical object in as much detail as possible, one attempts to
break it down into more elementary parts and describe the mutual relationships
between the parts. Quite often, this happens by recognising that the given
structure is isomorphic (or in some other way ‘equivalent’ or ‘similar’) to a
product of some kind. Smaller structures obtained as a result of such a
decomposition are often expected or required to be _sub_ structures of the
original, but not always.

A flagship instance of the phenomenon just described is the decomposition of
finite abelian groups into the direct product of cyclic (sub)groups of prime
power order. For groups, and indeed monoids, there is an exact correlation
between internal and external direct products. However, as soon as one moves
away from monoids, even just as far as semigroups, this correlation is lost.
In particular, an external direct product $S\times T$ of semigroups $S$ and
$T$ need not contain subsemigroups isomorphic to either $S$ or $T$, yet
clearly $S\times T$ is determined by $S$ and $T$.

Another natural class of examples illustrating the same phenomenon—but where
the smaller parts are not necessarily substructures—are the _almost-
factorisable inverse semigroups_ of Lawson [76]. A semigroup $S$ from this
class decomposes as a product $S=EG$, where $G$ is the group of units of some
_different_ inverse monoid $M$, and where $E$ is the semilattice of singular
(i.e., non-identity) idempotents of $M$. Here we have $E\subseteq S$, yet
$G\not\subseteq S$; in fact, we even have $G\cap S=\varnothing$. Nevertheless,
the structure of $S$ is closely governed by that of the constituents $E$ and
$G$, and in fact $S$ is an idempotent-separating homomorphic image of a
semidirect product $E\rtimes G$.

Yet another natural example is the semigroup $S$ of singular (i.e., non-
bijective) partial transformations of a finite set $X$. This semigroup can be
decomposed as $S=ET$, where $E$ is the (monoid) semilattice of all partial
identities, and $T$ is the semigroup of singular (full) transformations of
$X$. This time we have $T\subseteq S$ but $E\not\subseteq S$, although we do
not have the extreme $E\cap S=\varnothing$; rather, $E\setminus S$ consists
only of the identity map. _Many_ more such examples exist.

The main goal of the current study is to provide a unified framework for
working with a large class of structures including the kind just described;
presentations will form an important set of tools. The central notion
throughout is that of an _action pair_ $(U,S)$. Such a pair consists of
subsemigroups $U$ and $S$ of a common monoid $M$, such that $S$ acts on the
monoid completion $U^{1}=U\cup\\{1\\}$ in a way that is suitably compatible
with the product in $M$. The formal definitions are given later in the paper,
including various other conditions that are required. But it is worth noting
that ‘compatibility’ here means that

$s\cdot u={}^{s}u\cdot s\qquad\text{for all $s\in S$ and $u\in U^{1}$.}$

(In the above expression, $\cdot$ is the product in $M$, and the action of $S$
on $U^{1}$ is denoted by $(s,u)\mapsto{}^{s}u$.) It follows quickly from this
that the set product

$US=\\{{us}:{u\in U,\ s\in S}\\}$

is a subsemigroup of $M$. It is important to note that $US$ might contain
neither $U$ nor $S$, nor even isomorphic copies of either. On the other hand,
if $U$ and $S$ are both submonoids of $M$, then $US$ contains both $U$ and $S$
as submonoids.

An important natural class of examples extending those discussed above come
from (subsemigroups of) inverse semigroups [77], and more generally left
restriction semigroups [56] . If $S$ is an inverse semigroup with semilattice
of idempotents $E$, then $(E,S)$ is an action pair in the monoid completion
$S^{1}$, with respect to the conjugation action: ${}^{s}e=ses^{-1}$ for $e\in
E^{1}$ and $s\in S$. More generally, $(F,T)$ is an action pair for any
subsemigroup $T\leq S$ and any subsemilattice $F\leq E$ for which $F^{1}$ is
closed under conjugation by elements of $T$. (Almost-)factorisable inverse
monoids and semigroups [14, 76] are (very) special cases of this. Other
similar examples involve products of idempotents and one- or two-sided units
of arbitrary monoids [35]. Many other examples are considered throughout the
paper, some with very different behaviour to those discussed so far. The class
of semigroups arising from action pairs is extremely rich, and part of our
motivation stems from the desire to obtain a comprehensive understanding of
such semigroups, analogous to that of more specialised classes such as inverse
and (left) restriction semigroups.

As well as this introduction, and the preliminary Chapter 2, the paper
consists of two main parts:

  * •

Part I constitutes a detailed structural analysis of semigroups arising from
action pairs, and

  * •

Part II concerns presentations by generators and relations for semigroups in
this class.

We now give a very brief overview of the main highlights of the paper. More
detailed summaries can be found in the opening remarks of both parts, and in
the introductions to the chapters within.

As we have just indicated, the first part of the paper is devoted to analysing
the structure of a semigroup $US$ arising from an action pair $(U,S)$. Chapter
3 contains a number of preliminary results on actions and semidirect products
that will be used extensively throughout the rest of the paper. Our notion of
semidirect product is slightly more general than the usual one, which is
essential to capture the widest possible collection of applications. Chapter 4
introduces action pairs, provides an extensive collection of examples (and
non-examples for contrast), and contains the first main structural result:

  * •

Theorem 4.53 shows that $US$ is a homomorphic image of a semidirect product
$U\rtimes S$, and classifies all congruences on semidirect products that
correspond to action pairs.

Chapter 5 concerns the special class of _proper_ action pairs; these extend
the notion of proper (a.k.a. $E$-unitary) inverse semigroups, and more
generally proper left restriction semigroups. The main results of this chapter
are broad generalisations of McAlister’s celebrated Covering Theorem [86] and
P-Theorem [87] for inverse semigroups. These new results encapsulate many
extensions of McAlister’s theorems to other more specialised classes:

  * •

Theorem 5.39 shows that $US$ has a _proper cover_ , by which we mean that
there is a proper action pair $(U^{\prime},S^{\prime})$ in a suitable monoid,
with ${U^{\prime}\cong U}$ and $S^{\prime}\cong S$, and a natural surmorphism
${U^{\prime}S^{\prime}\to US}$. Further, if $S$ or $U$ is a submonoid of the
monoid containing $US$, then the surmorphism ${U^{\prime}S^{\prime}\to US}$
restricts to an isomorphism $U^{\prime}\to U$ or $S^{\prime}\to S$,
respectively.

  * •

Theorem 5.60 shows that a proper _monoid_ $US$ embeds in a semidirect product
${\mathcal{U}\rtimes(S/\sigma)}$, where $\mathcal{U}$ is a monoid containing
$U$, and where $\sigma$ is a special congruence on $S$ akin to the least group
congruence on an inverse semigroup. Theorems 5.47, 5.82 and 5.102 are
variations on this result, and show that when $U$ has certain additional
structural properties (such as being commutative, a semilattice or a left-
regular band), then an embedding $US\to\mathcal{U}\rtimes(S/\sigma)$ of the
above kind exists where $\mathcal{U}$ also has the relevant structural
property.

As applications, we give new proofs of the generalisations of McAlister’s
above-mentioned theorems to the class of left restriction semigroups [98, 75,
47, 9].

In the second part of the paper we turn our attention to presentations by
generators and relations. As discussed above, presentations are extremely
important tools when working with any kind of algebraic structure, and several
results exist for building presentations for algebras that arise from others
via natural constructions. See for example [106, 74, 24, 41, 105, 65, 21,
107]; the introduction to [36] contains a fuller discussion and many more
references. Chapter 6 contains many general results on presentations for a
semigroup $US$ arising from an action pair $(U,S)$ in several important cases:

  * •

Theorems 6.4, 6.12, 6.27, 6.29 and 6.31 concern the case in which $U$ and $S$
are both submonoids of the over-monoid. As we will see, one complication that
must be overcome is the fact that the semidirect product $U\rtimes S$ need not
be a monoid in this case.

  * •

Theorems 6.35, 6.43 and 6.49 only assume that one of $U$ or $S$ is a
submonoid. Other conditions must be satisfied in these cases in order to
obtain usable results.

The general machinery developed in Chapter 6 is then applied to a number of
important examples in the remaining chapters:

  * •

Free left restriction monoids are treated in Chapter 7; see Theorem 7.13.

  * •

Chapter 8 concerns several monoids and semigroups of (partial) endomorphisms
of an independence algebra. See especially Theorem 8.40, which gives a
presentation for the $\cap$-semilattice of finite-codimensional subalgebras of
an arbitrary strong independence algebra.

  * •

A number of transformational wreath products are covered in Chapter 9, the
main results here being Theorems 9.33, 9.58 and 9.67–9.69.

The potential for further applications is vast.

The article also includes several other results of independent interest. For
example:

  * •

In Section 3.2 we prove a number of results on (our more general) semidirect
products, including the fact that a semidirect product $U\rtimes S$ is a
monoid if and only if $U$ and $S$ are both monoids, with $S$ acting monoidally
on $U$ by monoid morphisms; see Corollary 3.13.

  * •

Theorem 6.49 gives a presentation for a semidirect product $U\rtimes S$ in the
case that $S$ is a monoid and $U$ an arbitrary semigroup. This complements
[41, Theorem 3.1], which treats the reverse case, where $U$ is a monoid and
$S$ a semigroup. By contrast, [74, Corollary 2] gives a presentation for
$U\rtimes S$ when it is a monoid (which occurs when the conditions discussed
in the previous point hold). Our Theorem 6.29 extends this result, by relaxing
the assumption that $S$ acts by _monoid_ morphisms, and applies to the largest
submonoid of $U\rtimes S$ with identity $(1,1)$.

  * •

In Section 7.2 we show that the free left restriction monoid over any set has
a unique minimum (monoid) generating set; see Proposition 7.11, Theorem 7.13
and Remark 7.14.

  * •

In Section 8.3, we prove several results on independence algebras with
suitably large subalgebras, and show that these are precisely the algebras
displaying the most ‘freedom’ in (partial) automorphisms; see especially
Propositions 8.26–8.30.

  * •

In Section 8.5, we give a number of results on generating sets for groups of
(finitary) automorphisms of independence algebras, which provide a common
generalisation of the following two facts: (i) finite symmetric groups are
generated by transpositions, and (ii) finite-dimensional general linear groups
are generated by ‘elementary row operation’ matrices. See Theorems 8.47 and
8.50.

Throughout the text we pose a number of open problems and enticing directions
for future studies.

#### Acknowledgements

The first author was supported by an EPSRC doctoral fellowship. The second
author is partially supported by the Ministry of Education, Science, and
Technological Development of the Republic of Serbia. The third author is
supported by Future Fellowship FT190100632 of the Australian Research Council.

###  2 Preliminaries

This article is intended to be largely self-contained, and accessible to a
broad audience of algebraists. Accordingly, we now gather the basic
definitions and facts concerning semigroups and presentations we need, and fix
most of the notation we will be using. For more background on semigroups, and
for proofs of the various assertions below, see for example [64, 62, 16]. For
inverse semigroups in particular, see [77] or [64, Chapter 5]; for (left)
restriction semigroups, see [56].

####  2.1 Semigroups

A _semigroup_ is a set with an associative binary operation, typically denoted
by juxtaposition. A _monoid_ is a semigroup with an identity element. Unless
otherwise specified, the identity of any monoid is denoted by $1$.

A _semigroup morphism_ is a map $\phi:S\to T$, where $S$ and $T$ are
semigroups, and where $(xy)\phi=(x\phi)(y\phi)$ for all $x,y\in S$. A monoid
morphism is a semigroup morphism between monoids that additionally maps the
identity to the identity.

If $S$ is a semigroup, then $S^{1}$ denotes the _monoid completion_ of $S$.
Formally:

  * •

$S^{1}=S$ if $S$ is already a monoid;

  * •

otherwise, $S^{1}=S\sqcup\\{1\\}$, where $1$ is a symbol not belonging to $S$,
and acting as an identity for $S^{1}$. (Here and elsewhere $\sqcup$ denotes
disjoint union.)

If $S$ happens to be a subsemigroup of a specified monoid $M$, then we often
assume the identity of $S^{1}$ is the same as that of $M$; we will always be
clear about this.

An element $x$ of a semigroup $S$ is _(von Neumann) regular_ if $x=xax$ for
some $a\in S$. For $b=axa$, we then have $x=xbx$ and $b=bxb$. A semigroup is
_(von Neumann) regular_ if all its elements are regular. An element $x\in S$
is an _idempotent_ if $x=x^{2}$. We generally write $E(S)$ for the set of all
idempotents of $S$. Clearly any idempotent is regular. A semigroup $S$ is a
_band_ if $S=E(S)$: i.e., if every element of $S$ is an idempotent. A
_semilattice_ is a commutative band. A band $S$ is _left-regular_ if $xyx=xy$
for all $x,y\in S$; right-regular bands are defined symmetrically, but they
will not arise in our investigations. Left-regular bands are of course
generalisations of semilattices, but they are important for many reasons, one
of which is that they include many semigroups of geometric significance [82,
81].

A semigroup $S$ is _inverse_ if for every $x\in S$ there is a _unique_ element
$a\in S$ such that $x=xax$ and $a=axa$; this element is then denoted
$a=x^{-1}$. Equivalently, a semigroup $S$ is inverse if it is regular and its
idempotents commute; the set $E(S)$ is therefore a semilattice. Inverse
semigroups can also be defined as a variety of _unary semigroups_ (i.e.,
semigroups equipped with an additional basic unary operation); they are
precisely the class of semigroups with a unary operation $x\mapsto x^{-1}$
satisfying the identities

$(x^{-1})^{-1}=x=xx^{-1}x,\qquad(xy)^{-1}=y^{-1}x^{-1}\qquad\text{and}\qquad
xx^{-1}yy^{-1}=yy^{-1}xx^{-1}.$

A relation $\sigma$ on a semigroup $S$ is _left-compatible_ if
$(x,y)\in\sigma\ \Rightarrow\ (ax,ay)\in\sigma$ for all $a\in S$. _Right-
compatibility_ is defined symmetrically, and a relation is _compatible_ if it
is both left- and right-compatible. A _(left or right) congruence_ is an
equivalence relation that is (left- or right-) compatible. We will need the
following basic result a number of times; it is surely well known, but we
include a short proof for convenience.

######  Lemma 2.1.

If $\sigma$ is a right congruence on a monoid $M$, then the $\sigma$-class of
$1$ is a submonoid of $M$.

###### Proof.

Let $T$ be the $\sigma$-class of $1$. Clearly $1\in T$. Now let $a,b\in T$.
Since $a,b\mathrel{\sigma}1$, and since $\sigma$ is a right congruence, we
have $ab\mathrel{\sigma}1b=b\mathrel{\sigma}1$. ∎

As in [6, 118], a _mid-identity_ (a.k.a.  _mid-unit_ or _middle unit_) of a
semigroup $S$ is an element $a\in S$ such that $xay=xy$ for all $x,y\in S$.

A _retraction_ is a morphism $\phi:S\to T$, where $S$ is a semigroup, $T\leq
S$ a subsemigroup, and $\phi\mathord{\restriction}_{T}=\mathrm{id}_{T}$.
Obviously retractions are surjective.

######  Lemma 2.2.

Suppose $a$ is a mid-identity of a semigroup $S$, and let $T\leq S$.

  1. (i)

The following diagram commutes, with all sets subsemigroups of $S$, and all
maps surmorphisms:

${T}$${Ta}$${aT}$${aTa}$$\scriptstyle{f_{1}:x\mapsto
xa}$$\scriptstyle{f_{2}:x\mapsto ax}$$\scriptstyle{g_{1}:x\mapsto
ax}$$\scriptstyle{g_{2}:x\mapsto xa}$

  2. (ii)

If $Ta\leq T$, then $aTa\leq aT$, and $f_{1}$ and $g_{2}$ are retractions.

  3. (iii)

If $aT\leq T$, then $aTa\leq Ta$, and $f_{2}$ and $g_{1}$ are retractions.

###### Proof.

This is all easily checked, and we just give a couple of sample calculations.
To show that $f_{1}$ is a morphism, let $x,y\in T$. Then

$(xf_{1})\cdot(yf_{1})=xa\cdot ya=xay\cdot a=xy\cdot a=(xy)f_{1}.$

If $Ta\leq T$, then for any $t\in T$ we have $(ta)f_{1}=taa=ta$, so that
$f_{1}$ is a retraction. ∎

######  Remark 2.3.

Although $Ta$, $aT$ and $aTa$ are subsemigroups of $S$, they might not be
subsemigroups of $T$. See Remark 3.24 for a specific example of this, where
$aT\cap T=\varnothing$; in fact, $aT=S\setminus T$.

######  Remark 2.4.

Lemma 2.2 also follows from [19, Theorem 2.15], which concerns _sandwich
semigroups_ in (locally small) categories. The simplest case of a sandwich
semigroup is a _semigroup variant_ [61, 60]. The variant of a semigroup $S$
with respect to an element $a\in S$ is the semigroup $S^{a}=(S,\star_{a})$,
where the ‘sandwich operation’ $\star_{a}$ is defined by $x\star_{a}y=xay$ for
all $x,y\in S$. Of course if $a$ is a mid-identity of $S$, then $\star_{a}$ is
precisely the original operation in $S$.

We also need the following simple result.

######  Lemma 2.5.

Let $S$ be a semigroup, and suppose $P\leq S$ is a semilattice. If $x,y\in S$
are such that $[x\in Py$ and $y\in Px]$ or $[x\in yP$ and $y\in xP]$, then
$x=y$.

###### Proof.

By symmetry, we assume that $x\in Py$ and $y\in Px$, so that $x=py$ and $y=qx$
for some $p,q\in P$. But then

$x=py=pqx=qpx=(qq)px=q(qpx)=qx=y.\qed$

Green’s $\mathrel{\mathscr{R}}$, $\mathrel{\mathscr{L}}$,
$\mathrel{\mathscr{J}}$, $\mathrel{\mathscr{H}}$ and $\mathrel{\mathscr{D}}$
relations on a semigroup $S$ [58] will not play an explicit role in our study,
but since we do mention them in passing from time to time it is convenient to
define them here. The first three of these relations are defined, for $x,y\in
S$, by

$x\mathrel{\mathscr{R}}y\ \Leftrightarrow\ xS^{1}=yS^{1},\qquad
x\mathrel{\mathscr{L}}y\ \Leftrightarrow\ S^{1}x=S^{1}y\qquad\text{and}\qquad
x\mathrel{\mathscr{J}}y\ \Leftrightarrow\ S^{1}xS^{1}=S^{1}yS^{1}.$

These can also be characterised in terms of divisibility. For example,
$x\mathrel{\mathscr{R}}y$ if and only if $x=ya$ and $y=xb$ for some $a,b\in
S^{1}$; this is in turn equivalent to having either $x=y$ or else $x=ya$ and
$y=xb$ for some $a,b\in S$. The final two of Green’s relations are defined by

${\mathrel{\mathscr{H}}}={\mathrel{\mathscr{R}}}\cap{\mathrel{\mathscr{L}}}\qquad\text{and}\qquad{\mathrel{\mathscr{D}}}={\mathrel{\mathscr{R}}}\vee{\mathrel{\mathscr{L}}},$

where the latter denotes the join of $\mathrel{\mathscr{R}}$ and
$\mathrel{\mathscr{L}}$ in the lattice of all equivalence relations on $S$:
i.e., $\mathrel{\mathscr{D}}$ is the least equivalence containing
${\mathrel{\mathscr{R}}}\cup{\mathrel{\mathscr{L}}}$. It is well known that in
fact
${\mathrel{\mathscr{D}}}={\mathrel{\mathscr{R}}}\circ{\mathrel{\mathscr{L}}}={\mathrel{\mathscr{L}}}\circ{\mathrel{\mathscr{R}}}$.
For some classes of semigroups (some of) Green’s relations can be
characterised _equationally_. For example, if $S$ is inverse, then
$x\mathrel{\mathscr{R}}y\ \Leftrightarrow\ xx^{-1}=yy^{-1}$.

####  2.2 Transformation semigroups

Certain semigroups of (partial) transformations will play an important role
throughout, so we revise their definitions here. Let $X$ be an arbitrary set.
A _partial transformation_ of $X$ is a function $A\to X$ for some $A\subseteq
X$. The set $\mathcal{P}\mathcal{T}_{X}$ of all such partial transformations
of $X$ is a monoid under ordinary relational composition, called the _partial
transformation monoid_.

As usual, we denote by $\operatorname{dom}(\alpha)$ and
$\operatorname{im}(\alpha)$ the _domain_ and _image_ of a partial
transformation $\alpha\in\mathcal{P}\mathcal{T}_{X}$, with their standard
meanings. For $x\in\operatorname{dom}(\alpha)$ we write $x\alpha$ for the
image of $x$ under $\alpha$, and we compose partial transformations from left
to right. Note, for example, that

$\operatorname{dom}(\alpha\beta)=\operatorname{dom}(\beta)\alpha^{-1}=\big{\\{}{x\in\operatorname{dom}(\alpha)}:{x\alpha\in\operatorname{dom}(\beta)}\big{\\}}\qquad\text{for
$\alpha,\beta\in\mathcal{P}\mathcal{T}_{X}$.}$

The _kernel_ of $\alpha\in\mathcal{P}\mathcal{T}_{X}$ is the equivalence

$\ker(\alpha)=\big{\\{}{(x,y)\in\operatorname{dom}(\alpha)\times\operatorname{dom}(\alpha)}:{x\alpha=y\alpha}\big{\\}},$

and the _rank_ of $\alpha$ is the cardinal

$\operatorname{rank}(\alpha)=\big{|}{\operatorname{im}(\alpha)}\big{|}=\big{|}{\operatorname{dom}(\alpha)/\ker(\alpha)}\big{|}.$

Here $\operatorname{dom}(\alpha)/\ker(\alpha)$ is the quotient of the set
$\operatorname{dom}(\alpha)$ by the equivalence $\ker(\alpha)$: i.e., the set
of all $\ker(\alpha)$-classes. Important submonoids of
$\mathcal{P}\mathcal{T}_{X}$ include:

  * •

$\mathcal{T}_{X}=\\{{\alpha\in\mathcal{P}\mathcal{T}_{X}}:{\operatorname{dom}(\alpha)=X}\\}$,
the _full transformation monoid_ ,

  * •

$\mathcal{I}_{X}=\\{{\alpha\in\mathcal{P}\mathcal{T}_{X}}:{\alpha\text{ is
injective}}\\}$, the _symmetric inverse monoid_ ,

  * •

$\mathcal{G}_{X}=\\{{\alpha\in\mathcal{T}_{X}}:{\alpha\text{ is
bijective}}\\}$, the _symmetric group_.

As the name suggests, $\mathcal{I}_{X}$ is an inverse monoid. The inverse of
$\alpha\in\mathcal{I}_{X}$ is the ordinary inverse mapping $\alpha^{-1}$, and
the idempotents of $\mathcal{I}_{X}$ are precisely the partial identity maps
$\mathrm{id}_{A}$ ($A\subseteq X$). These are not the only idempotents of
$\mathcal{P}\mathcal{T}_{X}$ when $|X|\geq 2$.

When $X$ is finite, the sets

$\operatorname{Sing}(\mathcal{P}\mathcal{T}_{X})=\mathcal{P}\mathcal{T}_{X}\setminus\mathcal{G}_{X},\qquad\operatorname{Sing}(\mathcal{T}_{X})=\mathcal{T}_{X}\setminus\mathcal{G}_{X}\qquad\text{and}\qquad\operatorname{Sing}(\mathcal{I}_{X})=\mathcal{I}_{X}\setminus\mathcal{G}_{X}$

are subsemigroups, indeed two-sided ideals, of $\mathcal{P}\mathcal{T}_{X}$,
$\mathcal{T}_{X}$ and $\mathcal{I}_{X}$, respectively. For arbitrary $X$, we
also have the subsemigroup
$\mathcal{P}\mathcal{T}_{X}\setminus\mathcal{T}_{X}$ of all _strictly_ partial
transformations of $X$, which is in fact a right ideal of
$\mathcal{P}\mathcal{T}_{X}$. When $X=\\{1,2,\ldots,n\\}$ for some integer
$n$, we typically denote $\mathcal{P}\mathcal{T}_{X}$ by
$\mathcal{P}\mathcal{T}_{n}$, and similarly for $\mathcal{T}_{n}$,
$\mathcal{I}_{n}$ and so on. We often make use of the standard two-line
notation for partial transformations. For example,
$\alpha=\left(\begin{smallmatrix}1&2&3&4&5&6\\\
2&-&3&2&6&6\end{smallmatrix}\right)\in\mathcal{P}\mathcal{T}_{6}$ has domain
$\\{1,3,4,5,6\\}$, and maps $1\mapsto 2$, $3\mapsto 3$, and so on.

####  2.3 Left restriction semigroups

_Left restriction semigroups_ arise in many different contexts and have many
different names. Essentially, one can define them in three different ways:

  * •

by a representation: up to isomorphism, they are precisely subsemigroups of
partial transformation semigroups closed under the map
$\alpha\mapsto\mathrm{id}_{\operatorname{dom}(\alpha)}$;

  * •

as a generalisation of inverse semigroups: they possess a distinguished
semilattice of idempotents $E$ (not necessarily _all_ the idempotents), such
that every $\widetilde{\mathrel{\mathscr{R}}}_{E}$-class contains an
idempotent of $E$, and the so-called ‘ample condition’ holds; here
$\widetilde{\mathrel{\mathscr{R}}}_{E}$ is a relation containing Green’s
$\mathrel{\mathscr{R}}$-relation;

  * •

they form a variety of unary semigroups (see below).

For more details and background we refer the reader to [56]. It is convenient
here to take the third option:

######  Definition 2.6.

A _left restriction semigroup_ is a unary semigroup $S$ satisfying the
following identities, where we write the unary operation as $s\mapsto s^{+}$:

  1.   2. (L1)

$x^{+}x=x$,

  3. (L2)

$x^{+}y^{+}=y^{+}x^{+}$,

  4. (L3)

$(x^{+}y)^{+}=x^{+}y^{+}$,

  5. (L4)

$xy^{+}=(xy)^{+}x$.

_Right_ restriction semigroups are defined dually, and there is also a notion
of a _(two-sided) restriction semigroup_ ; the latter have two interacting
unary operations, but it is not necessary to give the definitions here.
Archetypal examples of left restriction semigroups include:

  * •

any inverse semigroup $S$, with $s^{+}=ss^{-1}$,

  * •

the partial transformation monoid $\mathcal{P}\mathcal{T}_{X}$ for any set
$X$, with $\alpha^{+}=\mathrm{id}_{\operatorname{dom}(\alpha)}$.

If the left restriction semigroup $S$ is a monoid with identity $1$, it
follows immediately from (L1) that $1^{+}=1$. Any monoid trivially becomes a
left restriction monoid, upon defining $s^{+}=1$ for all $s$, but such
structures are generally not of interest to us.

The identities (L1)–(L4) are rather compact, and some important properties are
not immediately apparent. For example (writing $=_{1}$ to indicate an
application of (L1), etc.), we have

  1. (L5)

$x^{+}x^{+}=_{3}(x^{+}x)^{+}=_{1}x^{+}$,

so that each $x^{+}$ is an idempotent. We then also have

  1. (L6)

$(x^{+})^{+}=_{5}(x^{+}x^{+})^{+}=_{3}x^{+}(x^{+})^{+}=_{2}(x^{+})^{+}x^{+}=_{1}x^{+}$,

so that the operation + is itself idempotent.

It follows from (L3) that the set

$P(S)=\\{{s^{+}}:{s\in S}\\}$

is a subsemigroup of $S$. By (L2) and (L5), $P(S)$ is in fact a semilattice.
By (L6), $P(S)$ is closed (indeed, fixed pointwise) under +. The elements of
$P(S)$ are called _projections_.

####  2.4 Presentations

We conclude this preliminary chapter by establishing the notation we use for
presentations. Presentations exist for any variety of universal algebras; here
we explain how they work for semigroups and monoids, both for the sake of
concreteness, and since these are our intended applications.

Let $X$ be a set, and $X^{+}$ the _free semigroup_ on $X$, which consists of
all non-empty words over $X$ under the operation of concatenation. Let
$R\subseteq X^{+}\times X^{+}$ be a set of pairs of words, and write
$R^{\sharp}$ for the congruence on $X^{+}$ generated by $R$. We say a
semigroup $S$ has _(semigroup) presentation_
${\textup{{Sgp}}\langle{X}:{R}\rangle}$ if $S\cong X^{+}/R^{\sharp}$, or
equivalently if there is a surmorphism $X^{+}\to S$ with kernel $R^{\sharp}$;
if $\phi$ is such a surmorphism, we say $S$ has _presentation
$\textup{{Sgp}}\langle{X}:{R}\rangle$ via $\phi$_. The elements of $X$ and $R$
are called _generators_ and _relations_ , respectively, and a relation
$(u,v)\in R$ is sometimes displayed as an equation, $u=v$. A _normal form
function_ is a function $N:S\to X^{+}$ such that $N(s)\phi=s$ for all $s\in
S$. If we write ${\sim}=R^{\sharp}=\ker(\phi)$, then it follows from the
definitions that

$N(u\phi)\sim u\qquad\text{and}\qquad N(st)\sim N(s)N(t)\qquad\text{for all
$u\in X^{+}$ and $s,t\in S$.}$ (2.7)

There is also a corresponding notion of monoid presentations. The _free
monoid_ on the set $X$ is denoted by $X^{*}$, and is defined by
$X^{*}=X^{+}\cup\\{\iota\\}$. Here $\iota$ is the empty word, which is the
identity of $X^{*}$. We say a monoid $S$ has _(monoid) presentation_
$\textup{{Mon}}\langle{X}:{R}\rangle$, where $R\subseteq X^{*}\times X^{*}$,
if $S\cong X^{*}/R^{\sharp}$. We also speak of monoid presentations via
surmorphisms $X^{*}\to S$ (with kernel $R^{\sharp}$), and normal form
functions $N:S\to X^{*}$; typically we assume that $N(1)=\iota$.

On occasions, we will identify $\textup{{Sgp}}\langle{X}:{R}\rangle$ with the
semigroup $X^{+}/R^{\sharp}$ itself, and similarly for monoid presentations.
Consider such a semigroup $\textup{{Sgp}}\langle{X}:{R}\rangle$, and another
semigroup $T$. Any function $f:X\to T$ extends uniquely to a morphism
$\phi:X^{+}\to T$, which is defined by $(x_{1}\cdots
x_{k})\phi=(x_{1}f)\cdots(x_{k}f)$ for any $x_{1},\ldots,x_{k}\in X$. Standard
notions of universal algebra tell us that if $\phi$ _preserves $R$_, in the
sense that $u\phi=v\phi$ for every $(u,v)\in R$, then
$R^{\sharp}\subseteq\ker(\phi)$. In this case, $\phi$ induces a morphism
$\Phi:\textup{{Sgp}}\langle{X}:{R}\rangle\to T$, defined by $[w]\Phi=w\phi$
for $w\in X^{+}$, where $[w]$ denotes the $R^{\sharp}$-class of $w$.

The next result has a simple proof, and is essentially folklore. Although we
state it for monoids (as that is the context in which we will most often apply
it), it holds more generally for arbitrary classes of (universal) algebras
admitting presentations, so in particular there is a semigroup version as
well, whose explicit statement we omit.

######  Lemma 2.8.

Suppose a monoid $S$ has presentation $\textup{{Mon}}\langle{X}:{R}\rangle$
via $\phi:X^{*}\to S$, and let ${N:S\to X^{*}}$ be a normal form function.
Suppose $\pi:S\to T$ is a surmorphism onto a monoid $T$, and suppose
$\ker(\pi)=\Omega^{\sharp}$, where $\Omega\subseteq S\times S$. Then $T$ has
presentation $\textup{{Mon}}\langle{X}:{R\cup R_{\Omega}}\rangle$ via
$\phi\pi:X^{*}\to T$, where
$R_{\Omega}=\big{\\{}{(N(s),N(t))}:{(s,t)\in\Omega}\big{\\}}$. ∎

The next result also has a semigroup version, which we omit. For the
statement, recall that left-regular bands were defined in Section 2.1.

######  Lemma 2.9.

Let $M=\textup{{Mon}}\langle{X}:{R}\rangle$, where $X$ is an alphabet and
$R\subseteq X^{*}\times X^{*}$. Also, let

$R_{1}=\big{\\{}{(x^{2},x)}:{x\in X}\big{\\}},\ \ \
R_{2}=\big{\\{}{(xy,yx)}:{x,y\in X}\big{\\}}\quad\text{and}\quad
R_{3}=\big{\\{}{(xyx,xy)}:{x,y\in X}\big{\\}}.$

  1. (i)

If $R_{2}\subseteq R^{\sharp}$, then $M$ is commutative.

  2. (ii)

If $R_{1}\cup R_{2}\subseteq R^{\sharp}$, then $M$ is a semilattice.

  3. (iii)

If $R_{1}\cup R_{3}\subseteq R^{\sharp}$, then $M$ is a left-regular band.

###### Proof.

(i). This is clear.

(ii). Since $R_{2}\subseteq R^{\sharp}$, $M$ is commutative. Since
$R_{1}\subseteq R^{\sharp}$, $M$ is idempotent-generated. Since any
idempotent-generated commutative semigroup is a semilattice, the result
follows.

(iii). Write ${\sim}=R^{\sharp}$, and for $w\in X^{*}$ write $\overline{w}$
for the $\sim$-class of $w$. We must show that

  1.   2. (a)

$u^{2}\sim u$ for all $u\in X^{*}$, and

  3. (b)

$uvu\sim uv$ for all $u,v\in X^{*}$.

In fact, it suffices to prove (b), as (a) follows upon taking $v=\iota$. We
first show that

  1. (c)

$xvx\sim xv$ for all $x\in X$ and $v\in X^{*}$.

We prove this by induction on $k=\ell(v)$, the length (number of letters) of
$v$. If $k=0$ then $v=\iota$, and (c) says $x^{2}\sim x$, which holds because
$R_{1}\subseteq R^{\sharp}$. So now suppose $k\geq 1$, and write
$v=v^{\prime}y$ where $v^{\prime}\in X^{*}$ and $y\in X$. Since
$\ell(v^{\prime})=k-1$, we have $xv^{\prime}\sim xv^{\prime}x$ by induction.
Since $R_{3}\subseteq R^{\sharp}$, we have $xy\sim xyx$. It then follows that

$xv=xv^{\prime}y\sim xv^{\prime}xy\sim xv^{\prime}xyx\sim xv^{\prime}yx=xvx,$

completing the proof of (c).

We now prove (b) by induction on $l=\ell(u)$. If $l=0$ then $u=\iota$, and (b)
says $v\sim v$, which is obviously true. So now suppose $l\geq 1$, and write
$u=zu^{\prime}$, where $u^{\prime}\in X^{*}$ and $z\in X$. Since
$\ell(u^{\prime})=l-1$, we have $u^{\prime}v\sim u^{\prime}vu^{\prime}$ by
induction. By (c) we have $zu^{\prime}v\sim zu^{\prime}vz$. It then follows
that

$uv=zu^{\prime}v\sim zu^{\prime}vu^{\prime}\sim zu^{\prime}vzu^{\prime}=uvu,$

completing the proof of (b), and hence of the lemma. ∎

######  Remark 2.10.

If $R_{1}\subseteq R^{\sharp}$, then of course $M$ is idempotent-generated,
but it need not be a band.

##  Part I Structure

This paper concerns a certain class of semigroups arising from what we will
call _action pairs_. This first part of the paper is devoted to a structural
analysis of such semigroups and pairs.

Roughly speaking, an action pair $(U,S)$ consists of two subsemigroups of a
common over-monoid $M$, with an action of $S$ on $U^{1}$ that is suitably
compatible with the operation of $M$. Associated to such a pair is another
subsemigroup of $M$, namely the (internal) product

$US=\\{{us}:{u\in U,\ s\in S}\\}.$

The semigroup $US$ does not always contain $U$ or $S$ as subsemigroups, and
does not even need to contain isomorphic copies of either. Nevertheless, we
will see that the structure of $US$ can be concisely described in terms of the
structure of $U$ and $S$, the action of $S$ on $U^{1}$, and some other data.
We will also see that action pairs provide a natural context in which to
extend a number of important classical results on inverse semigroups, left
restriction semigroups, and more general structures.

There are three chapters in this part of the paper. Chapter 3 contains
preliminary material on actions and semidirect products, which will underpin
all that follows.

In Chapter 4, we introduce action pairs (and more generally weak action
pairs), discuss a number of examples (including the class of left restriction
semigroups), and prove a structure theorem that will be used extensively in
the second part of the paper when we study presentations. This result, Theorem
4.53, states that a semigroup $US$ arising from an action pair $(U,S)$ is a
homomorphic image of a semidirect product $U\rtimes S$, so that
$US\cong(U\rtimes S)/\theta$ for a suitable congruence $\theta$. The theorem
also classifies the congruences on semidirect products leading to action
pairs, showing how they are built from natural families of right congruences
on $S$.

Chapter 5 contains a number of further structure theorems, all of which
involve so-called _proper_ action pairs. The resulting ‘product semigroups’
$US$ include all proper left restriction semigroups, and in particular all
proper (a.k.a. $E$-unitary) inverse semigroups. A very brief summary of the
main results is as follows:

  1. (i)

A semigroup $US$ arising from an arbitrary (weak) action pair $(U,S)$ can be
_covered_ by a proper semigroup $U^{\prime}S^{\prime}$ with $U^{\prime}\cong
U$ and $S^{\prime}\cong S$. See Theorem 5.39.

  2. (ii)

Any proper semigroup $US$ can be naturally embedded in a semidirect product
${\mathcal{U}\rtimes(S/\sigma)}$, where $\mathcal{U}$ contains $U$, and where
$\sigma$ is a conguence on $S$ akin to the least group congruence on an
inverse semigroup. See Theorem 5.60. When $U$ has additional structural
properties (such as being commutative, a semilattice or a left-regular band),
we can adjust our construction to ensure that $\mathcal{U}$ has the same
structural property as well; see Theorems 5.47, 5.82 and 5.102.

As applications of these results, we provide short proofs of the above-
mentioned classical results on left restriction semigroups; this class
includes inverse semigroups, but with a different interpretation of the unary
operation.

###  3 Actions and semidirect products

Actions and semidirect products are important tools in practically every part
of algebra, and they will play a crucial role throughout the current work. In
this chapter we gather various facts that we need in the rest of the paper.

####  3.1 Actions

A _(left) action_ of a semigroup $S$ on a set $M$ is a map $S\times M\to
M:(s,x)\mapsto{}^{s}x$ such that

${}^{s}({}^{t}x)={}^{st}x\qquad\text{for all $s,t\in S$ and $x\in M$.}$

All actions are assumed to be left actions. Our choice of the (left)
superscript notation for actions is for readability, particularly as many
expressions below involve both actions and products; it is convenient to keep
the former ‘vertical’ and the latter ‘horizontal’. If $S$ happens to be a
monoid with identity $1$, then we say the action is _monoidal_ if

${}^{1}x=x\qquad\text{for all $x\in M$.}$

The actions that arise in later chapters will almost always be monoidal when
$S$ is a monoid. (A notable exception is Theorem 4.53.) We will also generally
be interested in the case that the action preserves an algebraic structure on
$M$. If $M$ happens to be a semigroup, then the action is _by semigroup
morphisms_ if

${}^{s}(xy)={}^{s}x\cdot{}^{s}y\qquad\text{for all for all $s\in S$ and
$x,y\in M$.}$

If $M$ happens to be a monoid with identity $1$, then the action is _by monoid
morphisms_ if

${}^{s}(xy)={}^{s}x\cdot{}^{s}y\qquad\text{and}\qquad{}^{s}1=1\qquad\text{for
all for all $s\in S$ and $x,y\in M$.}$

On many occasions in the paper, we will be concerned with the case in which a
semigroup $S$ acts on a monoid $M$ by semigroup morphisms, but _not_ by monoid
morphisms; see Example 3.3(ii) for an important/motivating case. In such
cases, the elements ${}^{s}1$ ($s\in S$) play a crucial role, so it is
convenient to record some of their properties. (All of these properties are
trivial when the action is by monoid morphisms.)

######  Lemma 3.1.

Suppose a semigroup $S$ acts on a monoid $M$ by semigroup morphisms, and for
$s\in S$ write $s^{+}={}^{s}1$. Then for all $s,t\in S$ and $x\in M$ we have

  1.   2. (i)

$s^{+}s^{+}=s^{+}$,

  3. (ii)

${}^{s}(t^{+})=(st)^{+}$,

  4. (iii)

$s^{+}(st)^{+}=(st)^{+}=(st)^{+}s^{+}$,

  5. (iv)

$s^{+}\cdot{}^{s}x={}^{s}x={}^{s}x\cdot s^{+}$.

###### Proof.

These are all easily established, and there are some interdependencies; for
example (i) follows from (iv) with $x=1$. As an example calculation,
$s^{+}\cdot{}^{s}x={}^{s}1\cdot{}^{s}x={}^{s}(1\cdot x)={}^{s}x$, giving the
first half of (iv). ∎

######  Remark 3.2.

The reader will note that we have used the notation $s^{+}$ for the unary
operation in a left restriction semigroup (cf. Section 2.3), and also for the
elements ${}^{s}1$ arising from a semigroup action on a monoid. This choice is
deliberate, and it will transpire that there are deep connections between the
two ideas; see especially Section 4.3.

We conclude this section with some useful examples that we will often revisit.

######  Example 3.3.

  1. (i)

Consider arbitrary semigroups $S$ and $M$, such that $M$ contains an
idempotent $e$. Then ${}^{s}x=e$ ($s\in S$, $x\in M$) defines an action of $S$
on $M$ by semigroup morphisms, which we call the _constant action with image
$e$_. If $S$ is a monoid, this action is monoidal if and only if $M=\\{e\\}$.
If $M$ is a monoid, the action is by monoid morphisms if and only if $e=1$.

  2. (ii)

Consider an inverse semigroup $S$, and let $E=E(S)$ be the semilattice of
idempotents. Then ${}^{s}e=ses^{-1}$ defines an action of $S$ on $E^{1}$ by
semigroup morphisms, as follows from standard facts about inverse semigroups;
cf. Section 2.1. For example,

${}^{s}(ef)=sefs^{-1}=ss^{-1}sefs^{-1}=ses^{-1}sfs^{-1}={}^{s}e\cdot{}^{s}f\qquad\text{for
all $s\in S$ and $e,f\in E^{1}$,}$

where we used the fact that the idempotents $e$ and $s^{-1}s$ commute. (Taking
$f=e=e^{2}$ in the above shows that indeed ${}^{s}e\in E^{1}$ for $s\in S$ and
$e\in E^{1}$.) With respect to this action, we have $s^{+}={}^{s}1=ss^{-1}$
for all $s\in S$. Thus, the action is by monoid morphisms if and only if
$ss^{-1}=1$ for all $s\in S$, which is equivalent to $S$ being a group. If $S$
happens to be a monoid, then the action is monoidal.

  3. (iii)

Let $X$ be a non-empty set, and consider the power set
${\mathcal{P}}(X)=\\{{A}:{A\subseteq X}\\}$ as a monoid (indeed, semilattice)
under $\cap$, with identity $X$. The partial transformation monoid
$\mathcal{P}\mathcal{T}_{X}$ (cf. Section 2.2) acts monoidally on
${\mathcal{P}}(X)$ by semigroup morphisms, via

${}^{\alpha}\\!A=A\alpha^{-1}=\\{{x\in\operatorname{dom}(\alpha)}:{x\alpha\in
A}\\}\qquad\text{for $A\subseteq X$ and
$\alpha\in\mathcal{P}\mathcal{T}_{X}$.}$

This action is not by _monoid_ morphisms, however, as
${}^{\alpha}\\!X=\operatorname{dom}(\alpha)$ for all
$\alpha\in\mathcal{P}\mathcal{T}_{X}$. On the other hand, the induced action
of the _full_ transformation monoid
$\mathcal{T}_{X}\leq\mathcal{P}\mathcal{T}_{X}$ _is_ by monoid morphisms, as
$\operatorname{dom}(\alpha)=X$ for all $\alpha\in\mathcal{T}_{X}$.

####  3.2 Semidirect products

Typically, to define a semidirect product $U\rtimes S$, for semigroups $U$ and
$S$, one begins with an action of $S$ on $U$, or equivalently a representation
of $S$ by endomorphisms of $U$; see for example [119, 103]. However, in order
to encompass all of our desired applications, we need a slightly more general
notion:

######  Definition 3.4.

Suppose $U$ and $S$ are semigroups, and suppose $S$ has a left action on
$U^{1}$ (the monoid completion of $U$) by semigroup morphisms, denoted

$\varphi:S\times U^{1}\to U^{1}:(s,u)\mapsto{}^{s}u.$

The _semidirect product_ $U\rtimes S=U\rtimes_{\varphi}S$ is the semigroup

$U\rtimes S=\big{\\{}{(u,s)}:{u\in U,\ s\in S}\big{\\}}\qquad\text{with
operation}\qquad(u,s)\cdot(v,t)=(u\cdot{}^{s}v,st).$ (3.5)

Throughout the paper, we omit the $\varphi$ subscript, and write $U\rtimes
S=U\rtimes_{\varphi}S$. Context will always make it clear which action is in
play.

######  Remark 3.6.

In (3.5), it is possible to have ${}^{s}v=1\not\in U$, but we always have
$u\cdot{}^{s}v\in U$.

If $S$ acts on $U$ itself, then it also acts on $U^{1}$; if $U\not=U^{1}$,
then we additionally define ${}^{s}1=1$ for all $s\in S$. In this case,
$U\rtimes S$ as in Definition 3.4 agrees with the ordinary semidirect product.
Throughout the paper, we will frequently be interested in actions on $U^{1}$
that do _not_ arise in this way. (An action of a semigroup on a monoid $M$
might not restrict to an action on $M\setminus\\{1\\}$, even if the latter is
a subsemigroup of $M$.)

As an example, let $S=\mathcal{T}_{X}$ for an arbitrary set $X$ with $|X|\geq
2$, and let $U={\mathcal{P}}(X)\setminus\\{X\\}$ be the $\cap$-semilattice of
all _proper_ subsets of $X$. Then the action of $S$ on
$U^{1}={\mathcal{P}}(X)$ given in Example 3.3(iii) does not arise from an
action on $U$. (If $\alpha\in\mathcal{T}_{X}$ has image $A\subsetneq X$, then
$A\in U$, but ${}^{\alpha}\\!A=X\not\in U$.) In this case, we will see that
the semidirect product $U\rtimes S$ maps naturally onto
$\mathcal{P}\mathcal{T}_{X}\setminus\mathcal{T}_{X}$, the semigroup of all
_strictly_ partial transformations of $X$; cf. Example 4.30 and Proposition
4.46.

For the rest of this section, we fix semigroups $S$ and $U$, and assume that
$S$ acts on $U^{1}$ by semigroup morphisms. We begin the following
observation.

######  Lemma 3.7.

If $U\rtimes S$ is a monoid, then $U$ and $S$ are both monoids, and the
identity of $U\rtimes S$ is $(1,1)$, where as usual we write $1$ for the
identity of both $U$ and $S$.

###### Proof.

Suppose $U\rtimes S$ is a monoid with identity $(e,f)$. So for all $u\in U$
and $s\in S$,

$(u,s)=(u,s)\cdot(e,f)=(u\cdot{}^{s}e,sf)\qquad\text{and
similarly}\qquad(u,s)=(e\cdot{}^{f}u,fs).$ (3.8)

Looking at the second coordinates of (3.8), it follows that $S$ is a monoid
with identity $f=1$. Looking at the first coordinates, and taking $s=1\in S$
(and remembering $f=1$), we have

$u=u\cdot{}^{1}e=e\cdot{}^{1}u\qquad\text{for all $u\in U$.}$ (3.9)

But then for any $u\in U$, several applications of (3.9) gives

$ue=e\cdot{}^{1}(ue)=e\cdot{}^{1}u\cdot{}^{1}e=u\cdot{}^{1}e=u,$

so that $e\in U$ is a right identity for $U$. In particular, $e$ is an
idempotent; combining this with (3.9), it quickly follows that $e$ is also a
left identity for $U$. ∎

So $U\rtimes S$ being a monoid implies that $U$ and $S$ are both monoids, but
the next result shows that the converse is not true in general.

Recall that we have a fixed action of $S$ on $U^{1}$. This can be extended to
an action of $S^{1}$ on $U^{1}$ in an obvious way: if $S\not=S^{1}$, then we
additionally define ${}^{1}u=u$ for all $u\in U^{1}$. This action of $S^{1}$
is of course monoidal if $S\not=S^{1}$, but need not be monoidal if $S=S^{1}$
is a monoid. In any case, we also have the semidirect product $U^{1}\rtimes
S^{1}$, which contains $U\rtimes S$ as a subsemigroup.

We now define ${\bf 1}=(1,1)\in U^{1}\rtimes S^{1}$. One might initially
expect $U^{1}\rtimes S^{1}$ to be a monoid with identity ${\bf 1}$, but this
is not always true. (But if $U^{1}\rtimes S^{1}$ _is_ a monoid, then its
identity is ${\bf 1}$, by Lemma 3.7.) Rather, we have

${\bf 1}\cdot(u,s)=({}^{1}u,s)\qquad\text{and}\qquad(u,s)\cdot{\bf
1}=(us^{+},s)\qquad\text{for any $u\in U^{1}$ and $s\in S^{1}$,}$ (3.10)

where again we write $s^{+}={}^{s}1$ for $s\in S^{1}$.

######  Lemma 3.11.

  1. (i)

${\bf 1}$ is a left identity for $U^{1}\rtimes S^{1}$ if and only if the
action of $S^{1}$ on $U^{1}$ is monoidal,

  2. (ii)

${\bf 1}$ is a right identity for $U^{1}\rtimes S^{1}$ if and only if the
action of $S^{1}$ on $U^{1}$ is by monoid morphisms,

  3. (iii)

$U^{1}\rtimes S^{1}$ is a monoid with identity ${\bf 1}$ if and only if the
action of $S^{1}$ on $U^{1}$ is monoidal and by monoid morphisms.

###### Proof.

This all follows from (3.10). The only (slightly) non-obvious part is the
forwards implication in (ii). For this, we note that if ${\bf 1}$ is a right
identity, then $(1,s)=(1,s)\cdot{\bf 1}=(s^{+},s)\ \Rightarrow\ s^{+}=1$ for
all $s\in S$. ∎

######  Remark 3.12.

In general, ${\bf 1}$ need not be a left or right identity for $U^{1}\rtimes
S^{1}$, and in fact it need not even be an idempotent, as ${\bf 1}\cdot{\bf
1}=({}^{1}1,1)=(1^{+},1)$. It is quite possible to have $1^{+}\not=1$. For
example, this is the case for the constant action of $S=S^{1}$ on $U^{1}$
whose image is a non-identity idempotent; cf. Example 3.3(i).

Combining Lemmas 3.7 and 3.11, we immediately obtain the following neat
characterisation of monoid semidirect products:

######  Corollary 3.13.

The following are equivalent:

  1. (i)

$U\rtimes S$ is a monoid,

  2. (ii)

$U$ and $S$ are both monoids, and the action of $S(=S^{1})$ on $U(=U^{1})$ is
monoidal and by monoid morphisms. ∎

Even though ${\bf 1}$ need not be a left or right identity for $U^{1}\rtimes
S^{1}$ (cf. Remark 3.12), it does nevertheless have the following very
important property.

######  Lemma 3.14.

The element ${\bf 1}$ is a mid-identity for $U^{1}\rtimes S^{1}$.

###### Proof.

For any $u,v\in U^{1}$ and $s,t\in S^{1}$, we have

$(u,s)\cdot{\bf
1}\cdot(v,t)=(us^{+},s)\cdot(v,t)=(us^{+}\cdot{}^{s}v,st)=(u\cdot{}^{s}v,st)=(u,s)\cdot(v,t),$

where we used Lemma 3.1(iv) in the third step. ∎

######  Remark 3.15.

We noted in Remark 3.12 that ${\bf 1}$ might not be an idempotent of
$U^{1}\rtimes S^{1}$, but now we have seen that it is always a mid-identity.
It is well known, and easy to see, that any _regular_ mid-identity must be an
idempotent. Thus, $U^{1}\rtimes S^{1}$ gives a neat example of a semigroup
with a non-regular mid-identity (when $1^{+}\not=1$). The square of any mid-
identity is always an idempotent mid-identity, and here this is ${\bf
1}\cdot{\bf 1}=(1^{+},1)$.

The next result is not essential for our purposes, but it seems worth
recording; it of course leads to an alternative proof of Lemma 3.14.

######  Proposition 3.16.

The mid-identities of $U^{1}\rtimes S^{1}$ are precisely the elements of the
form $(e,1)$ for which ${}^{1}e$ is a left identity for the subsemigroup
$\\{{{}^{1}u}:{u\in U^{1}}\\}\leq U^{1}$.

###### Proof.

Throughout the proof we write $V=\\{{{}^{1}u}:{u\in U^{1}}\\}$. (This is a
subsemigroup because ${}^{1}u\cdot{}^{1}v={}^{1}(uv)$ for all $u,v\in U^{1}$.)

Suppose first that $(e,f)$ is a mid-identity for $U^{1}\rtimes S^{1}$, and let
$u\in U^{1}$ be arbitrary. Then

$({}^{1}u,1)={\bf 1}\cdot(u,1)={\bf
1}\cdot(e,f)\cdot(u,1)=({}^{1}e\cdot{}^{f}u,f).$

It follows that $f=1$, and then also that
${}^{1}u={}^{1}e\cdot{}^{f}u={}^{1}e\cdot{}^{1}u$, so that ${}^{1}e$ is a left
identity for $V$.

Conversely, suppose $e\in U^{1}$ is such that ${}^{1}e$ is a left identity for
$V$, and let $(u,s),(v,t)\in U^{1}\rtimes S^{1}$ be arbitrary. Then

$(u,s)\cdot(v,t)=(u\cdot{}^{s}v,st)\qquad\text{and}\qquad(u,s)\cdot(e,1)\cdot(v,t)=(u\cdot{}^{s}e\cdot{}^{s}v,st)=(u\cdot{}^{s}(ev),st),$

so we must show that $u\cdot{}^{s}(ev)=u\cdot{}^{s}v$. But this follows
quickly from

${}^{s}(ev)={}^{s\cdot
1}(ev)={}^{s}({}^{1}(ev))={}^{s}({}^{1}e\cdot{}^{1}v)={}^{s}({}^{1}v)={}^{s\cdot
1}v={}^{s}v.\qed$

The remaining results of this section concern certain natural subsemigroups of
$U\rtimes S$. We begin by defining

$M_{1}=\big{\\{}{(u,s)\in U\rtimes
S}:{u=us^{+}}\big{\\}}\qquad\text{and}\qquad M_{2}=\big{\\{}{(u,s)\in U\rtimes
S}:{u={}^{1}u}\big{\\}}.$

(We will soon see that these are indeed subsemigroups. In $M_{2}$, note that
${}^{1}u$ is defined even if $1\not\in S$.) We also define

$M=M_{1}\cap M_{2}=\big{\\{}{(u,s)\in U\rtimes
S}:{{}^{1}u=u=us^{+}}\big{\\}}.$

######  Lemma 3.17.

We have $M=\big{\\{}{(u,s)\in U\rtimes S}:{u={}^{1}us^{+}}\big{\\}}$.

###### Proof.

We must show that for any $u\in U$ and $s\in S$, we have ${}^{1}u=u=us^{+}\
\Leftrightarrow\ u={}^{1}us^{+}$. The forwards implication being clear,
suppose $u={}^{1}us^{+}$. Then

$\displaystyle{}^{1}u$
$\displaystyle={}^{1}({}^{1}us^{+})={}^{1}({}^{1}u)\cdot{}^{1}(s^{+})={}^{1\cdot
1}u\cdot(1\cdot s)^{+}={}^{1}u\cdot s^{+}=u$
$\displaystyle\qquad\text{and}\qquad us^{+}$ $\displaystyle={}^{1}us^{+}\cdot
s^{+}={}^{1}us^{+}=u,$

where we used parts (i) and (ii) of Lemma 3.1. ∎

The statements below refer to the element ${\bf 1}=(1,1)\in U^{1}\rtimes
S^{1}$, which might not belong to $U\rtimes S$, and in particular might not
belong to any of the subsets $M_{1}$, $M_{2}$ or $M$. When ${\bf 1}$ does
belong to $U\rtimes S$ (i.e., when $U$ and $S$ are monoids), it belongs to any
of these subsets if and only if it belongs to all of them, and this is
equivalent to having $1=1^{+}$.

######  Lemma 3.18.

  1. (i)

$M_{1}=(U\rtimes S)\cdot{\bf 1}$ is a left ideal (and hence a subsemigroup) of
$U\rtimes S$, and ${\bf 1}$ is a right identity for $M_{1}$ (but not
necessarily an element of $M_{1}$).

  2. (ii)

$M_{1}=U\rtimes S$ if and only if

$u=us^{+}\qquad\text{for all $u\in U$ and $s\in S$,}$ (3.19)

in which case $M=M_{2}$.

###### Proof.

(i). First let $(u,s)\in M_{1}$. Then $(u,s)\cdot{\bf 1}=(us^{+},s)=(u,s)$.
This shows that ${M_{1}\subseteq(U\rtimes S)\cdot{\bf 1}}$, and also that
${\bf 1}$ is a right identity for $M_{1}$.

Conversely, suppose $(u,s)\in(U\rtimes S)\cdot{\bf 1}$, so that
$(u,s)=(v,t)\cdot{\bf 1}=(vt^{+},t)$ for some $v\in U$ and $t\in S$. It
follows that $u=vt^{+}\in U$ (as $v\in U$) and $s=t\in S$, so that $(u,s)\in
U\rtimes S$. Using Lemma 3.1(i), it also follows that $us^{+}=vt^{+}\cdot
t^{+}=vt^{+}=u$, so that $(u,s)\in M_{1}$.

This completes the proof that $M_{1}=(U\rtimes S)\cdot{\bf 1}$. Since
$(U\rtimes S)\cdot{\bf 1}$ is clearly closed under left multiplication by
elements of $U\rtimes S$, and since $M_{1}\subseteq U\rtimes S$ (by
definition), it follows that $M_{1}$ is a left ideal of $U\rtimes S$.

(ii). This is immediate from the definitions. ∎

######  Remark 3.20.

Condition (3.19) is satisfied, for example, when the action of $S$ (or
equivalently of $S^{1}$) on $U^{1}$ is by monoid morphisms (i.e., $s^{+}=1$
for all $s\in S$), but this is not necessary. For example, (3.19) still holds
if $U$ has a right identity $e$ that is not a left identity, and the action of
$S$ on $U^{1}(\not=U)$ is constant with image $e$, as then $us^{+}=ue=u$ for
all $u\in U$ and $s\in S$.

The situation for $M_{2}$ is slightly more complicated, as we do not always
have $M_{2}={\bf 1}\cdot(U\rtimes S)$:

######  Lemma 3.21.

  1. (i)

$M_{2}\subseteq{\bf 1}\cdot(U\rtimes S)$, with equality if and only if

${}^{1}u\in U\qquad\text{for all $u\in U$.}$ (3.22)

  2. (ii)

$M_{2}$ is a right ideal (and hence a subsemigroup) of $U\rtimes S$, and ${\bf
1}$ is a left identity for $M_{2}$ (but not necessarily an element of
$M_{2}$).

  3. (iii)

$M_{2}=U\rtimes S$ if and only if

$u={}^{1}u\qquad\text{for all $u\in U$,}$ (3.23)

in which case $M=M_{1}$. Moreover, (3.23) is equivalent to the action of
$S^{1}$ on $U^{1}$ being monoidal.

###### Proof.

(i). If $(u,s)\in M_{2}$, then ${\bf 1}\cdot(u,s)=({}^{1}u,s)=(u,s)$. This
shows that ${M_{2}\subseteq{\bf 1}\cdot(U\rtimes S)}$, and also that ${\bf 1}$
is a left identity for $M_{2}$.

Next we assume that (3.22) holds, aiming to prove the reverse inclusion in
this case. To do so, let ${(u,s)\in{\bf 1}\cdot(U\rtimes S)}$, so that
$(u,s)={\bf 1}\cdot(v,t)=({}^{1}v,t)$ for some $v\in U$ and $t\in S$, so
$(u,s)\in U\rtimes S$ by (3.22). We also have
${}^{1}u={}^{1}({}^{1}v)={}^{1\cdot 1}v={}^{1}v=u$, so that $(u,s)\in M_{2}$,
as required.

If, on the other hand, (3.22) does not hold, then ${}^{1}u\not\in U$ for some
$u\in U$. In this case, for any $s\in S$ we have ${{\bf
1}\cdot(u,s)=({}^{1}u,s)\not\in U\rtimes S}$, so certainly ${\bf
1}\cdot(u,s)\not\in M_{2}$.

(ii). Since the reverse inclusion in part (i) does not hold in general, we
must prove this directly, so let $(u,s)\in M_{2}$ and $(v,t)\in U\rtimes S$.
So ${}^{1}u=u$, and we have $(u,s)\cdot(v,t)=(u\cdot{}^{s}v,st)\in U\rtimes
S$, with

${}^{1}(u\cdot{}^{s}v)={}^{1}u\cdot{}^{1}({}^{s}v)=u\cdot{}^{1\cdot
s}v=u\cdot{}^{s}v.$

We have already seen that ${\bf 1}$ is a left identity for $M_{2}$.

(iii). The first claim is again immediate.

For the second, we first observe that monoidality clearly implies (3.23). For
the converse, suppose (3.23) holds. We must show that ${}^{1}1=1$. This is
clear if ${}^{1}1\not\in U$, as then ${{}^{1}1\in U^{1}\setminus U=\\{1\\}}$,
so we now assume that ${}^{1}1\in U$. For any $u\in U$, we use (3.23) to
calculate

${}^{1}1\cdot u={}^{1}1\cdot{}^{1}u={}^{1}(1\cdot u)={}^{1}u=u\qquad\text{and
similarly}\qquad u\cdot{}^{1}1=u.$

Thus, ${}^{1}1\in U$ is an identity element of $U$, so that $U^{1}=U$, and
${}^{1}1=1$ by definition. ∎

######  Remark 3.24.

Condition (3.22) does not hold in general. For example, when $S=S^{1}$ and
$U\not=U^{1}$, and the action of $S$ on $U^{1}$ is constant with image
$1(\not\in U)$, we have

$M_{2}=\varnothing\qquad\text{and}\qquad{\bf 1}\cdot(U\rtimes
S)=\\{1\\}\rtimes S=(U^{1}\rtimes S)\setminus(U\rtimes S).$

Clearly (3.23) implies (3.22); that is, (3.22) holds when $S^{1}$ acts
monoidally on $U^{1}$ (i.e., ${}^{1}u=u$ for all $u\in U^{1}$). However, the
reverse implication (3.22)$\ \Rightarrow\ $(3.23) need not hold. For example,
if $e$ is a non-identity idempotent of $U$, then the constant action with
image $e$ satisfies (3.22) but not (3.23).

We now give the corresponding result for $M=M_{1}\cap M_{2}$.

######  Lemma 3.25.

  1. (i)

$M\subseteq{\bf 1}\cdot(U\rtimes S)\cdot{\bf 1}$, with equality if and only if

${}^{1}us^{+}\in U\qquad\text{for all $u\in U$ and~{}$s\in S$.}$ (3.26)

  2. (ii)

$M$ is a subsemigroup of $U\rtimes S$, and ${\bf 1}$ is a two-sided identity
for $M$ (but not necessarily an element of $M$).

  3. (iii)

$M$ is a monoid with identity ${\bf 1}$ if and only if $U$ and $S$ are monoids
and $1=1^{+}$.

  4. (iv)

$M=U\rtimes S$ if and only if

$u={}^{1}us^{+}\qquad\text{for all $u\in U$ and $s\in S$,}$ (3.27)

and this is equivalent to (3.19) and (3.23) both holding.

###### Proof.

Throughout the proof we use Lemmas 3.1, 3.18 and 3.21 without explicit
mention. We begin with the second part.

(ii). Since $M_{1}$ and $M_{2}$ are subsemigroups of $U\rtimes S$, so too is
$M_{1}\cap M_{2}=M$. Since ${\bf 1}$ is a right and left identity for $M_{1}$
and $M_{2}$, respectively, it is a two-sided identity for $M_{1}\cap M_{2}=M$.

(i). By part (ii) we have $M={\bf 1}\cdot M\cdot{\bf 1}\subseteq{\bf
1}\cdot(U\rtimes S)\cdot{\bf 1}$.

Next, we assume that (3.26) holds, and let $(u,s)\in{\bf 1}\cdot(U\rtimes
S)\cdot{\bf 1}$. So

$(u,s)={\bf 1}\cdot(v,t)\cdot{\bf 1}=({}^{1}vt^{+},t)\qquad\text{for some
$v\in U$ and $t\in S$.}$

So $(u,s)\in U\rtimes S$ by (3.26). As in the proof of Lemma 3.17, one can
check that

${}^{1}us^{+}={}^{1}({}^{1}vt^{+})\cdot t^{+}={}^{1}vt^{+}=u,$

so that $(u,s)\in M$, by the same lemma.

If, on the other hand, (3.26) does not hold, then ${}^{1}us^{+}\not\in U$ for
some $u\in U$ and $s\in S$. In this case, ${\bf 1}\cdot(u,s)\cdot{\bf
1}=({}^{1}us^{+},s)\not\in U\rtimes S$, so certainly ${\bf
1}\cdot(u,s)\cdot{\bf 1}\not\in M$.

(iii). By part (ii), $M$ is a monoid with identity ${\bf 1}$ if and only if
${\bf 1}\in M$, and we noted just before Lemma 3.18 that this is equivalent to
the stated conditions.

(iv). The equivalence of $M=U\rtimes S$ with (3.27) follows from Lemma 3.17.
The equivalence of $M=U\rtimes S$ with (3.19) and (3.23) follows immediately
from the original definition of $M$. ∎

######  Remark 3.28.

As in Remark 3.24, condition (3.26) is not always satisfied, but it is (for
example) when $S^{1}$ acts monoidally on $U^{1}$.

As in Remark 3.20, condition (3.27) is satisfied (for example) when the action
of $S^{1}$ on $U^{1}$ is monoidal and by monoid morphisms, but this is again
not necessary. Indeed, consider the case in which $U$ has a right identity $e$
that is not a left identity, and suppose $S\not=S^{1}$. Consider the constant
action of $S$ on $U^{1}(\not=U)$ with image $e$, and extend this to a monoidal
action of $S^{1}(\not=S)$ on $U^{1}$ in the usual way. Then for any $u\in U$
and $s\in S$, we have ${}^{1}us^{+}=ue=u$.

By Lemmas 2.2 and 3.14, we have the following commutative diagram, with all
sets subsemigroups of $U^{1}\rtimes S^{1}$, and all maps surmorphisms:

${U\rtimes S}$${(U\rtimes S)\cdot{\bf 1}}$${{\bf 1}\cdot(U\rtimes S)}$${{\bf
1}\cdot(U\rtimes S)\cdot{\bf 1}}$$\scriptstyle{f_{1}\colon{\bf x}\ \mapsto\
{\bf x}\cdot{\bf 1}}$$\scriptstyle{f_{2}\colon{\bf x}\ \mapsto\ {\bf
1}\cdot{\bf x}}$$\scriptstyle{g_{1}\colon{\bf x}\ \mapsto\ {\bf 1}\cdot{\bf
x}}$$\scriptstyle{g_{2}\colon{\bf x}\ \mapsto\ {\bf x}\cdot{\bf 1}}$

By Lemma 3.18(i), the semigroup on the western side of this diagram is equal
to $M_{1}$. By Lemma 3.21(i) the eastern semigroup is $M_{2}$ precisely when
(3.22) holds. Since (3.22) clearly implies (3.26), it follows from Lemma
3.25(i) that the southern semigroup is $M$ in this case. Thus, we have the
following:

######  Proposition 3.29.

If ${}^{1}u\in U$ for all $u\in U$, then the following diagram commutes, with
all sets subsemigroups of $U\rtimes S$, and all maps retractions:

${U\rtimes S}$${M_{1}}$${M_{2}}$${M}$$\scriptstyle{f_{1}\colon{\bf x}\
\mapsto\ {\bf x}\cdot{\bf 1}}$$\scriptstyle{f_{2}\colon{\bf x}\ \mapsto\ {\bf
1}\cdot{\bf x}}$$\scriptstyle{g_{1}\colon{\bf x}\ \mapsto\ {\bf 1}\cdot{\bf
x}}$$\scriptstyle{g_{2}\colon{\bf x}\ \mapsto\ {\bf x}\cdot{\bf 1}}$

∎

######  Remark 3.30.

It follows that under the ${}^{1}u\in U$ assumption from Proposition 3.29, the
subsemigroups $M_{1}$, $M_{2}$ and $M$ of $U\rtimes S$ can also be identified
with quotients of $U\rtimes S$:

$M_{1}\cong(U\rtimes S)/\ker(f_{1}),\qquad M_{2}\cong(U\rtimes
S)/\ker(f_{2})\qquad\text{and}\qquad M\cong(U\rtimes S)/\ker(F),$

where $F=f_{1}g_{1}=f_{2}g_{2}$ is the retraction

$F:(U\rtimes S)\to M:{\bf x}\mapsto{\bf 1}\cdot{\bf x}\cdot{\bf
1}\qquad\text{given by}\qquad(u,s)F=({}^{1}us^{+},s)\qquad\text{for $u\in U$
and $s\in S$.}$

The kernels of these maps can be described easily. For example:

  * •

$(u,s)f_{1}=(v,t)f_{1}\ \Leftrightarrow\ us^{+}=vt^{+}\text{ and }s=t$,

  * •

$(u,s)f_{2}=(v,t)f_{2}\ \Leftrightarrow\ {}^{1}u={}^{1}v\text{ and }s=t$, and

  * •

$(u,s)F=(v,t)F\ \Leftrightarrow\ {}^{1}us^{+}={}^{1}vt^{+}\text{ and }s=t$.

######  Remark 3.31.

The ${}^{1}u\in U$ assumption from Proposition 3.29 is precisely condition
(3.22). When (3.19) additionally holds,

$f_{1}:U\rtimes S\to M_{1}=U\rtimes S\qquad\text{and}\qquad g_{2}:M=M_{2}\to
M$

are identity maps; cf. Lemma 3.18(ii). The analogous statement holds for
$f_{2}$ and $g_{1}$ when (3.23) holds.

As we have already mentioned, the actions considered in the rest of the paper
will almost always be monoidal. It is therefore convenient to record the next
result, which summarises the main points above in the case that the action of
$S^{1}$ on $U^{1}$ is monoidal. For the statement, recall (see for example
[104]) that the _local monoid_ at an idempotent $e$ of a semigroup $T$ is the
set

$eTe=eT\cap Te=\\{{ete}:{t\in T}\\}=\\{{t\in T}:{et=t=te}\\},$

and is the largest subsemigroup of $T$ that happens to be a monoid with
identity $e$.

######  Proposition 3.32.

Suppose $S$ acts on $U^{1}$ by semigroup morphisms, and suppose this action is
monoidal if $S$ is a monoid. Then

$M=\big{\\{}{(u,s)\in U\rtimes S}:{u=us^{+}}\big{\\}}$

is a subsemigroup of $U\rtimes S$, and the map

$F:U\rtimes S\to M:(u,s)\mapsto(us^{+},s)$

is a retraction. If $U$ and $S$ are monoids, then $M$ is a (local) monoid with
identity ${\bf 1}=(1,1)$.

###### Proof.

The monoidality assumption in the statement is equivalent to the action of
$S^{1}$ on $U^{1}$ being monoidal. It follows from Lemma 3.21(iii) that

$M=M_{1}=\big{\\{}{(u,s)\in U\rtimes S}:{u=us^{+}}\big{\\}}\leq U\rtimes S.$

Monoidality also means that Proposition 3.29 applies, and together with
$M=M_{1}$, this gives the assertion regarding the retraction $F=f_{1}$; cf.
Remark 3.31. The final assertion follows from parts (i) and (iii) of Lemma
3.25, and the fact that monoidality implies (3.26) and $1=1^{+}$. ∎

###  4 Action pairs

In this chapter we introduce the kinds of semigroups that are the focus of the
paper. These are certain products of the form $US$, where $U$ and $S$ are
subsemigroups of a common over-monoid $M$ satisfying natural axioms. After
giving the relevant definitions and basic properties in Section 4.1, we
discuss several examples in Section 4.2. Section 4.3 treats the class of left
restriction monoids, showing how these fit into our general framework, and
providing the tools for studying many of our motivating examples in Part II of
the paper. Section 4.4 contains the main structural result of the chapter,
Theorem 4.53, which characterises our product semigroups as quotients of
semidirect products by certain special congruences that we abstractly
axiomatise. Finally, Section 4.5 contains several results on generating sets
for these congruences, which then feed into presentations in Part II.

####  4.1 Basic definitions and properties

The general set-up we will be concerned with involves two subsemigroups of a
common semigroup; since the latter embeds in its monoid completion, we may
assume without loss of generality that it already is a monoid. Thus, we have
$U,S\leq M$, where $M$ is a monoid, subject to various natural assumptions to
be detailed below. We denote the identity of $M$ by $1$, and write
$U^{1}=U\cup\\{1\\}$ and $S^{1}=S\cup\\{1\\}$. Note that $U^{1}=U$ if $U$
happens to be a submonoid of $M$, and similarly for $S^{1}$. In any case, we
also have $U^{1},S^{1}\leq M$.

######  Lemma 4.1.

If $U,S\leq M$, for some monoid $M$, and if $SU\subseteq U^{1}S^{1}$, then
$US\leq M$.

###### Proof.

This follows from $US\cdot US\subseteq UU^{1}\cdot S^{1}S=US$. ∎

Note that the semigroups $U$ and $S$ need not be contained in the product
$US$. Indeed, this will be the case for several of our motivating examples. Of
course, if $U$ and $S$ are both submonoids of $M$, then $U,S\subseteq US$.

Here is the first key definition.

######  Definition 4.2.

A _weak (left) action pair_ in a monoid $M$ is a pair $(U,S)$ of subsemigroups
of $M$ for which the following condition holds:

  1. (A1)

$S$ has a left action on $U^{1}$ by semigroup morphisms, written
$(s,u)\mapsto{}^{s}u$, such that

$s\cdot u={}^{s}u\cdot s\qquad\text{for all $s\in S$ and $u\in U^{1}$.}$

If the following condition also holds, we call $(U,S)$ a _(left) action pair_
:

  1. (A2)

$us=vt\ \Rightarrow\ u\cdot{}^{s}1=v\cdot{}^{t}1$ for all $u,v\in U^{1}$ and
$s,t\in S$.

######  Remark 4.3.

Strictly speaking, it would be more precise to speak of a _(weak) action
triple_ $(U,S,\varphi)$, where $\varphi:S\times U^{1}\to
U^{1}:(s,u)\mapsto{}^{s}u$ is the specific action from (A1). However, as with
semidirect products in Definition 3.4, we omit the label $\varphi$ for
brevity, and the action will always be clear from context.

For the time being, we have resisted using the notation $s^{+}$ for the
elements ${}^{s}1$ ($s\in S$), for reasons that will soon become clear; see
Proposition 4.7.

It of course follows from Lemma 4.1 and (A1) that $US\leq M$ for any (weak)
action pair $(U,S)$ in $M$. Most of the results of this paper concern action
pairs, but some hold more generally for weak action pairs; see for example
Proposition 4.46, and Theorems 5.39 and 6.4.

The main role played by the monoid $M$ in Definition 4.2 is to provide a
common environment in which the semigroups $U$ and $S$ exist, or, more
specifically, a context in which to form products of the form $us$ and $su$,
for $u\in U$ and $s\in S$. Consequently, $M$ could be replaced in the
definition by any monoid containing $U$ and $S$, and we will often speak
simply of ‘an action pair $(U,S)$’, without explicitly specifying the common
over-monoid $M$.

######  Remark 4.4.

Note that we do not assume in (A1) that $S$ acts by _monoid_ morphisms (i.e.,
that ${}^{s}1=1$ for all $s\in S$), but we will soon see that this special
case is very important. On the other hand, if $S$ happens to be a submonoid of
$M$, then it follows from (A1) that $u=1\cdot u={}^{1}u\cdot 1={}^{1}u$ for
all $u\in U^{1}$, meaning that $S$ acts _monoidally_ in this case.

In any case, is important to note that $s=s\cdot 1={}^{s}1\cdot s$ for all
$s\in S$ in any weak action pair. When we return to the $s^{+}={}^{s}1$
notation later, this says that

$s=s^{+}s\qquad\text{for all $s\in S$.}$

Many of the action pairs we consider will satisfy stronger assumptions:

######  Definition 4.5.

A _strong (left) action pair_ in a monoid $M$ is a pair $(U,S)$ of
subsemigroups of $M$ for which the following conditions both hold:

  1. (SA1)

$sU^{1}\subseteq U^{1}s$ for all $s\in S$.

  2. (SA2)

$us=vt\ \Rightarrow\ u=v$ for all $u,v\in U^{1}$ and $s,t\in S$.

We call (SA2) the _left-uniqueness_ property.

######  Remark 4.6.

It is worth observing that $sU^{1}\subseteq U^{1}s$ in (SA1) is equivalent to
$sU\subseteq U^{1}s$. The condition $sU\subseteq Us$ of course implies
$sU^{1}\subseteq U^{1}s$, but not conversely in general; see Example 4.30.

The terminology obviously suggests that any strong action pair is an action
pair, even though no action was specified in Definition 4.5. This is indeed
the case, as we will show in Lemma 4.15. First, however, we give an
alternative characterisation of action pairs, which shows that certain key
properties of the elements ${}^{s}1$ ($s\in S$) determine the whole of the
action.

######  Proposition 4.7.

Let $U$ and $S$ be subsemigroups of a monoid $M$. Then $(U,S)$ is an action
pair if and only if (SA1) holds and there exists a map $S\to U^{1}:s\mapsto
s^{+}$ satisfying the following conditions:

$\displaystyle s$ $\displaystyle=s^{+}s$ for all $s\in S$, ($+$1)
$\displaystyle st^{+}$ $\displaystyle=(st)^{+}s$ for all $s,t\in S$, ($+$2)
$\displaystyle(st)^{+}$ $\displaystyle=(st)^{+}s^{+}$ for all $s,t\in S$,
($+$3) $\displaystyle us=vt\ \ $ $\displaystyle\Rightarrow\ \ us^{+}=vt^{+}$
for all $u,v\in U^{1}$ and $s,t\in S$. ($+$4)

###### Proof.

($\Rightarrow$). Suppose first that $(U,S)$ is an action pair. Since (A1)$\
\Rightarrow\ $(SA1), it remains to demonstrate the existence of a suitable map
$s\mapsto s^{+}$. For this, we define $s^{+}={}^{s}1$ for all $s\in S$.
Properties ($+$3) and ($+$4) then follow immediately from Lemma 3.1(iii) and
(A2), respectively, and we noted in Remark 4.4 that ($+$1) holds. For ($+$2),
let $s,t\in S$. Then (A1) and Lemma 3.1(ii) give

$st^{+}={}^{s}(t^{+})\cdot s=(st)^{+}s.$

($\Leftarrow$). Conversely, suppose (SA1) holds, and also that $S\to
U^{1}:s\mapsto s^{+}$ satisfies ($+$1)–($+$4).

We begin by defining an action of $S$ on $U^{1}$. To this end, let $s\in S$
and $u\in U^{1}$. By (SA1) we have $su=vs$ for some $v\in U^{1}$, and we
define ${}^{s}u=vs^{+}$. It follows quickly from ($+$4) that this is well
defined. Thus, for any $s\in S$ and $u\in U^{1}$, we have

${}^{s}u=vs^{+}\qquad\text{for any $v\in U^{1}$ such that $su=vs$.}$ (4.8)

We will soon show that this does indeed define an action by semigroup
morphisms. But first we observe that for $s,u,v$ as above, we have

$su=vs=vs^{+}s={}^{s}u\cdot s,$ (4.9)

where we used ($+$1) in the second step. Next we note that

${}^{s}1=s^{+}\qquad\text{for all $s\in S$.}$ (4.10)

Indeed, this follows immediately from (4.8) with $u=v=1$. Combining (4.10)
with ($+$4), we obtain (A2).

In light of (4.9), it remains to check that (4.8) does indeed define an action
of $S$ on $U^{1}$ by semigroup morphisms: i.e., that

${}^{st}u={}^{s}({}^{t}u)\qquad\text{and}\qquad{}^{s}(uv)={}^{s}u\cdot{}^{s}v\qquad\text{for
all $u,v\in U^{1}$ and $s,t\in S$.}$ (4.11)

To do this, we begin by showing that

${}^{s}u\cdot s^{+}={}^{s}u\qquad\text{for all $u\in U^{1}$ and $s\in S$.}$
(4.12)

To see this, first note that for any $s\in S$, ($+$1) and ($+$4) give

$1\cdot s=s=s^{+}\cdot s\ \ \ \Rightarrow\ \ \ 1\cdot s^{+}=s^{+}\cdot s^{+},$

so that each $s^{+}$ is an idempotent. Combining this with (4.8), we quickly
obtain (4.12).

Beginning with the second part of (4.11), let $s\in S$ and $u,v\in U^{1}$.
Several applications of (4.9) gives

${}^{s}(uv)\cdot s=s\cdot uv={}^{s}u\cdot s\cdot v={}^{s}u\cdot{}^{s}v\cdot
s,$

and it then follows from ($+$4) that ${}^{s}(uv)\cdot
s^{+}={}^{s}u\cdot{}^{s}v\cdot s^{+}$. We then apply (4.12) to both sides to
deduce ${}^{s}(uv)={}^{s}u\cdot{}^{s}v$.

For the first part of (4.11), let $s,t\in S$ and $u\in U^{1}$; we must show
that ${}^{st}u={}^{s}({}^{t}u)$. This time (4.9) gives

${}^{st}u\cdot st=st\cdot u=s\cdot{}^{t}u\cdot t={}^{s}({}^{t}u)\cdot st.$

It then follows from ($+$4) that
${{}^{st}u\cdot(st)^{+}={}^{s}({}^{t}u)\cdot(st)^{+}}$. Since
${}^{st}u\cdot(st)^{+}={}^{st}u$, by (4.12), the proof will be complete if we
can show that

${}^{s}({}^{t}u)\cdot(st)^{+}={}^{s}({}^{t}u).$ (4.13)

For this we use (4.9), (4.12), (4.9) and ($+$2) to calculate

${}^{s}({}^{t}u)\cdot s=s\cdot{}^{t}u=s\cdot{}^{t}u\cdot
t^{+}={}^{s}({}^{t}u)\cdot s\cdot t^{+}={}^{s}({}^{t}u)\cdot(st)^{+}\cdot s.$

It then follows from ($+$4) that ${}^{s}({}^{t}u)\cdot
s^{+}={}^{s}({}^{t}u)\cdot(st)^{+}\cdot s^{+}$. Combining this with (4.12) and
($+$3), we finally deduce that

${}^{s}({}^{t}u)={}^{s}({}^{t}u)\cdot s^{+}={}^{s}({}^{t}u)\cdot(st)^{+}\cdot
s^{+}={}^{s}({}^{t}u)\cdot(st)^{+}.$

This completes the proof of (4.13), and hence of the proposition. ∎

######  Remark 4.14.

Proposition 4.7 allows us to specify an action pair $(U,S)$ either by means
of:

  * •

an action $(s,u)\mapsto{}^{s}u$, as in Definition 4.2, or

  * •

an appropriate map $S\to U^{1}:s\mapsto s^{+}$.

The proof of the proposition shows how the two formulations are
interchangeable, but it is worth drawing this out explicitly here:

  * •

Given an action pair $(U,S)$, as in Definition 4.2, the map $S\to
U^{1}:s\mapsto s^{+}={}^{s}1$ satisfies ($+$1)–($+$4).

  * •

Conversely, if $(U,S)$ satisfies (SA1), and if $S\to U^{1}:s\mapsto s^{+}$
satisfies ($+$1)–($+$4), then $(U,S)$ is an action pair (as in Definition 4.2)
with respect to the action of $S$ on $U^{1}$ given in (4.8).

In all that follows, we will use either viewpoint, as convenient, typically
without further comment.

Here is the promised result justifying the terminology of strong action pairs;
it also gives a criterion for distinguishing the strong ones.

######  Lemma 4.15.

  1. (i)

Any strong action pair is an action pair.

  2. (ii)

An action pair $(U,S)$ is strong if and only if $s^{+}=1$ for all $s\in S$,
meaning that the action of $S$ on $U^{1}$ is by monoid morphisms.

###### Proof.

(i). The trivial map ${S\to U^{1}:s\mapsto s^{+}=1}$ always satisfies
conditions ($+$1)–($+$3). If $(U,S)$ is a strong action pair, then this
trivial map also satisfies condition ($+$4).

(ii). Suppose first that the action pair $(U,S)$ is strong, and let $s\in S$.
Then from $s^{+}\cdot s=s=1\cdot s$, it follows from (SA2) that $s^{+}=1$.

Conversely, suppose $s^{+}=1$ for all $s\in S$. Together with (A2), this
clearly implies (SA2). Since (SA1) always follows from (A1), it follows that
$(U,S)$ is strong. ∎

It follows that strong action pairs have the following equivalent definition:

######  Definition 4.16.

A _strong (left) action pair_ in a monoid $M$ is a pair $(U,S)$ of
subsemigroups of $M$ for which the following conditions both hold:

  1. (SA1)′

$S$ has a left action on $U^{1}$ by monoid morphisms, written
$(s,u)\mapsto{}^{s}u$, such that

$s\cdot u={}^{s}u\cdot s\qquad\text{for all $s\in S$ and $u\in U^{1}$.}$

  1. (SA2)${\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}^{\prime}}$

$us=vt\ \Rightarrow\ u=v$ for all $u,v\in U^{1}$ and $s,t\in S$.

######  Remark 4.17.

We have seen that a pair $(U,S)$ of subsemigroups of a monoid $M$ is a strong
action pair if and only if

  * •

(SA1) and (SA2) both hold (cf. Definition 4.5), or equivalently

  * •

(SA1)′ and (SA2) both hold (cf. Definition 4.16).

Since (SA1)′$\ \Rightarrow\ $(A1)$\ \Rightarrow\ $(SA1), $(U,S)$ is also
strong if and only if

  * •

(A1) and (SA2) both hold.

In Section 6.1, we will prove one result (Theorem 6.4) concerning pairs
$(U,S)$ satisfying (SA1)′, but not necessarily (SA2). This limited attention
does not seem to warrant naming such pairs.

The next lemma involves restricting action pairs to subsemigroups; it follows
quickly from an examination of Definitions 4.2 and 4.16.

######  Lemma 4.18.

If $(U,S)$ is a (strong) action pair, then so too is:

  1. (i)

$(U,T)$ for any $T\leq S$,

  2. (ii)

$(V,S)$ for any $V\leq U$ for which $V^{1}$ is closed under the action of $S$.
∎

For the next statement, we say that an action pair $(U,S)$ in $M$ _extends to_
$(V,T)$ if:

  * •

$(V,T)$ is an action pair in $M$,

  * •

$U\leq V$ and $S\leq T$, and

  * •

the map $S\to U^{1}:s\mapsto s^{+}$ is the restriction of the map $T\to
V^{1}:t\mapsto t^{+}$.

The third condition can be formulated equivalently in terms of actions of $S$
and $T$ on $U^{1}$ and $V^{1}$.

######  Lemma 4.19.

  1. (i)

Any action pair $(U,S)$ extends to $(U^{1},S)$.

  2. (ii)

An action pair $(U,S)$ extends to $(U,S^{1})$ and $(U^{1},S^{1})$ if and only
if the following condition holds:

     * •

$us=v\ \Rightarrow\ us^{+}=v$ for all $u,v\in U^{1}$ and $s\in S$.

  3. (iii)

Any strong action pair $(U,S)$ extends to $(U^{1},S)$, $(U,S^{1})$ and
$(U^{1},S^{1})$, and these are all strong.

###### Proof.

(i). This follows immediately from Definition 4.2, given that
$(U^{1})^{1}=U^{1}$.

(ii). Given part (i), we just need to prove the claim for the pair
$(U,S^{1})$. We first observe that (SA1) clearly holds for this pair. There is
only one way to extend the map $S\to U^{1}:s\mapsto s^{+}$ to a map $S^{1}\to
U^{1}$ satisfying ($+$1), since the latter implies $1^{+}=1$. Items
($+$1)–($+$3) all clearly hold for this extended map, while ($+$4) holds if
and only if the stated condition holds.

(iii). Item (SA1) is clear in each case. For (SA2), suppose $us=vt$ for some
$u,v\in U^{1}$ and $s,t\in S^{1}$. Then for any $x\in S$ we have
$u(sx)=v(tx)$, with $sx,tx\in S$, so it follows from left-uniqueness of
$(U,S)$ that $u=v$. ∎

The condition stated in part (ii) of the previous lemma leads to the following
useful fact.

######  Lemma 4.20.

If $(U,S)$ and $(U,S^{1})$ are both action pairs in $M$, then $US$ is a
submonoid of $M$ if and only if $U$ and $S$ are both submonoids.

###### Proof.

With the backwards implication being clear, suppose $US$ is a submonoid, so
that $us=1$ for some $u\in U$ and $s\in S$. Since $us=1\cdot 1$ with $u,1\in
U^{1}$ and $s,1\in S^{1}$, we can apply (A2) in the action pair $(U,S^{1})$ to
deduce that $us^{+}=1\cdot 1^{+}=1\cdot 1=1$. But then

$s=1\cdot s=us^{+}\cdot s=us=1,$

where we used ($+$1) in the third step. It follows that also $u=u\cdot
1=us=1$. Thus, ${1=u=s\in U\cap S}$. ∎

The next result provides a necessary condition for an action pair to be
strong.

######  Lemma 4.21.

If $(U,S)$ is a strong action pair, then

  1. (i)

$U\cap S\subseteq\\{1\\}$,

  2. (ii)

$1\in US\ \Leftrightarrow\ 1\in U\cap S\ \Leftrightarrow\ U\cap S=\\{1\\}\
\Leftrightarrow\ U\cap S\not=\varnothing$.

###### Proof.

(i). If $x\in U\cap S$, then from $x\cdot x=1\cdot x^{2}$, left-uniqueness
implies $x=1$.

(ii). The following implications are all obvious:

$U\cap S=\\{1\\}\ \Rightarrow\ 1\in U\cap S\ \Rightarrow\ U\cap
S\not=\varnothing\qquad\text{and}\qquad 1\in U\cap S\ \Rightarrow\ 1\in US.$

Part (i) gives $U\cap S\not=\varnothing\ \Rightarrow\ U\cap S=\\{1\\}$, so it
remains to show that $1\in US\ \Rightarrow\ 1\in U\cap S$. Since $(U,S)$ is
strong, $(U,S^{1})$ is a (strong) action pair, by Lemma 4.19(iii). The
required implication then follows from Lemma 4.20. ∎

######  Remark 4.22.

The assumption that $(U,S)$ is strong cannot be removed from Lemma 4.21. For
example, let $M$ be a left restriction monoid with semilattice of projections
$P$; see Section 2.3 for the definitions. We will show in Proposition 4.37
that $(P,M)$ is an action pair, and we note that $P\cap M=P$ need not contain
only the identity element.

Now we provide a _sufficient_ condition. Recall that a _right unit_ of a
monoid $M$ is an element $s\in M$ such that $1=st$ for some $t\in M$. We write
$R(M)=\\{{s\in M}:{1\in sM}\\}$ for the submonoid of all right units of $M$.
(The submonoid $R(M)$ is also Green’s $\mathrel{\mathscr{R}}$-class of $1\in
M$.)

######  Lemma 4.23.

Suppose $(U,S)$ is an action pair in a monoid $M$.

  1. (i)

If $s\in S\cap R(M)$, then $s^{+}=1$.

  2. (ii)

If $S\subseteq R(M)$, then $(U,S)$ is strong.

###### Proof.

By Lemma 4.15(ii), it suffices to prove the first part. For this, suppose
${s\in S\cap R(M)}$, so that $1=st$ for some $t\in M$. Combining this with
($+$1), we have ${1=st=s^{+}st=s^{+}}$. ∎

######  Remark 4.24.

It of course follows from Lemma 4.23(ii) that $(U,S)$ is strong if $S$ is
contained in the group of (two-sided) units of $M$.

The condition $S\subseteq R(M)$ is certainly not necessary for $(U,S)$ to be
strong; see Example 4.30 (out of _many_ other examples considered in the
paper).

The final result of this section shows that (A2) is equivalent to an
ostensibly stronger condition.

######  Lemma 4.25.

If $(U,S)$ is an action pair, then for any $u,v,w\in U^{1}$ and $s,t\in S$, we
have

$us=vt\ \Rightarrow\ u\cdot{}^{s}w=v\cdot{}^{t}w.$

###### Proof.

Combining $us=vt$ with (A1), we have

$u\cdot{}^{s}w\cdot s=us\cdot w=vt\cdot w=v\cdot{}^{t}w\cdot t.$

It then follows from (A2) that $u\cdot{}^{s}w\cdot s^{+}=v\cdot{}^{t}w\cdot
t^{+}$, and Lemma 3.1(iv) completes the proof. ∎

Before we move on, we note that there are obviously right-handed versions of
our action pairs. By symmetry/duality, results concerning _right_ pairs can be
directly obtained from their left-handed counterparts, so we focus only on the
left. The left-handed theory is our chosen focus, as it applies directly to
all of our motivating examples.

There is also a notion of _right_ -uniqueness (for _left_ action pairs), where
${us=vt\ \Rightarrow\ s=t}$, in contrast to the left-uniqueness property
(SA2). Right-uniqueness is satisfied, for example, by free left restriction
monoids, which are the topic of Chapter 7. However, right-uniqueness is
equivalent to certain right congruences on $S$ (introduced in Section 4.4) all
being trivial, so there is no need to develop a parallel theory in this case.

####  4.2 Examples

The following collection of examples and non-examples highlight some
subtleties in the definitions. They also show that the semigroups $US$ arising
from (weak/strong) action pairs $(U,S)$ include some well known classes.
Additional examples will be explored in greater depth in the next section, and
in Part II of the paper.

Several of the examples discussed below use the transformation monoids
$\mathcal{P}\mathcal{T}_{X}$, $\mathcal{T}_{X}$,
$\operatorname{Sing}(\mathcal{T}_{n})$, and so on, as defined in Section 2.2.
Some involve inverse semigroups, as defined in Section 2.1. For the
calculations in such examples, we make use of the previously-mentioned fact
that idempotents of an inverse semigroup commute. It is also clear that every
idempotent of an inverse semigroup is its own inverse. On a number of
occasions we make use of the easily checked fact that if $U$ and $S$ are
submonoids of a monoid $M$, and if $SU\subseteq US$, then $US=\langle U\cup
S\rangle$.

We begin with a somewhat degenerate construction:

######  Example 4.26.

Let $U$ and $S$ be arbitrary semigroups, and assume $U\cap S=\varnothing$. Let
$0$ and $1$ be symbols not belonging to $U\cup S$, and let $M=U\cup
S\cup\\{0,1\\}$. Then $M$ is a monoid under the product that extends the
original products in $U$ and $S$, and where additionally

$us=su=0,\ \ \ x0=0x=0\quad\text{and}\quad x1=1x=x\qquad\text{for all $u\in
U$, $s\in S$ and $x\in M$.}$

(So $M$ is the monoid completion of the _0-direct union_ of $U$ and $S$.) For
any action of $S$ on $U$ by morphisms, $(U,S)$ is a weak action pair in $M$
with respect to the extended action of $S$ on $U^{1}$. We then of course have
$US=SU=\\{0\\}$.

The next series of examples involve idempotents and one- or two-sided units.

######  Example 4.27.

Let $M$ be an inverse monoid, and let

$E=E(M)=\\{{e\in M}:{e=e^{2}}\\}\qquad\text{and}\qquad G=G(M)=\\{{g\in
M}:{gg^{-1}=g^{-1}g=1}\\}$

be the semilattice of idempotents, and group of units of $M$. It is easy to
see that ${(E,G)=(E^{1},G^{1})}$ is a strong action pair in $M$:

  1. (SA1)

For $g\in G$ and $e\in E(=E^{1})$, we have $ge=(geg^{-1})g$, with $geg^{-1}\in
E$. This shows that $gE\subseteq Eg$. In fact, it is easy to see that $gE=Eg$.
(The action of $G$ on $E(=E^{1})$ from (A1) or (SA1)′ is given by
${}^{g}e=geg^{-1}$.)

  2. (SA2)

Suppose $eg=fh$ for some $e,f\in E$ and $g,h\in G$. Then from $e=fhg^{-1}$ we
obtain $e=fe$. Similarly, $f=ef$, and since idempotents commute, it follows
that $e=f$.

(Alternatively, one could verify conditions (A1) and (A2), and then apply
Lemma 4.23(ii).) The resulting subsemigroup $EG=GE\leq M$ is the largest
_factorisable inverse submonoid_ of $M$. See [14, 42] for more on factorisable
inverse monoids, and [24] for presentations. By the symmetry afforded by the
inversion map, $(E,G)$ is also a strong _right_ action pair.

If we write $E^{\flat}=E\setminus\\{1\\}$, then $(E^{\flat},G)$ is also a
strong action pair in $M$ (cf. Lemma 4.18(ii)). The subsemigroup
$E^{\flat}G=GE^{\flat}=EG\setminus G$ is an _almost-factorisable inverse
semigroup_ in the sense of Lawson [76]; see also [88]. Note that this class of
inverse semigroups has an abstract definition, and it is a highly nontrivial
result that any almost-factorisable inverse semigroup has the form
$EG\setminus G$ for suitable $E$ and $G$ (and $M$).

Generalisations of Example 4.27 will be discussed in Section 4.3.

######  Example 4.28.

Following on from the previous example, let $M$ be an arbitrary monoid, $E$
the set of idempotents of $M$, and $G$ the group of units. Since $E$ need not
be a submonoid, we also write $U=\langle E\rangle$ for the submonoid generated
by the idempotents. Since the conjugate of an idempotent by a unit is an
idempotent, we have an action of $G$ on $U$ given by ${}^{g}u=gug^{-1}$, with
respect to which $g\cdot u={}^{g}u\cdot g$. Thus, (A1) holds, meaning that
$(U,G)$ is a weak action pair. Of course it follows that (SA1) holds as well.
We have $UG=GU=\langle E\cup G\rangle$ by [38, Lemma 32].

Since ${}^{g}1=1$ for all $g\in G$, conditions (A2) and (SA2) are equivalent
for the pair $(U,G)$, and we note that these need not hold in general
(although we have just seen in Example 4.27 that they do hold when $M$ is
inverse). For a specific example of this, let
$M=\mathcal{T}_{2}=\\{\alpha,\beta,\gamma,\delta\\}$, where

$\alpha=\left(\begin{smallmatrix}1&2\\\
1&1\end{smallmatrix}\right),\qquad\beta=\left(\begin{smallmatrix}1&2\\\
2&2\end{smallmatrix}\right),\qquad\gamma=\left(\begin{smallmatrix}1&2\\\
1&2\end{smallmatrix}\right)\qquad\text{and}\qquad\delta=\left(\begin{smallmatrix}1&2\\\
2&1\end{smallmatrix}\right).$

Then $U=E=\\{\alpha,\beta,\gamma\\}$ and $G=\\{\gamma,\delta\\}$, yet
$\alpha\gamma=\beta\delta$.

######  Example 4.29.

Again let $M$ be an arbitrary monoid, and let $G$, $E$ and $U=\langle
E\rangle$ be as in the previous example. This time let

$L=L(M)=\\{{s\in M}:{1\in Ms}\\}\qquad\text{and}\qquad R=R(M)=\\{{s\in
M}:{1\in sM}\\}$

be the submonoids of left and right units.

Let $s\in L$, so that $1=s^{\prime}s$ for some $s^{\prime}\in M$. For any
idempotent $e\in E$, we have $se=(ses^{\prime})s$, and it is easy to check
that $ses^{\prime}$ is an idempotent. This quickly leads to $sU\subseteq Us$,
so that (SA1) holds for the pair $(U,L)$. We have $UL=\langle E\cup L\rangle$
by [34, Lemma 2.5].

Condition (SA2) still does not hold for $(U,L)$ in general. Indeed, this is
again witnessed by $M=\mathcal{T}_{2}$ from Example 4.28, since when $M$ is
finite every left unit is a (two-sided) unit so that $L=G$ is the group of
units. Even though $(U,L)$ is not left-unique in general, we still always have
$U\cap L=\\{1\\}$; see [34, Lemma 2.1], and cf. Lemma 4.21.

Regarding condition (A1) for the pair $(U,L)$, for each $s\in L$ we fix some
$s^{\prime}\in M$ such that $1=s^{\prime}s$. Then for $s\in L$ and $u\in
U(=U^{1})$ we define ${}^{s}u=sus^{\prime}\in U$, and we have $su={}^{s}u\cdot
s$. We also have ${}^{s}(uv)={}^{s}u\cdot{}^{s}v$ for all $s\in L$ and $u,v\in
U$. However, for (A1) to hold, we also require that ${}^{st}u={}^{s}({}^{t}u)$
for all $s,t\in L$ and $u\in U$: i.e.,
$(st)u(st)^{\prime}=s(tut^{\prime})s^{\prime}$ for all such $s,t,u$. This
would be the case if we could choose the elements $s^{\prime}$ in such a way
that $(st)^{\prime}=t^{\prime}s^{\prime}$ for all $s,t\in L$, as is the case
for example when $M$ is inverse (and $s^{\prime}=s^{-1}$).

The above discussion concentrated on the pair $(U,L)$. We note that the pair
$(U,R)$ might not even satisfy (SA1). For a specific example of this, let
$M=\mathcal{PB}_{\mathbb{N}}$ be the partial Brauer monoid over
$\mathbb{N}=\\{0,1,2,\ldots\\}$. The definition of this monoid is somewhat
involved, so we refer the reader to [34]. Consider the elements

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{}{}{}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@curveto{8.5359pt}{15.18614pt}{10.06453pt}{13.65752pt}{11.95021pt}{13.65752pt}\pgfsys@moveto{11.95021pt}{13.6575pt}\pgfsys@lineto{13.6575pt}{13.6575pt}\pgfsys@moveto{13.6575pt}{13.6575pt}\pgfsys@curveto{15.54318pt}{13.6575pt}{17.07181pt}{15.18613pt}{17.07181pt}{17.07181pt}\pgfsys@stroke\pgfsys@invoke{
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}{}\pgfsys@moveto{51.21548pt}{0.0pt}\pgfsys@lineto{85.35913pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{0.4pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{68.28731pt}{17.07182pt}\pgfsys@lineto{85.35913pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
}}\qquad\text{and}\qquad\eta={\lower 6.02773pt\hbox{ \leavevmode\hbox
to78.73pt{\vbox
to20.49pt{\pgfpicture\makeatletter\hbox{\hskip-6.82875pt\lower-1.70715pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@moveto{10.24306pt}{17.07182pt}\pgfsys@curveto{10.24306pt}{18.01466pt}{9.47874pt}{18.77898pt}{8.5359pt}{18.77898pt}\pgfsys@curveto{7.59306pt}{18.77898pt}{6.82875pt}{18.01466pt}{6.82875pt}{17.07182pt}\pgfsys@curveto{6.82875pt}{16.12898pt}{7.59306pt}{15.36467pt}{8.5359pt}{15.36467pt}\pgfsys@curveto{9.47874pt}{15.36467pt}{10.24306pt}{16.12898pt}{10.24306pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@moveto{18.77898pt}{17.07182pt}\pgfsys@curveto{18.77898pt}{18.01466pt}{18.01466pt}{18.77898pt}{17.07182pt}{18.77898pt}\pgfsys@curveto{16.12898pt}{18.77898pt}{15.36467pt}{18.01466pt}{15.36467pt}{17.07182pt}\pgfsys@curveto{15.36467pt}{16.12898pt}{16.12898pt}{15.36467pt}{17.07182pt}{15.36467pt}\pgfsys@curveto{18.01466pt}{15.36467pt}{18.77898pt}{16.12898pt}{18.77898pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@moveto{27.3149pt}{17.07182pt}\pgfsys@curveto{27.3149pt}{18.01466pt}{26.55058pt}{18.77898pt}{25.60774pt}{18.77898pt}\pgfsys@curveto{24.6649pt}{18.77898pt}{23.90059pt}{18.01466pt}{23.90059pt}{17.07182pt}\pgfsys@curveto{23.90059pt}{16.12898pt}{24.6649pt}{15.36467pt}{25.60774pt}{15.36467pt}\pgfsys@curveto{26.55058pt}{15.36467pt}{27.3149pt}{16.12898pt}{27.3149pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@moveto{35.8508pt}{17.07182pt}\pgfsys@curveto{35.8508pt}{18.01466pt}{35.08649pt}{18.77898pt}{34.14365pt}{18.77898pt}\pgfsys@curveto{33.2008pt}{18.77898pt}{32.4365pt}{18.01466pt}{32.4365pt}{17.07182pt}\pgfsys@curveto{32.4365pt}{16.12898pt}{33.2008pt}{15.36467pt}{34.14365pt}{15.36467pt}\pgfsys@curveto{35.08649pt}{15.36467pt}{35.8508pt}{16.12898pt}{35.8508pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{42.67957pt}{17.07182pt}\pgfsys@moveto{44.38672pt}{17.07182pt}\pgfsys@curveto{44.38672pt}{18.01466pt}{43.6224pt}{18.77898pt}{42.67957pt}{18.77898pt}\pgfsys@curveto{41.73672pt}{18.77898pt}{40.97241pt}{18.01466pt}{40.97241pt}{17.07182pt}\pgfsys@curveto{40.97241pt}{16.12898pt}{41.73672pt}{15.36467pt}{42.67957pt}{15.36467pt}\pgfsys@curveto{43.6224pt}{15.36467pt}{44.38672pt}{16.12898pt}{44.38672pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{42.67957pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{51.21548pt}{17.07182pt}\pgfsys@moveto{52.92264pt}{17.07182pt}\pgfsys@curveto{52.92264pt}{18.01466pt}{52.15833pt}{18.77898pt}{51.21548pt}{18.77898pt}\pgfsys@curveto{50.27264pt}{18.77898pt}{49.50833pt}{18.01466pt}{49.50833pt}{17.07182pt}\pgfsys@curveto{49.50833pt}{16.12898pt}{50.27264pt}{15.36467pt}{51.21548pt}{15.36467pt}\pgfsys@curveto{52.15833pt}{15.36467pt}{52.92264pt}{16.12898pt}{52.92264pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{51.21548pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{59.75139pt}{17.07182pt}\pgfsys@moveto{61.45854pt}{17.07182pt}\pgfsys@curveto{61.45854pt}{18.01466pt}{60.69423pt}{18.77898pt}{59.75139pt}{18.77898pt}\pgfsys@curveto{58.80855pt}{18.77898pt}{58.04424pt}{18.01466pt}{58.04424pt}{17.07182pt}\pgfsys@curveto{58.04424pt}{16.12898pt}{58.80855pt}{15.36467pt}{59.75139pt}{15.36467pt}\pgfsys@curveto{60.69423pt}{15.36467pt}{61.45854pt}{16.12898pt}{61.45854pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{59.75139pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{10.24306pt}{0.0pt}\pgfsys@curveto{10.24306pt}{0.94284pt}{9.47874pt}{1.70715pt}{8.5359pt}{1.70715pt}\pgfsys@curveto{7.59306pt}{1.70715pt}{6.82875pt}{0.94284pt}{6.82875pt}{0.0pt}\pgfsys@curveto{6.82875pt}{-0.94284pt}{7.59306pt}{-1.70715pt}{8.5359pt}{-1.70715pt}\pgfsys@curveto{9.47874pt}{-1.70715pt}{10.24306pt}{-0.94284pt}{10.24306pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
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}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{25.60774pt}{0.0pt}\pgfsys@moveto{27.3149pt}{0.0pt}\pgfsys@curveto{27.3149pt}{0.94284pt}{26.55058pt}{1.70715pt}{25.60774pt}{1.70715pt}\pgfsys@curveto{24.6649pt}{1.70715pt}{23.90059pt}{0.94284pt}{23.90059pt}{0.0pt}\pgfsys@curveto{23.90059pt}{-0.94284pt}{24.6649pt}{-1.70715pt}{25.60774pt}{-1.70715pt}\pgfsys@curveto{26.55058pt}{-1.70715pt}{27.3149pt}{-0.94284pt}{27.3149pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.60774pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{34.14365pt}{0.0pt}\pgfsys@moveto{35.8508pt}{0.0pt}\pgfsys@curveto{35.8508pt}{0.94284pt}{35.08649pt}{1.70715pt}{34.14365pt}{1.70715pt}\pgfsys@curveto{33.2008pt}{1.70715pt}{32.4365pt}{0.94284pt}{32.4365pt}{0.0pt}\pgfsys@curveto{32.4365pt}{-0.94284pt}{33.2008pt}{-1.70715pt}{34.14365pt}{-1.70715pt}\pgfsys@curveto{35.08649pt}{-1.70715pt}{35.8508pt}{-0.94284pt}{35.8508pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14365pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{42.67957pt}{0.0pt}\pgfsys@moveto{44.38672pt}{0.0pt}\pgfsys@curveto{44.38672pt}{0.94284pt}{43.6224pt}{1.70715pt}{42.67957pt}{1.70715pt}\pgfsys@curveto{41.73672pt}{1.70715pt}{40.97241pt}{0.94284pt}{40.97241pt}{0.0pt}\pgfsys@curveto{40.97241pt}{-0.94284pt}{41.73672pt}{-1.70715pt}{42.67957pt}{-1.70715pt}\pgfsys@curveto{43.6224pt}{-1.70715pt}{44.38672pt}{-0.94284pt}{44.38672pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{42.67957pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{51.21548pt}{0.0pt}\pgfsys@moveto{52.92264pt}{0.0pt}\pgfsys@curveto{52.92264pt}{0.94284pt}{52.15833pt}{1.70715pt}{51.21548pt}{1.70715pt}\pgfsys@curveto{50.27264pt}{1.70715pt}{49.50833pt}{0.94284pt}{49.50833pt}{0.0pt}\pgfsys@curveto{49.50833pt}{-0.94284pt}{50.27264pt}{-1.70715pt}{51.21548pt}{-1.70715pt}\pgfsys@curveto{52.15833pt}{-1.70715pt}{52.92264pt}{-0.94284pt}{52.92264pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{51.21548pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{59.75139pt}{0.0pt}\pgfsys@moveto{61.45854pt}{0.0pt}\pgfsys@curveto{61.45854pt}{0.94284pt}{60.69423pt}{1.70715pt}{59.75139pt}{1.70715pt}\pgfsys@curveto{58.80855pt}{1.70715pt}{58.04424pt}{0.94284pt}{58.04424pt}{0.0pt}\pgfsys@curveto{58.04424pt}{-0.94284pt}{58.80855pt}{-1.70715pt}{59.75139pt}{-1.70715pt}\pgfsys@curveto{60.69423pt}{-1.70715pt}{61.45854pt}{-0.94284pt}{61.45854pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{59.75139pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@lineto{25.60774pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@lineto{34.14365pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67957pt}{17.07182pt}\pgfsys@lineto{42.67957pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{51.21548pt}{17.07182pt}\pgfsys@lineto{51.21548pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{59.75139pt}{17.07182pt}\pgfsys@lineto{59.75139pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{0.4pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{68.28731pt}{17.07182pt}\pgfsys@lineto{85.35913pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{0.4pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{68.28731pt}{0.0pt}\pgfsys@lineto{85.35913pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}$

from $\mathcal{PB}_{\mathbb{N}}$. Then $\eta\in E\subseteq U$, and since
$\alpha\beta=1$ we have $\alpha\in R$. We then have

$\alpha\eta={\lower 6.02773pt\hbox{ \leavevmode\hbox to78.73pt{\vbox
to20.49pt{\pgfpicture\makeatletter\hbox{\hskip-6.82875pt\lower-1.70715pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@moveto{10.24306pt}{17.07182pt}\pgfsys@curveto{10.24306pt}{18.01466pt}{9.47874pt}{18.77898pt}{8.5359pt}{18.77898pt}\pgfsys@curveto{7.59306pt}{18.77898pt}{6.82875pt}{18.01466pt}{6.82875pt}{17.07182pt}\pgfsys@curveto{6.82875pt}{16.12898pt}{7.59306pt}{15.36467pt}{8.5359pt}{15.36467pt}\pgfsys@curveto{9.47874pt}{15.36467pt}{10.24306pt}{16.12898pt}{10.24306pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@moveto{18.77898pt}{17.07182pt}\pgfsys@curveto{18.77898pt}{18.01466pt}{18.01466pt}{18.77898pt}{17.07182pt}{18.77898pt}\pgfsys@curveto{16.12898pt}{18.77898pt}{15.36467pt}{18.01466pt}{15.36467pt}{17.07182pt}\pgfsys@curveto{15.36467pt}{16.12898pt}{16.12898pt}{15.36467pt}{17.07182pt}{15.36467pt}\pgfsys@curveto{18.01466pt}{15.36467pt}{18.77898pt}{16.12898pt}{18.77898pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@moveto{27.3149pt}{17.07182pt}\pgfsys@curveto{27.3149pt}{18.01466pt}{26.55058pt}{18.77898pt}{25.60774pt}{18.77898pt}\pgfsys@curveto{24.6649pt}{18.77898pt}{23.90059pt}{18.01466pt}{23.90059pt}{17.07182pt}\pgfsys@curveto{23.90059pt}{16.12898pt}{24.6649pt}{15.36467pt}{25.60774pt}{15.36467pt}\pgfsys@curveto{26.55058pt}{15.36467pt}{27.3149pt}{16.12898pt}{27.3149pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@moveto{35.8508pt}{17.07182pt}\pgfsys@curveto{35.8508pt}{18.01466pt}{35.08649pt}{18.77898pt}{34.14365pt}{18.77898pt}\pgfsys@curveto{33.2008pt}{18.77898pt}{32.4365pt}{18.01466pt}{32.4365pt}{17.07182pt}\pgfsys@curveto{32.4365pt}{16.12898pt}{33.2008pt}{15.36467pt}{34.14365pt}{15.36467pt}\pgfsys@curveto{35.08649pt}{15.36467pt}{35.8508pt}{16.12898pt}{35.8508pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{42.67957pt}{17.07182pt}\pgfsys@moveto{44.38672pt}{17.07182pt}\pgfsys@curveto{44.38672pt}{18.01466pt}{43.6224pt}{18.77898pt}{42.67957pt}{18.77898pt}\pgfsys@curveto{41.73672pt}{18.77898pt}{40.97241pt}{18.01466pt}{40.97241pt}{17.07182pt}\pgfsys@curveto{40.97241pt}{16.12898pt}{41.73672pt}{15.36467pt}{42.67957pt}{15.36467pt}\pgfsys@curveto{43.6224pt}{15.36467pt}{44.38672pt}{16.12898pt}{44.38672pt}{17.07182pt}\pgfsys@closepath\pgfsys@moveto{42.67957pt}{17.07182pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@moveto{10.24306pt}{0.0pt}\pgfsys@curveto{10.24306pt}{0.94284pt}{9.47874pt}{1.70715pt}{8.5359pt}{1.70715pt}\pgfsys@curveto{7.59306pt}{1.70715pt}{6.82875pt}{0.94284pt}{6.82875pt}{0.0pt}\pgfsys@curveto{6.82875pt}{-0.94284pt}{7.59306pt}{-1.70715pt}{8.5359pt}{-1.70715pt}\pgfsys@curveto{9.47874pt}{-1.70715pt}{10.24306pt}{-0.94284pt}{10.24306pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@moveto{18.77898pt}{0.0pt}\pgfsys@curveto{18.77898pt}{0.94284pt}{18.01466pt}{1.70715pt}{17.07182pt}{1.70715pt}\pgfsys@curveto{16.12898pt}{1.70715pt}{15.36467pt}{0.94284pt}{15.36467pt}{0.0pt}\pgfsys@curveto{15.36467pt}{-0.94284pt}{16.12898pt}{-1.70715pt}{17.07182pt}{-1.70715pt}\pgfsys@curveto{18.01466pt}{-1.70715pt}{18.77898pt}{-0.94284pt}{18.77898pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{25.60774pt}{0.0pt}\pgfsys@moveto{27.3149pt}{0.0pt}\pgfsys@curveto{27.3149pt}{0.94284pt}{26.55058pt}{1.70715pt}{25.60774pt}{1.70715pt}\pgfsys@curveto{24.6649pt}{1.70715pt}{23.90059pt}{0.94284pt}{23.90059pt}{0.0pt}\pgfsys@curveto{23.90059pt}{-0.94284pt}{24.6649pt}{-1.70715pt}{25.60774pt}{-1.70715pt}\pgfsys@curveto{26.55058pt}{-1.70715pt}{27.3149pt}{-0.94284pt}{27.3149pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.60774pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{34.14365pt}{0.0pt}\pgfsys@moveto{35.8508pt}{0.0pt}\pgfsys@curveto{35.8508pt}{0.94284pt}{35.08649pt}{1.70715pt}{34.14365pt}{1.70715pt}\pgfsys@curveto{33.2008pt}{1.70715pt}{32.4365pt}{0.94284pt}{32.4365pt}{0.0pt}\pgfsys@curveto{32.4365pt}{-0.94284pt}{33.2008pt}{-1.70715pt}{34.14365pt}{-1.70715pt}\pgfsys@curveto{35.08649pt}{-1.70715pt}{35.8508pt}{-0.94284pt}{35.8508pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14365pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{42.67957pt}{0.0pt}\pgfsys@moveto{44.38672pt}{0.0pt}\pgfsys@curveto{44.38672pt}{0.94284pt}{43.6224pt}{1.70715pt}{42.67957pt}{1.70715pt}\pgfsys@curveto{41.73672pt}{1.70715pt}{40.97241pt}{0.94284pt}{40.97241pt}{0.0pt}\pgfsys@curveto{40.97241pt}{-0.94284pt}{41.73672pt}{-1.70715pt}{42.67957pt}{-1.70715pt}\pgfsys@curveto{43.6224pt}{-1.70715pt}{44.38672pt}{-0.94284pt}{44.38672pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{42.67957pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{51.21548pt}{0.0pt}\pgfsys@moveto{52.92264pt}{0.0pt}\pgfsys@curveto{52.92264pt}{0.94284pt}{52.15833pt}{1.70715pt}{51.21548pt}{1.70715pt}\pgfsys@curveto{50.27264pt}{1.70715pt}{49.50833pt}{0.94284pt}{49.50833pt}{0.0pt}\pgfsys@curveto{49.50833pt}{-0.94284pt}{50.27264pt}{-1.70715pt}{51.21548pt}{-1.70715pt}\pgfsys@curveto{52.15833pt}{-1.70715pt}{52.92264pt}{-0.94284pt}{52.92264pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{51.21548pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{59.75139pt}{0.0pt}\pgfsys@moveto{61.45854pt}{0.0pt}\pgfsys@curveto{61.45854pt}{0.94284pt}{60.69423pt}{1.70715pt}{59.75139pt}{1.70715pt}\pgfsys@curveto{58.80855pt}{1.70715pt}{58.04424pt}{0.94284pt}{58.04424pt}{0.0pt}\pgfsys@curveto{58.04424pt}{-0.94284pt}{58.80855pt}{-1.70715pt}{59.75139pt}{-1.70715pt}\pgfsys@curveto{60.69423pt}{-1.70715pt}{61.45854pt}{-0.94284pt}{61.45854pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{59.75139pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{8.5359pt}{17.07182pt}\pgfsys@lineto{25.60774pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@lineto{34.14365pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{25.60774pt}{17.07182pt}\pgfsys@lineto{42.67957pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{34.14365pt}{17.07182pt}\pgfsys@lineto{51.21548pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67957pt}{17.07182pt}\pgfsys@lineto{59.75139pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{0.4pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{51.21548pt}{17.07182pt}\pgfsys@lineto{85.35913pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{0.4pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{68.28731pt}{0.0pt}\pgfsys@lineto{85.35913pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}\in\alpha U,$

but we claim that $\alpha\eta\not\in U\alpha$, which then of course implies
that $\alpha U\not\subseteq U\alpha$. In fact, $\alpha\eta$ does not even
belong to $\mathcal{PB}_{\mathbb{N}}\cdot\alpha$, as every element of
$\mathcal{PB}_{\mathbb{N}}\cdot\alpha$ contains at least one lower ‘hook’