Left reductive regular semigroups

P. A. Azeef Muhammed Centre for Research in Mathematics and Data Science, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia. azeefp@gmail.com, A.ParayilAjmal@WesternSydney.edu.au and Gracinda M. S. Gomes Faculdade de Ciências, Departamento de Matemática, Universidade de Lisboa, 1749-016, Lisboa, Portugal gracindamsgmcunha@gmail.com, gmcunha@fc.ul.pt

Abstract.

In this paper, we develop an ideal structure theory for the class of left reductive regular semigroups and apply it to several subclasses of popular interest. In these classes, we observe that the right ideal structure of the semigroup is ‘embedded’ inside the left ideal one, and so we can construct these semigroups starting with only one object (unlike in other more general cases). To this end, we introduce an upgraded version of the Nambooripad’s normal category [41] as our building block, which we call a connected category.

The main theorem of the paper describes a category equivalence between the category of left reductive regular semigroups and the category of connected categories. Then, we specialise our result to describe constructions of $\mathrel{\mathscr{L}}$ -unipotent semigroups, right regular bands, inverse semigroups and arbitrary regular monoids. Exploiting the left-right duality of semigroups, we also construct right reductive regular semigroups and use that to describe the more particular subclasses of $\mathrel{\mathscr{R}}$ -unipotent semigroups and left regular bands. Finally, we provide concrete (and rather simple) descriptions to the connected categories that arise from finite transformation semigroups, linear transformation semigroups (over a finite dimensional vector space) and symmetric inverse monoids.

Key words and phrases:

This work was partially developed within the activities of Centro de Matemática Computacional e Estocástica, CEMAT, and Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa, within the projects UIDB/04621/2020 and UIDP/04621/2020, financed by Fundação para a Ciência e a Tecnologia, FCT

1. Introduction

The most important algebraic invariants of any given semigroup are its Green relations which describe the ideal structure of the semigroup. Introduced in a seminal paper [22] in 1951, Green relations are certain equivalence relations defined on a semigroup which partition the semigroup elements into an ‘egg-box’ diagram (for example, see Figures 3, 4 and 5). In this partitioning, the elements generating the same principal left ideals fall in the same column of the egg-box and those generating the same principal right ideal fall in the same row of the diagram. This captures a lot of information regarding the local and global structure of the semigroup. In fact, it is precisely this partitioning that makes semigroups manageable, in spite of them being rather general objects. Hence, it is no surprise that Howie would remark the Green relations are “so all-pervading that, on encountering a new semigroup, almost the first question one asks is ‘What are the Green relations like?’” [30]. Therefore, it is very natural that any structure theorem for semigroups may aim to begin with building blocks that abstract the principal left (and right) ideals. To make any headway in this direction, one needs to understand the inter-relationship between principal left and principal right ideal structures. In general, this is quite complicated and invariably involves two ‘ordered objects’ (each abstracting left and right ideal structures of the semigroup) interconnected in a non-trivial fashion. Unsurprisingly, this rather difficult question is still open in many general cases. Such a quest leads naturally towards the special class of regular semigroups.

In regular semigroups, each principal left (or right) ideal is generated by idempotent elements, giving some control over the structure of the semigroup. Indeed there is a very close relationship between the ideal structure and the idempotent structure, and in fact, we can obtain one from the other [38, 6, 7]. Recall that these semigroups were introduced by Green in [22], wherein he credited Rees for the suggestion to adopt von Neumann’s definition [51] from ring theory.

Definition 1.1.

A semigroup $S$ is said to be (von Neumann) regular semigroup if for every element $a$ in $S$ , there exists $x\in S$ such that $axa=a$ .

Historically, one of the first major leaps into regular semigroups was by Hall [25] who extended Munn’s [35] construction of fundamental inverse semigroups to fundamental regular semigroups generated by idempotents. To this end, Hall considered certain transformations on the partially ordered set (poset) of the principal left ideals and on that of the principal right ideals. Later, Grillet [23] gave an abstract characterisation of these posets as regular posets, and introduced the notion of cross-connection to describe the exact relationship between the left and the right posets of a regular semigroup. Simultaneously, Nambooripad [36, 37, 39] developed the idea of (regular) biordered set, as an abstract model of the set of idempotents of a (regular) semigroup, and using groupoids gave a general structure theorem for regular semigroups. This seminal work also described an equivalence between the category of regular semigroups and the category of certain groupoids, and in the process, it puts final touches to the celebrated ESN (Ehresmann-Schein-Nambooripad) Theorem [32]. Although extremely clever, Nambooripad’s description of the sets of idempotents as biordered sets is still complicated and pretty cumbersome to work with, especially for constructions. In 1978, Nambooripad [38] showed that regular biordered sets and cross-connected regular posets are equivalent. Elaborating on this fact, he developed his theory of cross-connections [41] by replacing regular posets with normal categories (which contain regular posets as subcategories). In this way, he proved that the category of regular semigroups is equivalent to the category of cross-connected normal categories.

A major problem with such a general approach is that the theory developed is too heavy to be applied to the vast majority of the objects which may have a rather simple structure! In fact, we believe this is one of the major reasons why Nambooripad’s cross-connection theory has not achieved the popularity and acclaim that such a deep work deserves. Addressing this wide gap in the literature is one of the main motivations behind this paper.

As the reader may observe, the entire discussion in this paper can be traced back to the Cayley’s theorem for groups. Just as any group $G$ may be realised as a subgroup of the symmetric group on the set $G$ , it is well-known that any semigroup $S$ can be looked at as a transformation semigroup on the set $S^{1}:=S\cup{1}$ , [29]. This may be achieved by considering the regular representation of $S$ [13, Section 1.3], which is the homomorphism $\rho\colon S\mapsto\mathscr{T}_{S}$ , $a\mapsto\rho_{a}$ , where $\rho_{a}\colon x\mapsto xa$ , for every $x\in S$ . Adjoining the element $1$ to the set $S$ (if $S$ is not a monoid) is sufficient to ensure injectivity of the representation, and so in this case $S$ is isomorphic $S\rho$ .

So, a question arises: for which classes of semigroups do we have injectivity without adjoining $1$ ? This leads us to left reductive semigroups.

Definition 1.2.

A semigroup $S$ is said to be left reductive if the regular representation $\rho$ is injective.

In this paper, we discuss the ideal structure of the class of left reductive regular semigroups¹¹1All semigroups considered in this paper are regular. So, in the sequel, we often write just left reductive semigroups instead of left reductive regular semigroups. and apply our structural result to obtain constructions for several popular subclasses.

Left reductive semigroups include in particular, all regular monoids, the $\mathrel{\mathscr{L}}$ -unipotent semigroups, the inverse semigroups, the right regular bands, the full (linear) transformation semigroups, the singular transformation semigroups on a finite set, and the semigroup of singular linear transformations over a finite dimensional vector space. Each of these classes occurs naturally in various branches of mathematics, statistics or physics. We will discuss these in detail. Observe that left reductive semigroups exclude several ‘simple’ classes like completely simple semigroups [4], bands [44] and regular- $*$ semigroups [15].

In left reductive semigroups, the relationship between the left and the right ideals is relatively simpler and more transparent than in arbitrary regular semigroups. Roughly speaking, in the left reductive case, the left and the right ideals are very tightly interconnected, and the $\mathrel{\mathscr{L}}$ -structure totally restricts the $\mathrel{\mathscr{R}}$ -structure. More precisely, given a left reductive semigroup $S$ with the category $\mathbb{L}(S)$ of principal left ideals, we may show that the poset of principal right ideals of $S$ is isomorphic to an order ideal $\mathfrak{R}$ of the poset of right ideals of the semigroup $\widehat{\mathbb{L}(S)}$ that arises from the category $\mathbb{L}(S)$ (Proposition 3.17). As a result, we can avoid the use of several complicated notions needed in a general discussion of arbitrary regular semigroups (see [41]).

Just as an element of a group $G$ may be represented as a permutation of the set $G$ , in this paper we represent an element of a left reductive semigroup $S$ by a cone in the category $\mathbb{L}(S)$ (see Definition 2.3). In an arbitrary regular semigroup [41], Nambooripad used a pair of cross-connected cones to represent a typical element. Notice that in [41], Nambooripad also briefly outlined the construction of left reductive semigroups²²2Nambooripad had reversed the convention and called these semigroups as right reductive in [41]. We shall follow Clifford and Preston [13, Section 1.3] wherein these semigroups are named left reductive. This will coincide with the conventions in $\mathrel{\mathscr{L}}$ -unipotent semigroups later (see Section 4). inside his framework of cross-connections. However, his construction involved two normal categories and their duals, certain $\mathbf{Set}$ -valued functors, and the rather sophisticated definition of cross-connection relating all these categories (see [5, Section 5] for a concrete construction of a left reductive semigroup using cross-connections).

In contrast, taking advantage of the less complicated structure on hand, our construction uses just one normal category and bypasses most of the complicated tools of [41] including the cross-connections. We believe that our construction may drastically reduce the entry threshold of the ideal approach to the theory of the structure of semigroups.

The paper is divided into eight sections. After this introduction, in Section 2, we briefly recall the essential preliminary notions regarding semigroups and categories. We also discuss the initial layer of our construction including the notion of normal category from Nambooripad’s treatise [41]. In Section 3, we bifurcate ourselves from [41] and introduce connected categories. We describe the structure of left reductive semigroups using connected categories in Section 3.3 and prove a category equivalence in Section 3.4. We specialise the discussion to the category of $\mathrel{\mathscr{L}}$ -unipotent semigroups in Section 4, culminating in another category equivalence with the category of supported categories. In Section 5, we further particularise to describe a category adjunction between the category of supported categories and the category of right regular bands. As mentioned earlier, the left and the right Green relations induce a natural left-right duality in semigroups. We use this duality to show that the category of connected categories is also equivalent to the category of right reductive semigroups in Section 6. In Section 7, we discuss the arguably most important class of semigroups: the class of inverse semigroups. Here, we introduce self-supported categories which capture the isomorphism between the left and the right ideal structures in inverse semigroups. This leads to a category equivalence just as in [8] but this is much simpler and sheds more light on the symmetry of these semigroups. Finally, in Section 8, we discuss regular monoids and totally left reductive semigroups. One of the interesting results in this section is Theorem 8.4 which identifies the category of semigroups corresponding to the category of normal categories. We describe regular monoids using bounded above normal categories and provide concrete descriptions of connected categories in some of important semigroups like transformation semigroups, linear transformation semigroups and symmetric inverse monoids. These new descriptions subsume (and improve) the discussions in [5, 2] and also illustrate the precision of our construction.

The following table lists the various categories of left reductive regular semigroups considered in the paper and their corresponding categories of connected categories:

Semigroups		Categories
$\mathbf{LRS}$ - left reductive regular semigroups		$\mathbf{CC}$ - connected categories
$\mathbf{LUS}$ - $\mathrel{\mathscr{L}}$ -unipotent semigroups		$\mathbf{SC}$ - supported categories
$\mathbf{RRB}$ - right regular bands		—
$\mathbf{RRS}$ - right reductive regular semigroups		$\mathbf{CC}$ - connected categories
$\mathbf{RUS}$ - $\mathrel{\mathscr{R}}$ -unipotent semigroups		$\mathbf{SC}$ - supported categories
$\mathbf{LRB}$ - left regular bands		—
Inverse semigroups		Self-supported categories
Totally left reductive semigroups		Normal categories
Regular monoids		Bounded above normal categories
$\mathscr{T}_{n}$ - full transformation monoid		${\mathbb{P}}_{\Pi}$ (or just ${\mathbb{P}}$ ) - full powerset category
$\mathscr{T}_{n}\backslash\mathscr{S}_{n}$ - singular transformation semigroup		$\mathbb{PS}$ - powerset category
$\mathscr{L\mspace{-5.0mu}T}_{V}$ - linear transformation monoid		$\mathbb{SV}$ - full subspace category
$\mathscr{L\mspace{-5.0mu}T}_{V}\backslash\mathscr{G\mspace{-5.0mu}L}_{V}$ - singular linear transformation semigroup		$\mathbb{PV}$ - subspace category
$\mathscr{I}_{X}$ - symmetric inverse monoid		$\mathbb{X}$ - partial bijection subsets category

2. Preliminaries

We assume familiarity with some basic ideas from category theory and semigroup theory. For undefined notions, we refer to [33, 28] for category theory and [13, 29, 24] for semigroups and biordered sets. In the sequel, all mappings, morphisms and functors shall be written in the order of their composition, i.e., from left to right.

2.1. Semigroups and categories

As mentioned earlier, all the semigroups we discuss in this paper are regular. Given a regular semigroup $S$ , we can define two quasi-orders $\mathrel{\leqslant_{\ell}}$ and $\mathrel{\leqslant_{r}}$ on $S$ as follows. For $a,b\in S$ :

a\mathrel{\leqslant_{\ell}}b\iff a\in Sb\>\text{ and }\>a\mathrel{\leqslant_{r}}b\iff a\in bS.

Then the Green relations $\mathrel{\mathscr{L}}$ and $\mathrel{\mathscr{R}}$ are equivalence relations defined on the semigroup $S$ as follows:

\mathrel{\mathscr{L}}~{}:=~{}\mathrel{\leqslant_{\ell}}\cap(\mathrel{\leqslant_{\ell}})^{-1}\text{ and }\mathrel{\mathscr{R}}~{}:=~{}\mathrel{\leqslant_{r}}\cap(\mathrel{\leqslant_{r}})^{-1}.

Given an element $a\in S$ , we shall denote the $\mathrel{\mathscr{L}}$ and $\mathrel{\mathscr{R}}$ classes of $a$ using $L_{a}$ and $R_{a}$ , respectively. The natural partial order $\leqslant$ on a regular semigroup $S$ can be given by $\leqslant~{}:=~{}\mathrel{\leqslant_{\ell}}\cap\mathrel{\leqslant_{r}}$ [40]. In the sequel, we shall denote the restrictions of the above mentioned relations to the set $E(S)$ of idempotents of $S$ , also by the same symbols. Given an idempotent $e\in E(S)$ , we shall denote the downset of $e$ by $\omega(e):=\{f\in E(S):f\leqslant e\}$ .

In this paper, we shall deal with two types of categories. The first type are locally small categories whose objects are algebraic structures and morphisms are structure preserving mappings (for example, the category $\mathbf{LRS}$ of left reductive semigroups). These categories will be dealt with in a standard way and we will be concerned about adjunctions and equivalences between such categories. The second type are small categories which are treated as algebraic structures by themselves (for example, a normal category). So, when comparing such small categories, we shall employ stronger notions like normal category isomorphisms. It is worth mentioning that we shall also consider locally small categories whose objects are small categories (for example, the category $\mathbf{CC}$ of connected categories).

Given any category $\mathcal{C}$ , the class of objects of $\mathcal{C}$ is denoted by $v\mathcal{C}$ , and the set of morphisms by $\mathcal{C}$ itself. Hence, given two objects $c,c^{\prime}\in v\mathcal{C}$ , the set of all morphisms from $c$ to $c^{\prime}$ is denoted by $\mathcal{C}(c,c^{\prime})$ .

2.2. Regular semigroups and normal categories

Now, we proceed to give a quick introduction regarding the notion of normal categories and how these categories characterise the principal left ideals of a regular semigroup.

Recall that a morphism in a category is called a monomorphism if it is right cancellable and an epimorphism if it is left cancellable. A morphism $f\colon c\to c^{\prime}$ in a category $\mathcal{C}$ is said to be an isomorphism if there exists a morphism $g\colon c^{\prime}\to c$ in $\mathcal{C}$ such that $fg=1_{c}$ and $gf=1_{c^{\prime}}$ .

A preorder is a category such that there is at most one morphism between any two given objects (possibly equal). A strict preorder $\mathcal{P}$ is a preorder in which the only isomorphisms are the identity morphisms. In a strict preorder, the relation $\preceq$ on the class $v\mathcal{P}$ defined by:

p\preceq q\iff\mathcal{P}(p,q)\neq\phi\text{ for }p,q\in v\mathcal{P}

is a partial order. Hence, a small strict preorder category $\mathcal{P}$ is equivalent to a partially ordered set (poset) $(v\mathcal{P},\preceq)$ .

Definition 2.1.

Let $\mathcal{C}$ be a small category and $\mathcal{P}$ be a subcategory of $\mathcal{C}$ . Then the pair $(\mathcal{C},\mathcal{P})$ (often denoted by just $\mathcal{C}$ ) is said to be a category with subobjects if:

(1)

$\mathcal{P}$ is a strict preorder with $v\mathcal{P}=v\mathcal{C}$ .
(2)

Every $f\in\mathcal{P}$ is a monomorphism in $\mathcal{C}$ .
(3)

If $f,g\in\mathcal{P}$ and if $f=hg$ for some $h\in\mathcal{C}$ , then $h\in\mathcal{P}$ .

In a category $(\mathcal{C},\mathcal{P})$ with subobjects, the morphisms in $\mathcal{P}$ are called inclusions. If $c\to c^{\prime}$ is an inclusion, we have $c\preceq c^{\prime}$ , and we denote this inclusion by $j(c,c^{\prime})$ . An inclusion $j(c,c^{\prime})$ splits if there exists $q\colon c^{\prime}\to c\in\mathcal{C}$ such that $j(c,c^{\prime})q=1_{c}$ , and a morphism $q$ satisfying such an equality is called a retraction.

Definition 2.2.

Let $\mathcal{C}$ be a category with subobjects. A morphism $f$ in $\mathcal{C}$ is said to have a normal factorisation if $f=quj$ , where $q$ is a retraction, $u$ is an isomorphism and $j$ is an inclusion, respectively in $\mathcal{C}$ .

Given a normal factorisation $f=quj$ the morphism $qu$ does not depend on the factorisation, it is known as the epimorphic component of the morphism $f$ and is denoted in the sequel by $f^{\circ}$ . Similarly, the morphism $j$ is known as the inclusion of $f$ and denoted by $j_{f}$ . The codomain of $f^{\circ}$ is called the image of $f$ and shall be denoted as $\operatorname{im}f$ . Likewise, the codomain of the retraction $q$ is called the coimage of $f$ and denoted by $\operatorname{coim}f$ . We collect the following results as a lemma which will be quite useful in the sequel for manipulating expressions involving morphisms.

Lemma 2.1 ([41, Corollary II.4, Proposition II.5 and II.7]).

Let $\mathcal{C}$ be a category with normal factorisation property where inclusions split.

(1)

Every morphism $f$ has a unique epimorphic component $f^{\circ}$ i.e., $f^{\circ}$ is independent of the chosen normal factorisation of $f$ .
(2)

If $p$ is an epimorphism, then the epimorphic component $p^{\circ}=p$ .
(3)

If $f$ and $g$ are composable morphisms such that the inclusion of $f$ is $j_{f}$ , then

$(fg)^{\circ}=f^{\circ}(j_{f}g)^{\circ}.$
(4)

The inclusion of an epimorphism $p\in\mathcal{C}$ is the identity morphism, and so every normal factorisation of $p$ is of the form $p=qu$ , where $q$ is a retraction and $u$ is an isomorphism.
(5)

Dually, the retraction of a monomorphism $f\in\mathcal{C}$ is the identity morphism, and so every normal factorisation of $m$ is of the form $m=uj$ , where $u$ is an isomorphism and $j$ is an inclusion. In particular, the epimorphic component of an inclusion is the identity morphism.

Remark 2.1.

Given a morphism $f$ in a category with the normal factorisation property where inclusions split, by Lemma 2.1(1), the morphism $f$ has a unique factorisation of the form $f=pj$ where $p$ is an epimorphism and $j$ is an inclusion. Such a factorisation is called as a canonical factorisation of the morphism $f$ .

Definition 2.3.

Let $\mathcal{C}$ be a category with subobjects and $z\in v\mathcal{C}$ . A mapping $\gamma$ from $v\mathcal{C}$ to $\mathcal{C}$ defined by $\gamma\colon c\mapsto\gamma(c)\in\mathcal{C}(c,z)$ for each $c\in v\mathcal{C}$ , is said to be a cone³³3cones were called as normal cones in [41]. with vertex $z$ if:

(1)

whenever $a\preceq b$ , $j(a,b)\gamma(b)=\gamma(a)$ ;
(2)

there exists at least one $c\in v\mathcal{C}$ such that $\gamma(c)\colon c\to z$ is an isomorphism.

Given a cone $\gamma$ , we denote by $z_{\gamma}$ the vertex of $\gamma$ and the morphism $\gamma(c)$ is called the component of the cone $\gamma$ at the object $c$ . The figure 1 illustrates a typical cone $\gamma$ with vertex $z$ in a category $\mathcal{C}$ .

Figure 1. A cone

\gamma

with vertex

z_{\gamma}=z

Definition 2.4.

[41, Section III.1.3] A category $\mathcal{C}$ is said to be a normal category if:

(NC 1)

$\mathcal{C}$ is a category with subobjects;
(NC 2)

every inclusion in $\mathcal{C}$ splits;
(NC 3)

every morphism in $\mathcal{C}$ admits a normal factorisation;
(NC 4)

for each $c\in v\mathcal{C}$ there exists a cone $\mu$ with vertex $c$ such that $\mu(c)=1_{c}$ .

Naturally, for two normal categories to be isomorphic, we need an inclusion preserving functor which is fully-faithful such that the object map is a bijective order isomorphism.

Let $\mathcal{C}$ be a normal category and let $\gamma$ be a cone in $\mathcal{C}$ , if $f\in\mathcal{C}(z_{\gamma},z_{f})$ is an epimorphism with $\operatorname{im}f=z_{f}$ , then as in [41, Lemma I.1], we can easily see that the map

(1)

\gamma\ast f\colon c\mapsto\gamma(c)f\text{ for all }c\in v\mathcal{C}

is a cone such that the vertex $z_{\gamma\ast f}=z_{f}$ . Hence given cones $\gamma$ and $\delta$ ,

(2)

\gamma\cdot\delta=\gamma\ast(\delta(z_{\gamma}))^{\circ}

where $(\delta(z_{\gamma}))^{\circ}$ is the epimorphic component of the morphism $\delta(z_{\gamma})$ , defines a binary composition on the set of all cones in $\mathcal{C}$ . This binary composition on the set of cones is illustrated in Figure 2 wherein the components of the composed cone $\gamma\cdot\delta$ are drawn in red colour. So, for instance, the component of the cone $\gamma\cdot\delta$ at the object $a$ is the morphism $\gamma(a)(\delta(z_{\gamma}))^{\circ}$ . Observe that the vertices $z_{\gamma\cdot\delta}\preceq z_{\delta}$ but the inclusion $j(z_{\gamma\cdot\delta},z_{\delta})$ need not always be identity morphism. In the sequel, we may often denote the binary composition of cones by juxtaposition.

Figure 2. Binary composition of cones

\gamma

(on top) and

\delta

(below) in a normal category

\mathcal{C}

Lemma 2.2 ([41, Theorem I.2]).

Let $\mathcal{C}$ be a normal category. Then the set $\widehat{\mathcal{C}}$ of all cones forms a regular semigroup under the binary composition defined in (2). A cone $\mu$ in $\mathcal{C}$ is an idempotent if and only if $\mu(z_{\mu})=1_{z_{\mu}}$ .

The next two lemmas follow from the discussion in [41, Section III.2].

Lemma 2.3.

Let $\gamma,\delta$ be cones in the regular semigroup $\widehat{\mathcal{C}}$ . Then the quasi-orders in $\widehat{\mathcal{C}}$ are characterised as follows.

(1)

$\gamma\mathrel{\leqslant_{\ell}}\delta\text{ if and only if }z_{\gamma}\preceq z_{\delta}$ , and so $\gamma\mathrel{\mathscr{L}}\delta\text{ if and only if }z_{\gamma}=z_{\delta}$ .
(2)

$\gamma\mathrel{\leqslant_{r}}\delta\text{ if and only if the component }\gamma(z_{\delta})\text{ is an epimorphism such that }\gamma=\delta\ast\gamma(z_{\delta}).$ Hence, we have $\gamma\>\mathrel{\mathscr{R}}\>\delta$ if and only if $\gamma(z_{\delta})$ is an isomorphism such that $\gamma=\delta\ast\gamma(z_{\delta})$

Lemma 2.4.

Let $\nu,\mu$ be idempotent cones in the semigroup $\widehat{\mathcal{C}}$ and let $\leqslant$ be the natural partial order on the set of idempotents of $\widehat{\mathcal{C}}$ . Then $\nu\leqslant\mu$ if and only if $\nu(z_{\mu})$ is a retraction such that $\nu=\mu\ast\nu(z_{\mu}).$

Now, we briefly describe how normal categories come from regular semigroups. Given a regular semigroup $S$ , we can define the category $\mathbb{L}(S)$ of the principal left ideals, called the left category, by

v\mathbb{L}(S)=\{Se:e\in E(S)\},

and the set of all morphisms from the object $Se$ to the object $Sf$ is the set

\mathbb{L}(S)(Se,Sf)=\{r(e,u,f):u\in eSf\},

where $r(e,u,f)\colon x\mapsto xu$ , for each $x\in Se$ .

Given any morphisms $r(e,u,f)$ and $r(g,v,h)$ , they are equal if and only if $e\mathrel{\mathscr{L}}g$ , $f\mathrel{\mathscr{L}}h$ and $v=gu$ (or $u=ev$ ) ; and they are composable if $Sf=Sg$ (i.e., if $f\mathrel{\mathscr{L}}g$ ) in which case

r(e,u,f)\>r(g,v,h):=r(e,uv,h).

Observe that $\mathbb{L}(S)$ has a particular subcategory $\mathcal{P}_{\mathbb{L}}$ defined by $v\mathcal{P}_{\mathbb{L}}=v\mathbb{L}(S)$ and there exists in $\mathcal{P}_{\mathbb{L}}$ a morphism from $Se$ to $Sf$ if and only if $Se\subseteq Sf$ , this morphism being exactly $r(e,e,f)$ . The morphisms of $\mathcal{P}_{\mathbb{L}}$ correspond to the inclusions of principal ideals. By definition, $\mathcal{P}_{\mathbb{L}}$ is therefore a strict preorder and $(\mathbb{L}(S),\mathcal{P}_{\mathbb{L}})$ is a category with subobjects. Given an inclusion $r(e,e,f)\in\mathcal{P}_{\mathbb{L}}$ , it has a right inverse $r(f,fe,e)$ in $\mathbb{L}(S)$ ; every inclusion in the category $\mathbb{L}(S)$ splits.

Let $r(e,u,f)$ be an arbitrary morphism in $\mathbb{L}(S)$ , then as shown in [41, Corollary III.14], we can see that there exists $h\in E(L_{u})$ and for $g\in E(R_{u})\cap\omega(e)$ ,

r(e,u,f)=r(e,g,g)r(g,u,h)r(h,h,f),

where $r(e,g,g)$ is a retraction, $r(g,u,h)$ is an isomorphism and $r(h,h,f)$ is an inclusion. This is a normal factorisation of the morphism $r(e,u,f)$ . Observe that the image of the morphism $r(e,u,f)$ is uniquely determined and it is the principal left ideal $Sh=Su$ , but there is a choice for the coimage $Sg$ . This shows that in an arbitrary regular semigroup, although the image of a morphism is unique, the coimage need not be unique.

Further, if $a$ is an arbitrary element of a regular semigroup $S$ , then for each $Se\in v\mathbb{L}(S)$ , the mapping $r^{a}\colon v\mathbb{L}(S)\to\mathbb{L}(S)$ defined by

(3)

r^{a}(Se):=r(e,ea,f),\text{ where }f\in E(L_{a})

is a cone with vertex $Sf$ , usually referred to as a principal cone in the category $\mathbb{L}(S)$ . Observe that, for an idempotent $e\in E(S)$ , we have a principal cone $r^{e}$ with vertex $Se$ such that $r^{e}(Se)=r(e,e,e)=1_{Se}$ . Summarising the above discussion, it can be easily verified that $\mathbb{L}(S)$ is a normal category [41, Theorem III.16].

Conversely, given an abstractly defined normal category $\mathcal{C}$ , we obtain a regular semigroup $\widehat{\mathcal{C}}$ of cones in $\mathcal{C}$ . Then the left category $\mathbb{L}(\widehat{\mathcal{C}})$ of the semigroup $\widehat{\mathcal{C}}$ is isomorphic to $\mathcal{C}$ [41, Theorem III.19]. We shall give an independent proof of this fact, later as a consequence of our results (see Proposition 3.14 and Proposition 8.1).

It is worth mentioning here that although $\mathbb{L}(\widehat{\mathcal{C}})\cong\mathcal{C}$ for a normal category $\mathcal{C}$ , we do not have $\widehat{\mathbb{L}(S)}$ isomorphic to $S$ for an arbitrary regular semigroup $S$ . This relationship in general is more subtle as described in Theorem 2.6 below. So, every normal category comes from a regular semigroup although not every regular semigroup can be constructed from a normal category.

Theorem 2.5 ([41, Corollary III.20]).

A small category $\mathcal{C}$ is normal if and only if $\mathcal{C}$ is isomorphic to a category $\mathbb{L}(S)$ , for some regular semigroup $S$ .

Now, we shift our focus to the subclass of regular semigroups which can indeed be constructed using just one normal category. Recall that a regular semigroup $S$ is said to be left reductive if the regular representation $\rho$ is injective. In the category $\mathbb{L}(S)$ , the cones are direct abstractions of the regular representation of a semigroup. In fact, it can be shown that the semigroup $\{r^{a}:a\in S\}$ of all principal cones in $\mathbb{L}(S)$ is isomorphic to the image $S_{\rho}$ of the regular representation of the semigroup $S$ . Roughly speaking, the left ‘part’ of the regular semigroup $S$ is captured by the normal category $\mathbb{L}(S)$ .

Theorem 2.6 ([41, Theorem III.16]).

Let $S$ be a regular semigroup. There is a homomorphism $\bar{\rho}\colon S\to\widehat{\mathbb{L}(S)}$ given by $a\mapsto r^{a}$ . Also, $S$ is isomorphic to a subsemigroup of $\widehat{\mathbb{L}(S)}$ (via the map $\bar{\rho}$ ) if and only if $S$ is left reductive.

Dually, we define the normal category $\mathbb{R}(S)$ of principal right ideals of a regular semigroup $S$ by:

(4)

v\mathbb{R}(S)=\{eS:e\in E(S)\}\>\text{ and }\>\mathbb{R}(S)(eS,fS)=\{l(e,u,f):u\in fSe\}

where a morphism from $eS$ to $fS$ is the mapping $l(e,u,f)\colon x\mapsto ux$ for each $x\in eS$ .

3. Left reductive regular semigroups

We proceed to give a construction for a left reductive regular semigroup as a subsemigroup of the semigroup $\widehat{\mathcal{C}}$ of cones of a normal category $\mathcal{C}$ . This is where we bifurcate ourselves from Nambooripad’s construction.

3.1. Connected categories

First, recall that given any regular semigroup $S$ , the set $S/\mathrel{\mathscr{R}}$ forms a poset under the usual set inclusion as follows:

(5)

R_{e}\sqsubseteq R_{f}\iff eS\subseteq fS\iff e\mathrel{\leqslant_{r}}f.

In fact, the poset $(S/\mathrel{\mathscr{R}},\sqsubseteq)$ has been characterised by Grillet as a regular poset⁴⁴4Since the definition involves several new notions and as we do not explicitly use any of the properties of regular posets, we omit the formal definition. in [23]. Now given a normal category $\mathcal{C}$ , since $\widehat{\mathcal{C}}$ is a regular semigroup (Lemma 2.2), the poset $(\widehat{\mathcal{C}}/\mathrel{\mathscr{R}},\sqsubseteq)$ is a regular poset. We are now ready to give the most important definition of this paper.

Definition 3.1.

Let $\mathcal{C}$ be a normal category and let $\mathfrak{D}$ be an order ideal of the poset $\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ . Then $\mathcal{C}$ is said to be connected by $\mathfrak{D}$ if for every $c\in v\mathcal{C}$ , there is some $\mathfrak{d}\in\mathfrak{D}$ such that $\mathfrak{d}$ contains some idempotent cone with vertex $c$ . We denote such a category by $\mathcal{C}_{\mathfrak{D}}$ and say that the regular poset $\mathfrak{D}$ connects the normal category $\mathcal{C}$ .

Given a normal category $\mathcal{C}_{\mathfrak{D}}$ , we define

\widehat{\mathcal{C}_{\mathfrak{D}}}:=\{\>\gamma\in\widehat{\mathcal{C}}:R_{\gamma}\in\mathfrak{D}\>\}.

Observe that each idempotent cone in the set $\widehat{\mathcal{C}_{\mathfrak{D}}}$ may be uniquely represented as $\epsilon(c,\mathfrak{d})$ such that the vertex $z_{\epsilon(c,\mathfrak{d})}=c$ and $R_{\epsilon(c,\mathfrak{d})}=\mathfrak{d}$ where $c\in v\mathcal{C}$ and $\mathfrak{d}\in\mathfrak{D}$ . In this case, we shall say that the object $c$ is connected by $\mathfrak{d}$ . Hence, we have:

(6)

E(\widehat{\mathcal{C}_{\mathfrak{D}}})=\{\>\epsilon(c,\mathfrak{d}):c\text{ is connected by }\mathfrak{d}\>\}.

Remark 3.1.

The definition of the set $\widehat{\mathcal{C}_{\mathfrak{D}}}$ involves picking certain $\mathrel{\mathscr{R}}$ -classes from the semigroup $\widehat{\mathcal{C}}$ , using the order ideal $\mathfrak{D}$ . So, we could have equivalently defined connected categories by letting $\mathfrak{D}$ be an order ideal isomorphic to an order ideal of the poset $\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ . Admittedly, this would complicate the discussion substantially and so we avoid it at this stage. However, we shall indeed use this identification in Example 3.1 below and later in Section 8, where we discuss concrete cases, as such an identification will lead to simpler descriptions of the connecting posets $\mathfrak{D}$ .

Remark 3.2.

By Definition 3.1, every $c\in v\mathcal{C}$ is connected by at least one $\mathfrak{d}\in\mathfrak{D}$ and conversely each $\mathfrak{d}$ connects at least one $c$ . As we shall see later, this is a reflection of the fact that every $\mathrel{\mathscr{L}}$ and $\mathrel{\mathscr{R}}$ class of a regular semigroup contains at least one idempotent. Notice that, in general, one object $c\in\mathcal{C}$ may be connected by multiple $\mathfrak{d}\in\mathfrak{D}$ , and also different objects in $\mathcal{C}$ may be connected to the same $\mathfrak{d}\in\mathfrak{D}$ .

Before proceeding further, we shall recall some ideas from [5, 42, 43] and use them to illustrate a concrete example of a connected category. We also rectify an error in these papers by assuming the underlying set to be finite rather than arbitrary. The following Lemmas 3.2 and 3.3 could have been obtained as a Corollary of [5, Theorem 3.1], we include their proofs here as they can act as a roadmap for the reader to follow the more abstract construction. This example will be revisited, later in the Section 8.3.

Example 3.1.

(Full power set category, ${\mathbb{P}}$ ) Let $\mathbf{n}:=\{1,\dots,n\}$ . Then the set of all subsets of $\mathbf{n}$ forms a small category ${\mathbb{P}}$ with mappings as morphisms, i.e.,

v{\mathbb{P}}:=\{A:A\subseteq\mathbf{n}\}\>\text{ and }\>{\mathbb{P}}(A,B):=\{f:f\text{ is a mapping from }A\text{ to }B\}.

Also, given $A\subseteq\mathbf{n}$ , the identity map $1_{A}$ is the identity morphism at the object $A$ in ${\mathbb{P}}$ . Observe that ${\mathbb{P}}$ is a small, full subcategory of the large (in fact, locally small) category $\mathbf{Set}$ .

Lemma 3.2.

${\mathbb{P}}$ is a normal category.

Proof.

To begin with, we can realise ${\mathbb{P}}$ as a category with subobjects $({\mathbb{P}},\mathcal{P})$ as follows. Let $v\mathcal{P}:=v{\mathbb{P}}$ and for subsets $A\subseteq B\subseteq\mathbf{n}$ , we define

\mathcal{P}(A,B):=i(A,B)

where $i(A,B)$ is the set inclusion map from $A$ to $B$ . Then $\mathcal{P}$ is a strict preorder category, or equivalently $(v\mathcal{P},\subseteq)$ is a partially ordered set. It is routine to verify that the pair $({\mathbb{P}},\mathcal{P})$ satisfies Definition 2.1 and hence forms a category with subobjects. Observe that given an inclusion map $i(A,B)$ in $\mathcal{P}$ , we can always find a retraction map $q\colon B\to A$ in the category ${\mathbb{P}}$ such that $i(A,B)\>q=1_{A}$ . Observe that the retraction $q$ need not be unique, in general. But any mapping $f$ has a uniquely defined image, and so the image of the morphism $f$ in the category ${\mathbb{P}}$ is the image $Af$ of the mapping.

Further, given a mapping $f$ in ${\mathbb{P}}$ from $A$ to $B$ , let $B^{\prime}:=Af$ be the image of the mapping $f$ so that $j:=i(B^{\prime},B)$ is an inclusion map. Now, the map $f$ determines a partition of $A$ given by:

\pi_{f}:=\{(x,y)\in A\times A~{}:~{}xf=yf\}.

Let $A^{\prime}$ be a cross-section of the partition $\pi_{f}$ , and given an arbitrary $a\in A^{\prime}$ , let $[a]$ be the equivalence class of $\pi_{f}$ in the set $A$ containing $a$ . Define $q\colon A\to A^{\prime}$ as the surjection given by $q\colon[a]\mapsto a$ and then we have $i(A^{\prime},A)q=1_{A^{\prime}}$ . Also, $u:=f_{|A^{\prime}}$ will be a bijection from $A^{\prime}$ to $B^{\prime}$ . Hence we have $f=quj$ as illustrated in the diagram below where $q$ is a retraction, $u$ is an isomorphism and $j$ is an inclusion in the category ${\mathbb{P}}$ .

Hence any morphism $f$ in ${\mathbb{P}}$ has a normal factorisation. Finally, for any object $A\subseteq\mathbf{n}$ , and $\alpha\colon\mathbf{n}\to A$ any mapping such that $\alpha_{|A}=1_{A}$ , for each subset $S\subseteq\mathbf{n}$ , letting $\epsilon(S):=\alpha_{|S}$ , we have that $\epsilon$ is a cone in ${\mathbb{P}}$ with vertex $A$ such that $\epsilon(A)=\alpha_{|A}=1_{A}$ . Moreover, using Lemma 2.2, we can see that $\epsilon$ is an idempotent cone. Hence ${\mathbb{P}}$ is a normal category. ∎

Observe that in the last part of the proof above, a cone in the category ${\mathbb{P}}$ is determined by a mapping from $\mathbf{n}$ to itself. We shall see below that in fact, this relationship is much stronger. Recall that the semigroup of all mappings from a finite set $\mathbf{n}$ to itself, under mapping composition is known as the full transformation monoid $\mathscr{T}_{n}$ .

Lemma 3.3.

The semigroup $\widehat{{\mathbb{P}}}$ of cones in the category ${\mathbb{P}}$ is isomorphic to the full transformation monoid $\mathscr{T}_{n}$ .

Proof.

First, observe that the normal category ${\mathbb{P}}$ has a largest object, namely $\mathbf{n}$ . So given a cone $\gamma$ in the category ${\mathbb{P}}$ with vertex $Z$ , we may define

\phi\colon\widehat{{\mathbb{P}}}\to\mathscr{T}_{n}\text{ given by }\gamma\mapsto\gamma(\mathbf{n})i(Z,\mathbf{n}),

where the mapping $\gamma(\mathbf{n})i(Z,\mathbf{n})\colon\mathbf{n}\to\mathbf{n}$ is an element of the semigroup $\mathscr{T}_{n}$ , and so $\phi$ is well-defined. We proceed to prove that $\phi$ is an isomorphism. For a cone $\gamma$ , by Definition 2.3(2) there is some $C\subseteq\mathbf{n}$ such that $\gamma(C)\colon C\to Z$ is a bijection. However, since $C\subseteq\mathbf{n}$ , the component $\gamma(\mathbf{n})$ is always a surjection and so by Lemma 2.1(2), we have $(\gamma(\mathbf{n}))^{\circ}=\gamma(\mathbf{n})$ . Hence the expression $\gamma(\mathbf{n})i(Z,\mathbf{n})$ is in fact the unique canonical factorisation of the mapping $\gamma$ . Also notice that by Definition 2.3(1), for each $A\subseteq\mathbf{n}$ , we have $\gamma(A)=i(A,\mathbf{n})\gamma(\mathbf{n})$ .

Now, to verify that $\phi$ is a homomorphism, let $\gamma_{1},\gamma_{2}$ be cones in the category ${\mathbb{P}}$ with vertices $Z_{1}$ and $Z_{2}$ , respectively. If we denote the vertex $z_{\gamma_{1}\gamma_{2}}$ of the cone $\gamma_{1}\gamma_{2}$ by $Z$ , we see that $Z\subseteq Z_{2}$ . Then using equations (1) and (2), and Lemma 2.1 (3), we have

(\gamma_{1}\>\gamma_{2})\phi=(\gamma_{1}\ast(\gamma_{2}(A_{1}))^{\circ})\phi=\gamma_{1}(\mathbf{n})\>(\gamma_{2}(A_{1}))^{\circ}i(Z,\mathbf{n})=\gamma_{1}(\mathbf{n})\>(\gamma_{2}(A_{1}))^{\circ}i(Z,A_{2})i(A_{2},\mathbf{n})=\gamma_{1}(\mathbf{n})\>\gamma_{2}(A_{1})i(A_{2},\mathbf{n}).

Also by the definition of $\phi$ , since

\gamma_{1}\phi\>\gamma_{2}\phi=\gamma_{1}(\mathbf{n})i(A_{1},\mathbf{n})\>\gamma_{2}(\mathbf{n})i(A_{2},\mathbf{n})=\gamma_{1}(\mathbf{n})\>\gamma_{2}(A_{1})i(A_{2},\mathbf{n}),

we see that $\phi$ is a homomorphism.

To show that $\phi$ is injective, let $\gamma_{1}(\mathbf{n})i(Z_{1},\mathbf{n})=\gamma_{2}(\mathbf{n})i(Z_{2},\mathbf{n})$ . Then as this is the unique canonical factorisation, we have $\gamma_{1}(\mathbf{n})=\gamma_{2}(\mathbf{n})$ . Now since ${\mathbb{P}}$ has a largest object $\mathbf{n}$ , every cone $\gamma$ is uniquely determined by its component $\gamma(\mathbf{n})$ , whence $\gamma_{1}=\gamma_{2}$ .

Finally, to verify that $\phi$ is a surjection, given an arbitrary mapping $\alpha$ in the monoid $\mathscr{T}_{n}$ , for each $S\subseteq\mathbf{n}$ , the map $\gamma(S):=\alpha_{|S}$ is a cone with vertex $\mathbf{n}\alpha$ such that $\gamma\phi=\gamma(\mathbf{n})i(\mathbf{n}\alpha,\mathbf{n})=\alpha$ . We conclude that $\phi$ is a semigroup isomorphism. ∎

To realise the category ${\mathbb{P}}$ as a connected category, we need the characterisation of the Green $\mathrel{\mathscr{R}}$ -relation on the regular semigroup $\widehat{{\mathbb{P}}}$ . By the above lemma, the poset $\widehat{{\mathbb{P}}}/\mathrel{\mathscr{R}}$ is order isomorphic to $\mathscr{T}_{n}/\mathrel{\mathscr{R}}$ . So, we recall the following well known results regarding the monoid $\mathscr{T}_{n}$ .

Lemma 3.4 ([13, Section 2.2]).

Let $\alpha,\beta$ be arbitrary mappings in $\mathscr{T}_{n}$ .

(1)

For principal left ideals, $\mathscr{T}_{n}\alpha\subseteq\mathscr{T}_{n}\beta$ if and only if $\mathbf{n}\alpha\subseteq\mathbf{n}\beta$ . Hence $\alpha\mathrel{\mathscr{L}}\beta$ if and only if $\mathbf{n}\alpha=\mathbf{n}\beta$ .
(2)

For principal right ideals, $\alpha\mathscr{T}_{n}\subseteq\beta\mathscr{T}_{n}$ if and only if $\pi_{\alpha}\supseteq\pi_{\beta}$ . Hence $\alpha\mathrel{\mathscr{R}}\beta$ if and only if $\pi_{\alpha}=\pi_{\beta}$ .

We denote the poset of all partitions of the set $\mathbf{n}$ by $(\Pi,\supseteq)$ . Given an idempotent cone $\epsilon$ in $\widehat{{\mathbb{P}}}$ , define a map $G\colon\widehat{{\mathbb{P}}}/\mathrel{\mathscr{R}}\to\Pi$ by $R_{\epsilon}\mapsto\pi_{\epsilon(\mathbf{n})}$ . Using Lemmas 3.3 and 3.4(2), we can routinely verify that $G$ is an order isomorphism. This leads to the following characterisation of the poset $(\widehat{{\mathbb{P}}}/\mathrel{\mathscr{R}},\sqsubseteq)$ .

Lemma 3.5.

Let $\gamma_{1},\gamma_{2}$ be cones in the semigroup $\widehat{{\mathbb{P}}}$ . Then $R_{\gamma_{1}}\sqsubseteq R_{\gamma_{2}}$ if and only if $\pi_{\gamma_{1}(\mathbf{n})}\supseteq\pi_{\gamma_{2}(\mathbf{n})}$ . Hence the regular poset $(\widehat{{\mathbb{P}}}/\mathrel{\mathscr{R}},\sqsubseteq)$ is order isomorphic to the poset $(\Pi,\supseteq)$ of all partitions of the set $\mathbf{n}$ .

By the above lemma, we may identify the $\mathrel{\mathscr{R}}$ -classes of the semigroup $\widehat{{\mathbb{P}}}$ by the partitions $\pi\in\Pi$ . Summarising, given a finite set $\mathbf{n}$ , the set of subsets of $\mathbf{n}$ forms a normal category ${\mathbb{P}}$ such that the poset of $\mathrel{\mathscr{R}}$ -classes of the semigroup $\widehat{{\mathbb{P}}}$ is isomorphic to the set $\Pi$ of partitions of $\mathbf{n}$ . This leads us to our first example of a connected category.

Proposition 3.6.

Given a finite set $\mathbf{n}$ with powerset ${\mathbb{P}}$ and partitions $\Pi$ , the category ${\mathbb{P}}$ is connected by $\Pi$ , and so ${\mathbb{P}}_{\Pi}$ is a connected category.

Proof.

Given any subset $A\subseteq\mathbf{n}$ , let $\alpha\in\mathscr{T}_{n}$ be such that $\alpha_{|A}=1_{A}$ . Then $\alpha_{|A}$ is an idempotent in $\mathscr{T}_{n}$ . Then as in the last part of the proof of Lemma 3.2, for each subset $S\subseteq\mathbf{n}$ , define $\epsilon(S):=\alpha_{|S}$ . Now, $\epsilon$ is an idempotent cone in $\widehat{{\mathbb{P}}}$ such that $\epsilon(\mathbf{n})=\alpha$ and $R_{\epsilon}\cong\pi_{\epsilon(\mathbf{n})}=\pi_{\alpha}\in\Pi$ . Hence, the subset $A$ is connected by $\pi_{\alpha}$ and so, the normal category ${\mathbb{P}}$ is connected by the poset $\Pi$ . ∎

Remark 3.3.

In the above example of a connected category $\mathcal{C}_{\mathfrak{D}}$ , we have $\mathfrak{D}\cong\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ . As discussed in Remark 3.1, the relaxation that the ideal $\mathfrak{D}$ is an isomorphic copy of $\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ (rather than $\mathfrak{D}=\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ ) leads to a concrete characterisation of $\mathfrak{D}$ as the poset $\Pi$ . Strictly speaking, with the terminology of Definition 3.1, Proposition 3.6 says that the category ${\mathbb{P}}$ is connected by the poset $\widehat{{\mathbb{P}}}/\mathrel{\mathscr{R}}$ such that $\widehat{{\mathbb{P}}}/\mathrel{\mathscr{R}}$ is isomorphic to the poset $\Pi$ .

3.2. The connection semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$

Having digressed a bit, we now return back to the abstract construction of a left reductive semigroup from connected category $\mathcal{C}$ . We shall see that the required semigroup is in fact $\widehat{\mathcal{C}_{\mathfrak{D}}}$ , which is realised as the subsemigroup of the semigroup $\widehat{\mathcal{C}}$ of cones in the category $\mathcal{C}$ . The following lemma is crucial for the sequel.

Lemma 3.7.

Let $\mathcal{C}_{\mathfrak{D}}$ be a connected category. Then every cone $\gamma$ in the set $\widehat{\mathcal{C}_{\mathfrak{D}}}$ can be expressed as $\epsilon(c,\mathfrak{d})\ast u$ , for some idempotent cone $\epsilon(c,\mathfrak{d})$ and an isomorphism $u$ . Conversely, every cone in $\widehat{\mathcal{C}}$ which can be expressed in this form belongs to $\widehat{\mathcal{C}_{\mathfrak{D}}}$ .

Proof.

First, observe that given a cone $\gamma$ in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ , we have $R_{\gamma}\in\mathfrak{D}$ and let $\mathfrak{d}:=R_{\gamma}$ . Now, since $\mathfrak{D}$ connects the category $\mathcal{C}$ , by Remark 3.2, there is some $c\in v\mathcal{C}$ such that $\mathfrak{d}$ connects $c$ . Let the associated idempotent cone be $\epsilon(c,\mathfrak{d})$ . So we have $\gamma\mathrel{\mathscr{R}}\epsilon(c,\mathfrak{d})$ in the semigroup $\widehat{\mathcal{C}}$ . Then by Lemma 2.3(2), we get $\gamma=\epsilon(c,\mathfrak{d})\ast\gamma(c)$ such that $\gamma(c)$ is an isomorphism.

Conversely, if $\gamma=\epsilon(c,\mathfrak{d})\ast u$ , then by Lemma 2.3(2), we obtain $\gamma\mathrel{\mathscr{R}}\epsilon(c,\mathfrak{d})$ and so $R_{\gamma}=R_{\epsilon(c,\mathfrak{d})}=\mathfrak{d}\in\mathfrak{D}$ . Thus $\gamma\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ . ∎

Remark 3.4.

Given a cone $\gamma$ with vertex $c$ , the decomposition of $\gamma$ as above is not unique, in general. In fact, given idempotents $\epsilon_{1}:=\epsilon(c_{1},\mathfrak{d})$ and $\epsilon_{2}:=\epsilon(c_{2},\mathfrak{d})$ in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ such that $\mathfrak{d}:=R_{\gamma}$ , we can see that $\gamma=\epsilon_{1}\ast\gamma(c_{1})=\epsilon_{2}\ast\gamma(c_{2})$ for isomorphisms $\gamma(c_{1})\colon c_{1}\to c$ and $\gamma(c_{2})\colon c_{2}\to c$ . Figure 3 illustrates this situation, wherein we consider the ‘egg-box’ diagram of a typical $\mathrel{\mathscr{D}}$ -class of the regular semigroup $\widehat{\mathcal{C}}$ . Observe that by Lemma 2.3, the $\mathrel{\mathscr{L}}$ -classes of $\widehat{\mathcal{C}}$ are determined by the vertices of the cones.

Figure 3. Decomposition of a cone

\gamma

as an idempotent cone and an isomorphism

Proposition 3.8.

Let $\mathcal{C}_{\mathfrak{D}}$ be a connected category. Then $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a regular semigroup.

Proof.

First we need to show that $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a closed subset of $\widehat{\mathcal{C}}$ . Let $\gamma_{1}$ and $\gamma_{2}$ be two cones in the set $\widehat{\mathcal{C}_{\mathfrak{D}}}$ . Then by Lemma 3.7, there are an idempotent cone $\epsilon_{1}:=\epsilon(c_{1},\mathfrak{d}_{1})$ and an isomorphism $u_{1}$ such that $\gamma_{1}:=\epsilon_{1}\ast u_{1}$ . Let $\gamma:=\gamma_{1}\gamma_{2}$ . Using equation (2), and Lemma 2.1(2) and (3), we see that

\gamma=\gamma_{1}\cdot\gamma_{2}=\gamma_{1}\ast(\gamma_{2}(z_{\gamma_{1}}))^{\circ}=\epsilon_{1}\ast u_{1}\ast(\gamma_{2}(z_{\gamma_{1}}))^{\circ}=\epsilon_{1}\ast(u_{1}(\gamma_{2}(z_{\gamma_{1}}))^{\circ}).

As $u_{1}$ is an isomorphism and $(\gamma_{2}(z_{\gamma_{1}}))^{\circ}$ is an epimorphism, their composition is an epimorphism. Since we are in a normal category, by Lemma 2.1(4), an epimorphism has a normal factorisation such that the inclusion component is the identity, thus $u_{1}(\gamma_{2}(z_{\gamma_{1}}))^{\circ}=qu$ where $q\colon c_{1}\to c$ is retraction and $u\colon c\to z_{\gamma}$ is an isomorphism. Hence $\gamma=\epsilon_{1}\ast qu=(\epsilon_{1}\ast q)\ast u$ . Since a retraction is, in particular an epimorphism, by equation (2), we see that $\epsilon_{1}\ast q$ is a cone in $\widehat{\mathcal{C}}$ with vertex $c$ . Let $\mu:=\epsilon_{1}\ast q$ and $\mathfrak{d}:=R_{\mu}$ in the semigroup $\widehat{\mathcal{C}}$ . Now observe that

\mu(c)=\epsilon_{1}(c)\>q=j(c,c_{1})q=1_{c}\quad\text{ and }\quad\mu(c_{1})=\epsilon_{1}\ast q(c_{1})=\epsilon_{1}(c_{1})\>q=1_{c_{1}}\>q=q.

Thus $\mu$ is an idempotent, and $\mu=\epsilon_{1}\ast\mu(c_{1})$ with $\mu(c_{1})$ a retraction. Therefore, by Lemma 2.4, we have $\mu\leqslant\epsilon_{1}$ in the semigroup $\widehat{\mathcal{C}}$ . In particular, $\mathfrak{d}=R_{\mu}\sqsubseteq R_{\epsilon_{1}}=\mathfrak{d}_{1}$ . Since $\mathfrak{D}$ is an order ideal and $\mathfrak{d}_{1}\in\mathfrak{D}$ , we get $\mathfrak{d}\in\mathfrak{D}$ . So $\mu=\epsilon(c,\mathfrak{d})\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ and as shown in Figure 4, we have $\gamma=\mu\ast u$ . Next, using Lemma 3.7, we obtain $\gamma=\gamma_{1}\cdot\gamma_{2}\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ and so $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a subsemigroup of $\widehat{\mathcal{C}}$ .

Figure 4. Composition of cones in the semigroup

\widehat{\mathcal{C}_{\mathfrak{D}}}

Finally, to see that $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is regular, let $\gamma\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ be a cone with vertex $z_{\gamma}$ . By definition of a connected category, there is an idempotent cone $\epsilon(z_{\gamma},\mathfrak{d}^{\prime})$ in the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ with vertex $z_{\gamma}$ . Using Lemma 3.7, we can write $\gamma=\epsilon(c^{\prime},\mathfrak{d}_{\gamma})\ast u_{\gamma}$ for some $c^{\prime}\in v\mathcal{C}$ , $\mathfrak{d}_{\gamma}:=R_{\gamma}$ and $u_{\gamma}$ an isomorphism from $c^{\prime}$ to $z_{\gamma}$ . Then let $\gamma^{\prime}:=\epsilon(z_{\gamma},\mathfrak{d}^{\prime})\ast u_{\gamma}^{-1}$ so that $z_{\gamma^{\prime}}=c^{\prime}$ and $R_{\gamma^{\prime}}=\mathfrak{d}^{\prime}$ . Since $\mathfrak{d}^{\prime},\mathfrak{d}_{\gamma}\in\mathfrak{D}$ , by Lemma 3.7 the cone $\gamma^{\prime}\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ . Also, observe that $\gamma(c^{\prime})=u_{\gamma}$ and $\gamma^{\prime}(z_{\gamma})=u_{\gamma}^{-1}$ , and are both isomorphisms. Then

\gamma\gamma^{\prime}\gamma=(\gamma\ast(\gamma^{\prime}(z_{\gamma}))^{\circ})\ast(\gamma(c^{\prime}))^{\circ}=(\gamma\ast\>u_{\gamma}^{-1})\ast\>u_{\gamma}=\gamma\ast\>(u_{\gamma}^{-1}\>u_{\gamma})=\gamma.

Similarly, $\gamma^{\prime}\gamma\gamma^{\prime}=\gamma^{\prime}$ , and so $\gamma^{\prime}$ is an inverse of $\gamma$ (see Figure 5). Hence $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a regular subsemigroup of $\widehat{\mathcal{C}}$ .

Figure 5. Locating an inverse

\gamma^{\prime}

of a cone

\gamma

in the semigroup

\widehat{\mathcal{C}_{\mathfrak{D}}}

∎

The following variant of the Lemma 3.7 uses the technique applied in the proof of Proposition 3.8 and will be useful in the sequel.

Lemma 3.9.

Given a connected category $\mathcal{C}_{\mathfrak{D}}$ , any cone in the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ has a representation of the form $\epsilon_{1}\ast p$ for an idempotent cone $\epsilon_{1}$ and an epimorphism $p$ . Conversely, any cone of this form belongs to $\widehat{\mathcal{C}_{\mathfrak{D}}}$ .

Proof.

Since every isomorphism is an epimorphism, the first part of the lemma is obvious from Lemma 3.7. Conversely, let $\gamma$ be a cone in $\widehat{\mathcal{C}}$ of the form $\epsilon_{1}\ast p$ , where $\epsilon_{1}:=\epsilon(c_{1},\mathfrak{d}_{1})$ is an idempotent cone with vertex $c_{1}$ and $p$ is an epimorphism. Using Lemma 2.1(4), let $p=qu$ be the normal factorisation of the epimorphism $p$ . Then $\gamma=\epsilon_{1}\ast p=\epsilon_{1}\ast q\ast u$ . Then, as argued above in the proof of Proposition 3.8 (see Figure 4), we see that $\mu:=\epsilon_{1}\ast q$ is an idempotent cone such that $\mu\leqslant\epsilon_{1}$ . So, we have $\gamma=\mu\ast u$ , where $\mu$ is an idempotent cone and $u$ is an isomorphism. Hence, by Lemma 3.7, the cone $\gamma$ belongs to the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ . ∎

Proposition 3.10.

Let $\mathcal{C}_{\mathfrak{D}}$ be a connected category. The semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is left reductive.

Proof.

Recall that, to prove that $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is left reductive, given $\gamma_{1},\gamma_{2}\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ such that

(7)

\gamma\gamma_{1}=\gamma\gamma_{2}\text{ , for every }\gamma\in\widehat{\mathcal{C}_{\mathfrak{D}}},

we need to show $\gamma_{1}=\gamma_{2}$ . Also, observe that by Definition 2.3 and Lemma 2.1(1), to prove that two cones $\gamma_{1}$ and $\gamma_{2}$ are equal in a category $\mathcal{C}$ , it suffices to show that:

(1)

the vertices of the cones are same, i.e., $z_{\gamma_{1}}=z_{\gamma_{2}}$ , and,
(2)

the epimorphic components of the respective morphisms are same, i.e.,

$(\gamma_{1}(c))^{\circ}=(\gamma_{2}(c))^{\circ}\text{ , for every }c\in v\mathcal{C}.$

To begin with, observe that since the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is regular, there exists an idempotent cone $\epsilon_{1}\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ such that $\gamma_{1}\mathrel{\mathscr{R}}\epsilon_{1}$ , and so $\gamma_{1}=\epsilon_{1}\gamma_{1}$ . Then using the assumption (7) and letting $\gamma:=\epsilon_{1}$ , we have $\epsilon_{1}\gamma_{1}=\epsilon_{1}\gamma_{2}$ . So, $\gamma_{1}=\epsilon_{1}\gamma_{2}$ , i.e., $\gamma_{1}\mathrel{\leqslant_{\ell}}\gamma_{2}$ . Hence, using Lemma 2.3(1), we see that the vertices of the cones, satisfy $z_{\gamma_{1}}\preceq z_{\gamma_{2}}$ . Similarly, using an idempotent $\epsilon_{2}\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ such that $\gamma_{2}\mathrel{\mathscr{R}}\epsilon_{2}$ , we can show that $z_{\gamma_{2}}\preceq z_{\gamma_{1}}$ . Thus $z_{\gamma_{1}}=z_{\gamma_{2}}$ .

Next, given an arbitrary $c\in v\mathcal{C}$ , by definition there is an idempotent cone $\epsilon=\epsilon(c,\mathfrak{d})$ in the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ with vertex $c$ such that $\epsilon(c)=1_{c}$ . Letting $\gamma:=\epsilon$ in the assumption of (7), we get $\epsilon\gamma_{1}=\epsilon\gamma_{2}$ , and so using equation (2) we get $\epsilon\ast(\gamma_{1}(c))^{\circ}=\epsilon\ast(\gamma_{2}(c))^{\circ}$ . Now, comparing the component of these cones at the object $c$ , we obtain for the morphisms, $\epsilon(c)(\gamma_{1}(c))^{\circ}=\epsilon(c)(\gamma_{2}(c))^{\circ}$ . However, since $\epsilon(c)=1_{c}$ , we have $(\gamma_{1}(c))^{\circ}=(\gamma_{2}(c))^{\circ}$ , for each object $c\in v\mathcal{C}$ . Thereby we conclude that the cones $\gamma_{1}$ and $\gamma_{2}$ coincide, and so the regular semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is left reductive. ∎

Summarising the above discussion, given a connected category $\mathcal{C}_{\mathfrak{D}}$ , we constructed a left reductive regular semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ . We shall refer to this semigroup as the connection semigroup of the category $\mathcal{C}_{\mathfrak{D}}$ . Now, to take the discussion forward, we need to explore the left and right ideal structure of $\widehat{\mathcal{C}_{\mathfrak{D}}}$ . Since $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a regular subsemigroup of the semigroup $\widehat{\mathcal{C}}$ , the Green relations in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ get inherited from $\widehat{\mathcal{C}}$ (see [29, Proposition 2.4.2]). So using Lemma 2.3, we have the following:

Lemma 3.11.

Let $\gamma,\delta$ be cones in the connection semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ . Then

(1)

$\gamma\mathrel{\leqslant_{\ell}}\delta\text{ if and only if }z_{\gamma}\preceq z_{\delta}$ , and $\gamma=\delta\text{ if and only if }z_{\gamma}=z_{\delta}$ ;
(2)

$\gamma\mathrel{\leqslant_{r}}\delta\text{ if and only if }\gamma(z_{\delta})\text{ is an epimorphism such that }\gamma=\delta\ast\gamma(z_{\delta})$ .

Since the Green relations in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ and in $\widehat{\mathcal{C}}$ coincide ( see Remark 3.1), we readily conclude that the poset $\widehat{\mathcal{C}_{\mathfrak{D}}}/\mathrel{\mathscr{R}}$ is order isomorphic to $\mathfrak{D}\subseteq\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ . For later use, we denote this order isomorphism by the map $G$ .

By Lemma 3.11(1), it is clear that the poset $\widehat{\mathcal{C}_{\mathfrak{D}}}/\mathrel{\mathscr{L}}$ is order isomorphic to the poset $(v\mathcal{C},\preceq)$ . To completely describe the left ideal structure of the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ , we need a deeper understanding. To this end, we employ normal categories and dive one additional layer deeper. Recall that since $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a regular semigroup, the principal left ideals of $\widehat{\mathcal{C}_{\mathfrak{D}}}$ form a normal category $\mathbb{L}(\widehat{\mathcal{C}_{\mathfrak{D}}})$ .

To simplify the notation, fix $T:=\widehat{\mathcal{C}_{\mathfrak{D}}}$ for the remaining of the section. We can define a functor $F\colon\mathbb{L}(T)\to\mathcal{C}$ as follows: given $\epsilon\in E(T)$ and a morphism $r(\epsilon_{1},\gamma,\epsilon_{2})$ in $\mathbb{L}(T)$ such that $\gamma\in\epsilon_{1}T\epsilon_{2}$ , let

(8)

vF(T\epsilon):=z_{\epsilon}\>\text{ and }\>F(r(\epsilon_{1},\gamma,\epsilon_{2})):=\gamma(z_{\epsilon_{1}})j(z_{\gamma},z_{\epsilon_{2}}),

where $j(z_{\gamma},z_{\epsilon_{2}})$ is the inclusion morphism in $\mathbb{L}(T)$ . Observe that given a morphism $r(\epsilon_{1},\gamma,\epsilon_{2})$ in the category $\mathbb{L}(T)$ from $T\epsilon_{1}$ to $T\epsilon_{2}$ , since $\gamma\in\epsilon_{1}T$ , we have $\gamma\mathrel{\leqslant_{r}}\epsilon_{1}$ . So, using Lemma 3.11(2), we see that $\gamma(z_{\epsilon_{1}})$ is an epimorphism in $\mathcal{C}$ such that $\gamma=\epsilon_{1}\ast\gamma(z_{\epsilon_{1}})$ . Also, by Remark 2.1, the expression $\gamma(z_{\epsilon_{1}})j(z_{\gamma},z_{\epsilon_{2}})$ is the unique canonical factorisation of the corresponding morphism belonging to $\mathcal{C}(z_{\epsilon_{1}},z_{\epsilon_{2}})$ .

Lemma 3.12.

$F$ is a well-defined functor from the normal category $\mathbb{L}(T)$ to $\mathcal{C}$ .

Proof.

Suppose that $T\epsilon=T\epsilon^{\prime}$ , then by Lemma 3.11(1) we have $z_{\epsilon}=z_{\epsilon^{\prime}}$ , and $vF$ is well-defined on objects. To verify that $F$ is well-defined on morphisms, suppose that $r(\epsilon_{1},\gamma,\epsilon_{2})=r(\epsilon^{\prime}_{1},\gamma^{\prime},\epsilon^{\prime}_{2})$ in the left category $\mathbb{L}(T)$ . Then, from Section 2.2, this equality of morphisms implies $\epsilon_{1}\mathrel{\mathscr{L}}\epsilon^{\prime}_{1}$ , $\epsilon_{2}\mathrel{\mathscr{L}}\epsilon^{\prime}_{2}$ and $\gamma=\epsilon_{1}\gamma^{\prime}$ . By Lemma 3.11(1), we have $z_{\epsilon_{1}}=z_{\epsilon^{\prime}_{1}}$ and $z_{\epsilon_{2}}=z_{\epsilon^{\prime}_{2}}$ . Further, since $\gamma^{\prime}(z_{\epsilon^{\prime}_{1}})$ is an epimorphism, by Lemma 2.1(2), we get $(\gamma^{\prime}(z_{\epsilon^{\prime}_{1}}))^{\circ}=\gamma^{\prime}(z_{\epsilon^{\prime}_{1}}).$ Hence, using equations (1) and (2), we see that

\gamma(z_{\epsilon_{1}})=(\epsilon_{1}\gamma^{\prime})(z_{\epsilon_{1}})=\epsilon_{1}(z_{\epsilon_{1}})\>(\gamma^{\prime}(z_{\epsilon_{1}}))^{\circ}=1_{z_{\epsilon_{1}}}(\gamma^{\prime}(z_{\epsilon^{\prime}_{1}}))^{\circ}=(\gamma^{\prime}(z_{\epsilon^{\prime}_{1}}))^{\circ}=\gamma^{\prime}(z_{\epsilon^{\prime}_{1}}).

Now, since every morphism in the category $\mathcal{C}$ has a unique canonical factorisation, we see that the morphism $\gamma(z_{\epsilon_{1}})j(z_{\gamma},z_{\epsilon_{2}})=\gamma^{\prime}(z_{\epsilon^{\prime}_{1}})j(z_{\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})$ and so $F$ is well-defined.

To see that $F$ is a functor from $\mathbb{L}(T)$ to $\mathcal{C}$ , first observe that given the identity morphism $1_{T\epsilon}=r(\epsilon,\epsilon,\epsilon)$ in the category $\mathbb{L}(T)$ , we have $F(r(\epsilon,\epsilon,\epsilon))=\epsilon(z_{\epsilon})j(z_{\epsilon},z_{\epsilon})=1_{z_{\epsilon}}$ . Hence $F(1_{T\epsilon})=1_{vF(T\epsilon)}$ and the identities are preserved by $F$ .

Next, let $r_{1}:=r(\epsilon_{1},\gamma,\epsilon_{2})$ and $r_{2}:=r(\epsilon^{\prime}_{1},\gamma^{\prime},\epsilon^{\prime}_{2})$ be morphisms in $\mathbb{L}(T)$ such that $\epsilon_{2}\mathrel{\mathscr{L}}\epsilon^{\prime}_{1}$ . Then $r_{1}r_{2}=r(\epsilon_{1},\gamma\gamma^{\prime},\epsilon^{\prime}_{2})$ is a morphism such that $F(r_{1}r_{2})=\gamma\gamma^{\prime}(z_{\epsilon_{1}})j(z_{\gamma\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})$ . Also, $F(r_{1})=\gamma(z_{\epsilon_{1}})j(z_{\gamma},z_{\epsilon_{2}})$ and $F(r_{2})=\gamma^{\prime}(z_{\epsilon^{\prime}_{1}})j(z_{\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})$ . Thus $z_{\gamma}\preceq z_{\epsilon^{\prime}_{1}}$ since $z_{\epsilon_{2}}=z_{\epsilon^{\prime}_{1}}$ , and so by Definition of cone 2.3(1), we have $j(z_{\gamma},z_{\epsilon^{\prime}_{1}})\gamma^{\prime}(z_{\epsilon^{\prime}_{1}})=\gamma^{\prime}(z_{\gamma})$ . Moreover, as $z_{\gamma\gamma^{\prime}}=\operatorname{im}\gamma^{\prime}(z_{\gamma})$ , using Remark 2.1, we can write $\gamma^{\prime}(z_{\gamma})=(\gamma^{\prime}(z_{\gamma}))^{\circ}j(z_{\gamma\gamma^{\prime}},z_{\gamma^{\prime}})$ . Therefore, the morphisms $F(r_{1})$ and $F(r_{2})$ are composable and

F(r_{1})F(r_{2})=\gamma(z_{\epsilon_{1}})j(z_{\gamma},z_{\epsilon_{2}})\gamma^{\prime}(z_{\epsilon^{\prime}_{1}})j(z_{\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})=\gamma(z_{\epsilon_{1}})\gamma^{\prime}(z_{\gamma})j(z_{\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})=\gamma(z_{\epsilon_{1}})(\gamma^{\prime}(z_{\gamma}))^{\circ}j(z_{\gamma\gamma^{\prime}},z_{\gamma^{\prime}})j(z_{\gamma^{\prime}},z_{\epsilon^{\prime}_{2}}).

Finally, from $\gamma(z_{\epsilon_{1}})(\gamma^{\prime}(z_{\gamma}))^{\circ}=\gamma\gamma^{\prime}(z_{\epsilon_{1}})$ and $j(z_{\gamma\gamma^{\prime}},z_{\gamma^{\prime}})j(z_{\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})=j(z_{\gamma\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})$ , we obtain

F(r_{1})F(r_{2})=\gamma\gamma^{\prime}(z_{\epsilon_{1}})j(z_{\gamma\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})=F(r_{1}r_{2}).

Thus, the assignment $F$ preserves the composition also, whence $F$ is a functor. ∎

Lemma 3.13.

The functor $F$ is a normal category isomorphism.

Proof.

By Lemma 3.11(1), the map $vF$ is clearly a bijection. Given an inclusion $j(T\epsilon_{1},T\epsilon_{2})$ in the category $\mathbb{L}(T)$ , we can easily see that $j(z_{\epsilon_{1}},z_{\epsilon_{2}})$ is an inclusion in the category $\mathcal{C}$ . Hence $F$ is inclusion preserving.

To see that $F$ is faithful, suppose that $F(r(\epsilon_{1},\gamma,\epsilon_{2}))=F(r(\epsilon^{\prime}_{1},\gamma^{\prime},\epsilon^{\prime}_{2}))$ , i.e., $\gamma(z_{\epsilon_{1}})j(z_{\gamma},z_{\epsilon_{2}})=\gamma^{\prime}(z_{\epsilon^{\prime}_{1}})j(z_{\gamma^{\prime}},z_{\epsilon^{\prime}_{2}})$ in the category $\mathcal{C}$ . Then $z_{\epsilon_{1}}=z_{\epsilon^{\prime}_{1}}$ and $z_{\epsilon_{2}}=z_{\epsilon^{\prime}_{2}}$ and so by Lemma 3.11(1), we have $\epsilon_{1}\mathrel{\mathscr{L}}\epsilon^{\prime}_{1}$ and $\epsilon_{2}\mathrel{\mathscr{L}}\epsilon^{\prime}_{2}$ . On another hand, using the canonical factorisation property of morphisms in $\mathcal{C}$ , we get $\gamma(z_{\epsilon_{1}})=\gamma^{\prime}(z_{\epsilon^{\prime}_{1}})$ . Then applying (2), we obtain

\epsilon_{1}\gamma^{\prime}=\epsilon_{1}\ast(\gamma^{\prime}(z_{\epsilon_{1}}))^{\circ}=\epsilon_{1}\ast(\gamma^{\prime}(z_{\epsilon^{\prime}_{1}}))^{\circ}=\epsilon_{1}\ast(\gamma(z_{\epsilon_{1}}))^{\circ}=\epsilon_{1}\gamma=\gamma.

Hence, $r(\epsilon_{1},\gamma,\epsilon_{2})=r(\epsilon^{\prime}_{1},\gamma^{\prime},\epsilon^{\prime}_{2})$ in the category $\mathbb{L}(T)$ , and so $F$ is faithful.

To show that $F$ is full, given a morphism $f\in\mathcal{C}(c_{1},c_{2})$ , let $f=qj$ be its canonical factorisation. Since $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a connected category, there exist idempotent cones $\epsilon_{1}\text{ and }\epsilon_{2}\in E(\widehat{\mathcal{C}_{\mathfrak{D}}})$ with vertices $c_{1}$ and $c_{2}$ , respectively. Let $\gamma:=\epsilon_{1}\ast q$ . By Lemma 3.9, the cone $\gamma$ is in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ and by Lemma 3.11, we have $\gamma\in\epsilon_{1}T\epsilon_{2}$ . So $r:=r(\epsilon_{1},\gamma,\epsilon_{2})$ is a morphism in the normal category $\mathbb{L}(T)$ such that

F(r)=\gamma(z_{\epsilon_{1}})j(z_{\gamma},z_{\epsilon_{2}})=\epsilon_{1}\ast q(z_{\epsilon_{1}})j(z_{\gamma},z_{\epsilon_{2}})=\epsilon_{1}\ast q(c_{1})j(z_{\gamma},c_{2})=\epsilon_{1}(c_{1})\>q\>j=1_{c_{1}}qj=qj=f.

We conclude that $F$ is a normal category isomorphism, as required. ∎

We now summarise the above discussion.

Proposition 3.14.

Given a connected category $\mathcal{C}_{\mathfrak{D}}$ , the left ideal category $\mathbb{L}(\widehat{\mathcal{C}_{\mathfrak{D}}})$ is isomorphic to $\mathcal{C}$ and the regular poset $\widehat{\mathcal{C}_{\mathfrak{D}}}/\mathrel{\mathscr{R}}$ is order isomorphic to $\mathfrak{D}$ .

Remark 3.5.

Since $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a regular semigroup, the principal right ideals of $\widehat{\mathcal{C}_{\mathfrak{D}}}$ form a normal category $\mathbb{R}(\widehat{\mathcal{C}_{\mathfrak{D}}})$ as defined in equation (4). The poset $v\mathbb{R}(\widehat{\mathcal{C}_{\mathfrak{D}}})$ of objects of the right normal category is, in fact, order isomorphic to the poset $\mathfrak{D}$ .

From the discussion above, it follows the next characterisation of the quasi-orders on the set of idempotents of $\widehat{\mathcal{C}_{\mathfrak{D}}}$ .

Lemma 3.15.

Let $\epsilon_{1}=\epsilon(c_{1},\mathfrak{d}_{1})$ and $\epsilon_{2}=\epsilon(c_{2},\mathfrak{d}_{2})$ be idempotents in the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ , then:

(1)

$\epsilon_{1}\mathrel{\leqslant_{\ell}}\epsilon_{2}\text{ if and only if }c_{1}\preceq c_{2};$
(2)

$\epsilon_{1}\mathrel{\leqslant_{r}}\epsilon_{2}\text{ if and only if }\mathfrak{d}_{1}\sqsubseteq\mathfrak{d}_{2}$ .

In this last lemma, observe the abuse of notation as $\sqsubseteq$ is the partial order on $\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ , however it coincides with the partial order on $\widehat{\mathcal{C}_{\mathfrak{D}}}/\mathrel{\mathscr{R}}$ .

3.3. Structure of left reductive regular semigroups

In the previous section, we saw a left reductive semigroup constructed using a connected category. Now, we proceed to show how a left reductive semigroup gives rise to a connected category. To this end, recall from Theorem 2.6 that given a left reductive semigroup $S$ , we have $S\xhookrightarrow{}\widehat{\mathbb{L}(S)}$ via the map $\bar{\rho}\colon a\mapsto r^{a}$ . In fact, more is true.

Figure 6. The injective homomorphism

\bar{\rho}

for a left reductive semigroup

S

Lemma 3.16.

If $S$ is a left reductive semigroup, for an element $a\in S$ , the map $\bar{\rho}_{|R_{a}}$ is a bijection onto $R_{r^{a}}$ , wherein $R_{r^{a}}$ denotes the $\mathrel{\mathscr{R}}$ -class of the cone $r^{a}$ in the semigroup $\widehat{\mathbb{L}(S)}$ .

Proof.

We have already seen that the map $\bar{\rho}$ is injective, when $S$ is a left reductive semigroup. To see that the map $\bar{\rho}_{|R_{a}}$ has image $R_{r^{a}}$ , let $\gamma\in R_{r^{a}}\subseteq\widehat{\mathbb{L}(S)}$ . Thus $\gamma\mathrel{\mathscr{R}}r^{a}$ and $z_{\gamma}=Sf$ for some $f\in E(S)$ . Then since $\bar{\rho}$ is an injective homomorphism, there exists $e\in E(S)$ such that $e\mathrel{\mathscr{R}}a$ in $S$ , and $r^{e}\mathrel{\mathscr{R}}r^{a}\mathrel{\mathscr{R}}\gamma$ in the semigroup $\widehat{\mathbb{L}(S)}$ . Now, by Lemma 2.3(2), there is an isomorphism in $\mathbb{L}(S)$ , say $r(e,x,f)$ such that $\gamma=r^{e}\ast r(e,x,f)=r^{x}$ . Then since $r(e,x,f)$ is an isomorphism in $\mathbb{L}(S)$ , we have $e\mathrel{\mathscr{R}}x$ in the semigroup $S$ (see Figure 6 and [41, Proposition III.13(c)]). So, we obtain $x\mathrel{\mathscr{R}}a$ and $\bar{\rho}(x)=r^{x}=\gamma$ , whence $\bar{\rho}_{|R_{a}}$ maps onto $R_{r^{a}}$ . ∎

From the discussion above and using still the notation $\bar{\rho}$ , we see that $\bar{\rho}$ : $S/\mathrel{\mathscr{R}}\>\xhookrightarrow{}\>\widehat{\mathbb{L}(S)}/\mathrel{\mathscr{R}}$ , where $R_{a}$ maps to $R_{r^{a}}$ , is such that, for any element $a\in S$ , the sets $R_{a}$ and $R_{r^{a}}$ are in bijection. Hence in the sequel, for ease of notation we shall denote the $\mathrel{\mathscr{R}}$ -class in $\widehat{\mathbb{L}(S)}$ containing the cone $r^{a}$ by just $\mathfrak{r}_{a}$ . Observe that there might be $\mathrel{\mathscr{R}}$ -classes in $\widehat{\mathbb{L}(S)}$ which are not of the form $\mathfrak{r}_{a}$ for some $a\in S$ . For instance, in the Figure 6, the bottom $\mathrel{\mathscr{R}}$ -class of the semigroup $\widehat{\mathbb{L}(S)}$ does not have a preimage under $\bar{\rho}$ . As the reader may have already realised (also see Remark 3.1), the definition of a connection semigroup involves excluding from the semigroup $\widehat{\mathbb{L}(S)}$ the $\mathrel{\mathscr{R}}$ -classes that are not of the form $\mathfrak{r}_{a}$ . So, let

(9)

\mathfrak{R}:=\bar{\rho}(S/\mathrel{\mathscr{R}})=\{R_{r^{e}}:e\in E(S)\}=\{\mathfrak{r}_{e}:e\in E(S)\}.

Then, $\mathfrak{R}\subseteq\widehat{\mathbb{L}(S)}/\mathrel{\mathscr{R}}$ and for each $Sf\in v\mathbb{L}(S)$ , there is $\mathfrak{r}_{f}=\bar{\rho}(R_{f})\in\mathfrak{R}$ such that $\mathfrak{r}_{f}$ contains an idempotent cone with vertex $Sf$ , namely $r^{f}$ . Observe that this idempotent cone may be denoted by $\epsilon(Sf,\mathfrak{r}_{f})$ , i.e., the object $Sf$ is connected by $\mathfrak{r}_{f}$ . So, we can get a connected category from a left reductive semigroup. Hence we have proved the following proposition:

Proposition 3.17.

Let $S$ be a left reductive semigroup. The normal category $\mathbb{L}(S)$ is connected by the regular poset $\mathfrak{R}$ , that is, $\mathbb{L}(S)_{\mathfrak{R}}$ is a connected category.

Given the connected category $\mathbb{L}(S)_{\mathfrak{R}}$ , by Propositions 3.8 and 3.10, we know that the connection semigroup $\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ is a left reductive semigroup. Moreover, by equation (6), the idempotents of $\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ are given by:

E(\widehat{\mathbb{L}(S)_{\mathfrak{R}}})=\{\epsilon(Se,\mathfrak{r}_{e}):e\in E(S)\}=\{r^{e}:e\in E(S)\}.

Proposition 3.18.

Given a left reductive semigroup $S$ , the connection semigroup

\widehat{\mathbb{L}(S)_{\mathfrak{R}}}=\{r^{a}:a\in S\}

is isomorphic to $S$ .

Proof.

First, recall that there is an injective homomorphism $\bar{\rho}:S\to\widehat{\mathbb{L}(S)}$ given by $a\mapsto r^{a}$ . Now given any $x\in S$ , for some $e\in E(R_{x})$ and $f\in E(L_{x})$ , we have $r^{x}=r^{e}\ast r(e,x,f)$ . Here $r^{e}=\epsilon(Se,\mathfrak{r}_{e})$ is an idempotent cone in $\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ and $r(e,x,f)$ is an isomorphism in $\mathbb{L}(S)$ . Now, using Lemma 3.7, we see that $\bar{\rho}:S\to\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ is an injective homomorphism which is also surjective (see Figure 6). Hence the connection semigroup $\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ is isomorphic to $S$ . ∎

3.4. Category equivalence

We have seen that given a connected category $\mathcal{C}_{\mathfrak{D}}$ , we get a left reductive semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ such that the category $\mathbb{L}(\widehat{\mathcal{C}_{\mathfrak{D}}})$ is isomorphic to $\mathcal{C}$ and the order ideal $\widehat{\mathcal{C}_{\mathfrak{D}}}/\mathrel{\mathscr{R}}$ is isomorphic to the regular poset $\mathfrak{D}$ (Proposition 3.14). Conversely, given a left reductive semigroup $S$ , we have obtained a connected category $\mathbb{L}(S)_{\mathfrak{R}}$ such that its connection semigroup $\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ is isomorphic to $S$ (Proposition 3.18). Next, we proceed to extend this correspondence to a category equivalence.

First observe that left reductive semigroups form a full subcategory, say $\mathbf{LRS}$ of the category $\mathbf{RS}$ of regular semigroups, with semigroup homomorphisms as morphisms.

Definition 3.2.

Given connected categories $\mathcal{C}_{\mathfrak{D}}$ and $\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}$ , we define a CC-morphism as an ordered pair $m:=(F,G)$ such that $F\colon\mathcal{C}\to\mathcal{C^{\prime}}$ is an inclusion preserving functor and $G\colon\mathfrak{D}\to\mathfrak{D^{\prime}}$ is an order preserving map satisfying:

(10)

c\text{ is connected to }\mathfrak{d}\implies F(c)\text{ is connected to }G(\mathfrak{d})\quad\text{ and }\quad F(\epsilon(c,\mathfrak{d})(c^{\prime}))=\epsilon(F(c),G(\mathfrak{d}))(F(c^{\prime}))

for every $c^{\prime}\in v\mathcal{C}$ .

Remark 3.6.

Given a CC-morphism $m:=(F,G)$ from $\mathcal{C}_{\mathfrak{D}}$ to $\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}$ , by definition the functor $F$ maps an idempotent cone $\epsilon:=\epsilon(c,\mathfrak{d})$ in the category $\mathcal{C}$ to the idempotent cone $\epsilon^{\prime}:=\epsilon(F(c),G(\mathfrak{d}))$ in the category $\mathcal{C^{\prime}}$ . This makes the relation between the categories $\mathcal{C}$ and $\mathcal{C^{\prime}}$ via the functor $F$ rather strong. Roughly speaking, the semigroup homomorphism associated with the morphism $m$ will be an ‘extension’ of this mapping $\epsilon\mapsto\epsilon^{\prime}$ .

Remark 3.7.

Also, given an arbitrary pair $m:=(F,G)$ from $\mathcal{C}_{\mathfrak{D}}$ to $\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}$ such that $F$ is a normal category isomorphism (i.e., a category isomorphism that preserves inclusions) from $\mathcal{C}$ to $\mathcal{C^{\prime}}$ and $G$ is an order isomorphism from $\mathfrak{D}$ to $\mathfrak{D^{\prime}}$ , we can easily construct examples such that the semigroups $\widehat{\mathcal{C}_{\mathfrak{D}}}$ and $\widehat{\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}}$ are not isomorphic. Hence the condition (10) is crucial in the definition of a CC-morphism, to obtain isomorphic semigroups.

It is routine to verify that the class of all connected categories with CC-morphisms form a category. This category will be denoted by $\mathbf{CC}$ in the sequel.

Recall from Proposition 3.17 that given an object $S$ in $\mathbf{LRS}$ , the connected category $\mathbb{L}(S)_{\mathfrak{R}}$ is an object in the category $\mathbf{CC}$ . Now we proceed to make this correspondence functorial.

Lemma 3.19.

Given a semigroup homomorphism $\phi\colon S\to S^{\prime}$ in the category $\mathbf{LRS}$ , define a functor $F_{\phi}\colon\mathbb{L}(S)\to\mathbb{L}(S^{\prime})$ and a map $G_{\phi}\colon\mathfrak{R}\to\mathfrak{R^{\prime}}$ as follows: for idempotents $e,f\in S$ and $u\in eSf$ ,

vF_{\phi}:Se\mapsto S^{\prime}(e\phi)\>,\quad F_{\phi}\colon r(e,u,f)\mapsto r(e\phi,u\phi,f\phi)\>\text{ and }\>G_{\phi}\colon\mathfrak{r}_{e}\mapsto\mathfrak{r}_{e\phi}.

Then $m_{\phi}:=(F_{\phi},G_{\phi})$ is a CC-morphism from $\mathbb{L}(S)_{\mathfrak{R}}$ to $\mathbb{L}(S^{\prime})_{\mathfrak{R^{\prime}}}$ .

Proof.

It is a routine matter to verify that $F_{\phi}$ is an inclusion preserving functor from $\mathbb{L}(S)$ to $\mathbb{L}(S^{\prime})$ and $G_{\phi}$ is an order preserving map from $\mathfrak{R}$ to $\mathfrak{R^{\prime}}$ (see equation (9)). To verify (10), let the object $Se$ be connected to $\mathfrak{r}_{e}$ in $\mathbb{L}(S)_{\mathfrak{R}}$ , we are taking $e$ to be an idempotent in $S$ such that $r^{e}$ is an idempotent in $\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ . Let $e^{\prime}:=e\phi$ . As $\phi$ is a homomorphism, $e^{\prime}$ is an idempotent in $S^{\prime}$ . Hence in the category $\mathbb{L}(S^{\prime})_{\mathfrak{R^{\prime}}}$ , we have $S^{\prime}e^{\prime}$ connected to $\mathfrak{r}_{e^{\prime}}$ , where $\mathfrak{r}_{e^{\prime}}$ denotes the $\mathrel{\mathscr{R}}$ -class $R_{r^{e^{\prime}}}$ of the cone $r^{e^{\prime}}$ in the semigroup $\widehat{\mathbb{L}(S^{\prime})}$ . Further, for every $Sf\in v\mathbb{L}(S)$ , observe that $\epsilon(Se,\mathfrak{r}_{e})(Sf)=r^{e}(Sf)=r(e,ef,f)$ . Since $\phi$ is a homomorphism we have $(ef)^{\prime}=e^{\prime}f^{\prime}$ so that

F_{\phi}(\epsilon(Se,\mathfrak{r}_{e})(Sf))=F_{\phi}(r(e,ef,f))=r(e^{\prime},(ef)^{\prime},f^{\prime})=r(e^{\prime},e^{\prime}f^{\prime},f^{\prime}).

Also, $\epsilon(F_{\phi}(Se),G_{\phi}(\mathfrak{r}_{e}))=\epsilon(S^{\prime}e^{\prime},\mathfrak{r}_{e^{\prime}})=r^{e^{\prime}}$ in $\mathbb{L}(S^{\prime})$ . Hence

\epsilon(F_{\phi}(Se),G_{\phi}(\mathfrak{r}_{e}))(F_{\phi}(Sf))=r^{e^{\prime}}(S^{\prime}f^{\prime})=r(e^{\prime},e^{\prime}f^{\prime},f^{\prime}).

Therefore $m_{\phi}:=(F_{\phi},G_{\phi})$ satisfies equation (10) and so $m_{\phi}$ is a CC-morphism. ∎

Further, it is routine to verify that this assignment preserves identities and compositions. Hence, we have the following proposition.

Proposition 3.20.

The assignment

S\mapsto\mathbb{L}(S)_{\mathfrak{R}}\>\text{ and }\>\phi\mapsto m_{\phi}=(F_{\phi},G_{\phi})

constitutes a functor, say $\mathtt{C}$ from the category $\mathbf{LRS}$ of left reductive semigroups to the category $\mathbf{CC}$ of connected categories.

To build a functor in the opposite direction, we have seen that given a connected category $\mathcal{C}_{\mathfrak{D}}$ in $\mathbf{CC}$ , by Propositions 3.8 and 3.10, the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}\in\mathbf{LRS}$ . Now, given a CC-morphism in $\mathbf{CC}$ , we need to construct a semigroup homomorphism. To this end, recall that given a connected category $\mathcal{C}_{\mathfrak{D}}$ , for each $c\in v\mathcal{C}$ , there is an associated idempotent cone $\epsilon=\epsilon(c,\mathfrak{d})$ in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ such that $R_{\epsilon(c,\mathfrak{d})}=\mathfrak{d}$ . By Lemma 3.7, every cone in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ may be written as $\epsilon\ast u$ , for an idempotent cone $\epsilon$ and an isomorphism $u$ in $\mathcal{C}$ . Let $m:=(F,G)$ be a morphism in $\mathbf{CC}$ from $\mathcal{C}_{\mathfrak{D}}$ to $\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}$ . Define $\phi_{m}\colon\widehat{\mathcal{C}_{\mathfrak{D}}}\to\widehat{\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}}$ by:

(11)

\phi_{m}\colon\epsilon(c,\mathfrak{d})\ast u\mapsto\epsilon(F(c),G(\mathfrak{d}))\ast F(u).

Lemma 3.21.

$\phi_{m}$ is a well-defined map from semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ to $\widehat{\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}}$ .

Proof.

First, observe that by Lemma 3.7, any cone in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ admits a representation, not necessarily unique, of the form $\epsilon(c,\mathfrak{d})\ast u$ . Then $\epsilon(F(c),G(\mathfrak{d}))$ will be an idempotent cone in $\widehat{\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}}$ . Note that given an inclusion preserving functor $F$ preserves normal factorisations (see [41, Proof of Lemma V.4] for a routine verification). So $F(u)$ will be an isomorphism and using Lemma 3.7, we see that $\epsilon(F(c),G(\mathfrak{d}))\ast F(u)$ will be a cone in $\widehat{\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}}$ . Now let $\gamma$ be a cone in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ with vertex $c$ and $R_{\gamma}=\mathfrak{d}$ . As in Remark 3.4, suppose $\gamma=\epsilon_{1}\ast u_{1}=\epsilon_{2}\ast u_{2}$ where $\epsilon_{1}:=\epsilon(c_{1},\mathfrak{d})$ and $\epsilon_{2}:=\epsilon(c_{2},\mathfrak{d})$ are idempotent cones, and $u_{1}:=\gamma(c_{1})$ and $u_{2}:=\gamma(c_{2})$ are isomorphisms. Let $c_{1}^{\prime}:=F(c_{1})$ , $c_{2}^{\prime}:=F(c_{2})$ , $\mathfrak{d}^{\prime}:=G(\mathfrak{d})$ , $\epsilon_{1}^{\prime}:=\epsilon(c_{1}^{\prime},\mathfrak{d}^{\prime})$ , $\epsilon_{2}^{\prime}:=\epsilon(c_{2}^{\prime},\mathfrak{d}^{\prime})$ , $u_{1}^{\prime}:=F(u_{1})$ and $u_{2}^{\prime}:=F(u_{2})$ . We need to show that $\epsilon_{1}^{\prime}\ast u_{1}^{\prime}=\epsilon_{2}^{\prime}\ast u_{2}^{\prime}$ , whence $(\epsilon_{1}\ast u_{1})\phi_{m}=(\epsilon_{2}\ast u_{2})\phi_{m}$ .

Since $\gamma\mathrel{\mathscr{R}}\epsilon_{1}$ , by Lemma 3.11(2), we have $\gamma=\epsilon_{1}\ast\gamma(c_{1})$ . So using equation (1), we get $\gamma(c_{2})=\epsilon_{1}(c_{2})\gamma(c_{1})$ . As $m$ is a CC-morphism, using equation (10),

u_{2}^{\prime}=F(\gamma(c_{2}))=F(\epsilon_{1}(c_{2})\gamma(c_{1}))=F(\epsilon(c_{1},\mathfrak{d})(c_{2}))F(\gamma(c_{1}))=\epsilon(F(c_{1}),G(\mathfrak{d}))(F(c_{2}))=\epsilon_{1}^{\prime}(c_{2}^{\prime})u_{1}^{\prime}.

Finally, by Lemma 3.15(2), we have $\epsilon_{1}^{\prime}\mathrel{\mathscr{R}}\epsilon_{2}^{\prime}$ , and so by Lemma 3.11, we obtain $\epsilon_{1}^{\prime}=\epsilon_{2}^{\prime}\ast\epsilon_{1}^{\prime}(c_{2}^{\prime})$ , whence

\epsilon_{1}^{\prime}\ast u_{1}^{\prime}=\epsilon_{2}^{\prime}\ast\epsilon_{1}^{\prime}(c_{2}^{\prime})\ast u_{1}^{\prime}=\epsilon_{2}^{\prime}\ast\epsilon_{1}^{\prime}(c_{2}^{\prime})u_{1}^{\prime}=\epsilon_{2}^{\prime}\ast u_{2}^{\prime}.

Thus $\phi_{m}$ is a well-defined map from the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ to the semigroup $\widehat{\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}}$ . ∎

Lemma 3.22.

$\phi_{m}$ is a semigroup homomorphism.

Proof.

Using Lemma 3.7, let $\gamma_{1}:=\epsilon(c_{1},\mathfrak{d}_{1})\ast\gamma_{1}(c_{1})=\epsilon_{1}\ast u_{1}$ and $\gamma_{2}:=\epsilon(c_{2},\mathfrak{d}_{2})\ast\gamma_{2}(c_{2})=\epsilon_{2}\ast u_{2}$ be two arbitrary cones in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ such that $u_{1}:=\gamma_{1}(c_{1})\colon c_{1}\to z_{\gamma_{1}}$ and $u_{2}:=\gamma_{2}(c_{2})\colon c_{2}\to z_{\gamma_{2}}$ are isomorphisms. We need to show that $(\gamma_{1}\cdot\gamma_{2})\phi_{m}=\gamma_{1}\phi_{m}\cdot\gamma_{2}\phi_{m}$ .

To this end, given a morphism $u\colon c_{1}\to c_{2}$ in the category $\mathcal{C}$ , we shall denote its image in $\mathcal{C^{\prime}}$ under $F$ by dashed versions without further comment, i.e., we shall denote $F(u)\colon F(c_{1})\to F(c_{2})$ by $u^{\prime}\colon c_{1}^{\prime}\to c_{2}^{\prime}$ . Then since $\gamma_{1}(c_{1})=(\gamma_{1}(c_{1}))^{\circ}$ , we have

\gamma_{1}\cdot\gamma_{2}=\epsilon_{1}\ast\gamma_{1}(c_{1})\cdot\epsilon_{2}\ast\gamma_{2}(c_{2})=\epsilon_{1}\ast\gamma_{1}(c_{1})\ast(\epsilon_{2}(z_{\gamma_{1}})\ast\gamma_{2}(c_{2}))^{\circ}=\epsilon_{1}\ast(\gamma_{1}(c_{1}))^{\circ}\ast(\epsilon_{2}(z_{\gamma_{1}})\ast\gamma_{2}(c_{2}))^{\circ}.

Now, using Lemma 2.1(3), we see that

(\gamma_{1}(c_{1}))^{\circ}\ast(\epsilon_{2}(z_{\gamma_{1}})\ast\gamma_{2}(c_{2}))^{\circ}=(\gamma_{1}(c_{1})\>\epsilon_{2}(z_{\gamma_{1}})\>\gamma_{2}(c_{2}))^{\circ}.

So if we let $qu:=(\gamma_{1}(c_{1})\>\epsilon_{2}(z_{\gamma_{1}})\>\gamma_{2}(c_{2}))^{\circ}$ and $\epsilon(c,\mathfrak{d}):=\epsilon_{1}\ast q$ , we obtain

\gamma_{1}\cdot\gamma_{2}=\epsilon_{1}\ast(\gamma_{1}(c_{1})\>\epsilon_{2}(z_{\gamma_{1}})\>\gamma_{2}(c_{2}))^{\circ}=\epsilon_{1}\ast qu=\epsilon(c,\mathfrak{d})\ast u.

Hence by definition of $\phi_{m}$ , we get

(\gamma_{1}\cdot\gamma_{2})\phi_{m}=(\epsilon(c,\mathfrak{d})\ast u)\phi_{m}=\epsilon(F(c),G(\mathfrak{d}))\ast F(u)=\epsilon(c^{\prime},\mathfrak{d}^{\prime})\ast u^{\prime}.

Then as $F$ is a functor, using equation (10), we reach

\epsilon(c^{\prime},\mathfrak{d}^{\prime})\ast u^{\prime}=\epsilon_{1}^{\prime}\ast q^{\prime}\ast u^{\prime}=\epsilon_{1}^{\prime}\ast F(qu).

Also, as $F$ preserves normal factorisations, using Lemma 2.1(3) and equation (10), we conclude that

F(qu)=F((\gamma_{1}(c_{1})\>\epsilon_{2}(z_{\gamma_{1}})\>\gamma_{2}(c_{2}))^{\circ})=(F(\gamma_{1}(c_{1})\>\epsilon_{2}(z_{\gamma_{1}})\>\gamma_{2}(c_{2})))^{\circ}=(u_{1}^{\prime}\>\epsilon_{2}^{\prime}(z_{\gamma_{1}}^{\prime})\>u_{2}^{\prime})^{\circ}=u_{1}^{\prime}\ast(\epsilon_{2}^{\prime}(z_{\gamma_{1}}^{\prime})\ast u_{2}^{\prime})^{\circ}.

Therefore putting everything together and using equation (2),

(\gamma_{1}\cdot\gamma_{2})\phi_{m}=\epsilon_{1}^{\prime}\ast F(qu)=\epsilon_{1}^{\prime}\ast u_{1}^{\prime}\ast(\epsilon_{2}^{\prime}(z_{\gamma_{1}}^{\prime})\ast u_{2}^{\prime})^{\circ}=\epsilon_{1}^{\prime}\ast u_{1}^{\prime}\cdot\epsilon_{2}^{\prime}\ast u_{2}^{\prime}=\gamma_{1}\phi_{m}\cdot\gamma_{2}\phi_{m},

as required. ∎

After having constructed a semigroup homomorphism from a CC-morphism, we may now routinely verify the following assertion.

Proposition 3.23.

The assignment

\mathcal{C}_{\mathfrak{D}}\mapsto\widehat{\mathcal{C}_{\mathfrak{D}}}\>\text{ and }\>m\mapsto\phi_{m}

constitutes a functor, say $\mathtt{S}$ from the category $\mathbf{CC}$ of connected categories to the category $\mathbf{LRS}$ of left reductive semigroups.

Now, we have all the ingredients to prove the category equivalence of $\mathbf{LRS}$ and $\mathbf{CC}$ .

Lemma 3.24.

The identity functor $1_{\mathbf{LRS}}$ is naturally isomorphic to the functor $\mathtt{C}\mathtt{S}\colon\mathbf{LRS}\to\mathbf{LRS}$ .

Proof.

We need to illustrate a natural transformation between the functors $1_{\mathbf{LRS}}$ and $\mathtt{C}\mathtt{S}$ such that each of its components is a semigroup isomorphism. By Proposition 3.18 we know that the map $\bar{\rho}\colon a\mapsto r^{a}$ is an isomorphism from $S$ to $\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ . So, for each object $S\in\mathbf{LRS}$ , we denote this map by $\bar{\rho}(S)$ . Then since

1_{\mathbf{LRS}}(S)=S\quad\text{ and }\quad\mathtt{C}\mathtt{S}(S):=\mathtt{S}(\mathtt{C}(S))=\mathtt{S}(\mathbb{L}(S)_{\mathfrak{R}})=\widehat{\mathbb{L}(S)_{\mathfrak{R}}},

we see that the map $\bar{\rho}(S)$ is a semigroup isomorphism from $1_{\mathbf{LRS}}(S)$ to $\mathtt{C}\mathtt{S}(S)$ . Further, given a semigroup homomorphism $\phi\colon S\to S^{\prime}$ , we can routinely verify that the following diagram commutes:

Hence the assignment $S\mapsto\bar{\rho}(S)$ constitutes a natural isomorphism between the functors $1_{\mathbf{LRS}}$ and $\mathtt{C}\mathtt{S}$ . ∎

Lemma 3.25.

The identity functor $1_{\mathbf{CC}}$ is naturally isomorphic to the functor $\mathtt{S}\mathtt{C}\colon\mathbf{CC}\to\mathbf{CC}$ .

Proof.

From Proposition 3.14, given a connected category $\mathcal{C}_{\mathfrak{D}}$ , the left ideal category $\mathbb{L}(\widehat{\mathcal{C}_{\mathfrak{D}}})$ is a normal category isomorphic to the category $\mathcal{C}$ and the regular poset $\widehat{\mathcal{C}_{\mathfrak{D}}}/\mathrel{\mathscr{R}}$ is order isomorphic to $\mathfrak{D}$ , via the functor $F$ and the map $G$ respectively. Further, we can routinely (but admittedly a bit cumbersome in terms of notation) verify that $\iota:=(F,G)$ satisfies condition (10), and both $F$ and $G$ are bijections; hence $\iota$ is a CC-isomorphism. So, for each object $\mathcal{C}_{\mathfrak{D}}\in\mathbf{CC}$ , if we denote this morphism by $\iota(\mathcal{C}_{\mathfrak{D}})$ , then since

1_{\mathbf{CC}}(\mathcal{C}_{\mathfrak{D}})=\mathcal{C}_{\mathfrak{D}}\quad\text{ and }\quad\mathtt{S}\mathtt{C}(\mathcal{C}_{\mathfrak{D}}):=\mathtt{C}(\mathtt{S}(\mathcal{C}_{\mathfrak{D}}))=\mathtt{C}(\widehat{\mathcal{C}_{\mathfrak{D}}})=\mathbb{L}(\widehat{\mathcal{C}_{\mathfrak{D}}})_{\mathfrak{R}}

we see that $\iota(\mathcal{C}_{\mathfrak{D}})$ is a CC-isomorphism from $1_{\mathbf{CC}}(S)$ to $\mathtt{S}\mathtt{C}(\mathcal{C}_{\mathfrak{D}})$ . Further, given a CC-morphism $m\colon\mathcal{C}_{\mathfrak{D}}\to\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}$ , we can routinely verify that the following diagram in the category $\mathbf{CC}$ commutes:

As a meticulous reader may see, the condition (10) is again crucial in this verification. Hence the assignment $\mathcal{C}_{\mathfrak{D}}\mapsto\iota(\mathcal{C}_{\mathfrak{D}})$ constitutes a natural isomorphism between the functors $1_{\mathbf{CC}}$ and $\mathtt{S}\mathtt{C}$ . ∎

Combining the Lemmas 3.24 and 3.25, we have the main theorem of this paper:

Theorem 3.26.

The category $\mathbf{LRS}$ of left reductive semigroups is equivalent to the category $\mathbf{CC}$ of connected categories.

4. $\mathrel{\mathscr{L}}$ -unipotent semigroups

The remaining of the paper is essentially dedicated to applications of the results in Section 3 to various classes of left reductive semigroups. In this section, we specialise the construction in the previous section to give an abstract construction of $\mathrel{\mathscr{L}}$ -unipotent semigroups using supported normal categories. $\mathrel{\mathscr{L}}$ -unipotent semigroups were introduced and studied initially by Venkatesan [47, 48, 49, 50] under the name of right inverse semigroups. Over the years, various facets of this class of semigroups (and of its dual, the class of $\mathrel{\mathscr{R}}$ -unipotent semigroups) have been studied by many people including the second author [11, 31, 27, 17, 16, 46, 9, 10, 26, 19, 21, 20]. We begin by recalling some basic properties of $\mathrel{\mathscr{L}}$ -unipotent semigroups.

As usual by an $\mathrel{\mathscr{L}}$ -unipotent semigroup we mean a regular semigroup in which each $\mathrel{\mathscr{L}}$ -class contains a unique idempotent.

Proposition 4.1 ([48, Theorem 1][16, Corollary 1.2, 1.3]).

Let $S$ be a regular semigroup. The following are equivalent:

(1)

$S$ is an $\mathrel{\mathscr{L}}$ -unipotent semigroup;
(2)

$eS\cap fS=efS=feS$ , for all $e,f\in E(S)$ ;
(3)

$efe=fe$ , for all $e,f\in E(S)$ , i.e., $E(S)$ is a right regular band;
(4)

$a^{\prime}a=a^{\prime\prime}a$ , for all $a\in S$ and $a^{\prime},a^{\prime\prime}\in V(a)$ ;
(5)

each $\mathrel{\mathscr{L}}$ -class $L_{a}$ contains a unique idempotent $aa^{\prime}$ , for all $a\in S$ and $a^{\prime}\in V(a)$ ;
(6)

$a^{\prime}ea=a^{\prime\prime}ea$ , for all $a\in S$ and $a^{\prime},a^{\prime\prime}\in V(a)$ .
(7)

$aa^{\prime}ea=ea$ , for all $a\in S$ , $a^{\prime}\in V(a)$ and $e\in E(S)$ .

By [48, Theorem 4(1)], it is known that an $\mathrel{\mathscr{L}}$ -unipotent semigroup is left reductive, i.e., the regular representation $a\mapsto\rho_{a}$ is injective. Since this is a cornerstone of this section, we record this formally amongst some other useful results regarding $\mathrel{\mathscr{L}}$ -unipotent semigroups and provide a more transparent proof for the left-reductivity. In the process, we also characterise the natural partial order on an $\mathrel{\mathscr{L}}$ -unipotent semigroup.

Recall that a regular semigroup whose idempotents form a band is said to be orthodox, and this is the case of any $\mathrel{\mathscr{L}}$ -unipotent semigroup. In any orthodox semigroup $S$ , for all $a\in S$ , $a^{\prime}\in V(a)$ and $e\in E(S)$ , the elements $a^{\prime}ea$ and $aea^{\prime}$ are idempotents (cf. [29, Proposition 6.2.2]).

On another hand, in any regular semigroup $S$ , the natural partial order [40] is given by, for all $a,b\in S$ ,

a\leqslant b\iff\exists{e,f\in E(S)},\>a=be=fb.

When $S$ is $\mathrel{\mathscr{L}}$ -unipotent, a possible form taken by $\leqslant$ is, for all $a,b\in S$ ,

a\leqslant b\iff\exists f\in E(S),\>a=fb.

Proposition 4.2.

Let $S$ be an $\mathrel{\mathscr{L}}$ -unipotent semigroup. For every pair of distinct elements $a,b$ in $S$ , there exists an idempotent $f$ in $S$ such that $fa\neq fb$ . In particular, the semigroup $S$ is left reductive.

Proof.

We prove by contradiction. Suppose $fa=fb$ for every idempotent $f$ . Since $aa^{\prime}$ is an idempotent, we have $a=(aa^{\prime})a=(aa^{\prime})b$ ; similarly we get $b=(bb^{\prime})b=(bb^{\prime})a$ . This implies $a\leqslant b$ and $b\leqslant a$ respectively, whence $a=b$ and we get a contradiction. This concludes the first part of the lemma.

Given $a\neq b$ in $S$ , we have an idempotent $f$ in $S$ such that $f\rho_{a}\neq f\rho_{b}$ . Hence, the map $\rho\colon a\mapsto\rho_{a}$ is injective, whence $S$ is left reductive. ∎

Next, given an $\mathrel{\mathscr{L}}$ -unipotent semigroup $S$ , we shift our attention to the normal category $\mathbb{L}(S)$ . The following lemma is a simple consequence of Proposition 4.1(2).

Lemma 4.3.

In the category $\mathbb{L}(S)$ , given morphisms $r(e,u,f)$ and $r(g,v,h)$ we have $r(e,u,f)=r(g,v,h)$ if and only if $e=g$ , $f=h$ and $v=u$ . Also, if $r(e,e,f)$ is an inclusion in $\mathbb{L}(S)$ , the corresponding unique retraction is $r(f,e,e)$ .

Recall also that in any regular semigroup $S$ , the $\mathrel{\mathscr{R}}$ -classes of $S$ form a regular poset $(S/\mathrel{\mathscr{R}},\sqsubseteq)$ under usual set inclusion. If $S$ is, in addition $\mathrel{\mathscr{L}}$ -unipotent, then by Proposition 4.1(3), we have $eS\cap fS=efS=feS$ , and so the regular poset becomes a semilattice $(S/\mathrel{\mathscr{R}},\wedge)$ where the meet operation is given by set intersection:

R_{e}\wedge R_{f}=R_{ef}.

The condition (3) in the Proposition 4.1 may make one wonder if a regular semigroup $S$ such that its poset $S/\mathscr{R}$ is a semilattice is always an $\mathrel{\mathscr{L}}$ -unipotent semigroup (also see equation (5)). This need not be the case as the following simple example shows. Consider a three element semigroup $T$ given by the following $\mathrel{\mathscr{D}}$ -class picture (on the left):

This semigroup $T$ is clearly not $\mathrel{\mathscr{L}}$ -unipotent (as $e$ and $f$ are distinct $\mathrel{\mathscr{L}}$ -related idempotents) although the regular poset $T/\mathrel{\mathscr{R}}$ forms a three element semilattice (given on the right side). Observe that $eT\cap fT=0T=\{0\}$ .

Notice that this semilattice which appears in the context of the left regular bands has been referred to as the support semilattice of the semigroup [34, 12]. We shall be discussing the case of right and left regular bands in Sections 5 and 6, respectively. With this terminology in mind, we proceed to give the construction of an $\mathrel{\mathscr{L}}$ -unipotent semigroup by introducing supported categories, as specialisations of connected categories.

Definition 4.1.

Let $\mathcal{C}$ be a normal category and let $\mathfrak{D}$ be a sub-semilattice of the poset $\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ . Then $\mathcal{C}$ is said to be supported by $\mathfrak{D}$ if each $c\in v\mathcal{C}$ is connected by a unique $\mathfrak{d}\in\mathfrak{D}$ .

Remark 4.1.

By the definition above, there is a well-defined mapping $\Gamma:v\mathcal{C}\to\mathfrak{D}$ . This is a surjection and will be referred to as the support map in the sequel. Observe that the support map $\Gamma$ need not be injective, in general, but we shall later see that $\Gamma$ is always order preserving (see Proposition 4.5). Also note that in contrast to the support map of [34], our map $\Gamma$ is not a homomorphism as there is no semigroup structure in the set $v\mathcal{C}$ .

Remark 4.2.

The assumption that $\mathfrak{D}$ is a semilattice is not necessary in Definition 4.1. One can show that the regular poset $\mathfrak{D}$ will necessarily form a semilattice, once we establish the following proposition.

Proposition 4.4.

Let $\mathcal{C}_{\mathfrak{D}}$ be a supported category. Then the connection semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is $\mathrel{\mathscr{L}}$ -unipotent.

Proof.

Given a supported category $\mathcal{C}_{\mathfrak{D}}$ , it is connected and so by Proposition 3.8, we know that $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is a regular semigroup. Now, using Lemmas 3.11 and 3.15, given a cone $\gamma\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ with vertex $c$ , we can see that any idempotent in the $\mathrel{\mathscr{L}}$ -class of $\gamma$ is of the form $\epsilon(c,\mathfrak{d})$ , for some $\mathfrak{d}\in\mathfrak{D}$ connecting $c$ . However, $\mathcal{C}_{\mathfrak{D}}$ is a supported category, there is a unique $\mathfrak{d}$ with this property. Hence each $\mathrel{\mathscr{L}}$ -class of $\widehat{\mathcal{C}_{\mathfrak{D}}}$ contains a unique idempotent $\epsilon(c,\mathfrak{d})$ and by Proposition 4.1(2), the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is $\mathrel{\mathscr{L}}$ -unipotent. ∎

We now point out to the following specialisations of Lemmas 3.11 and 3.15, in the case of supported categories.

Proposition 4.5.

Given a supported category $\mathcal{C}_{\mathfrak{D}}$ , let $\epsilon_{1}=\epsilon(c_{1},\mathfrak{d}_{1})$ and $\epsilon_{2}=\epsilon(c_{2},\mathfrak{d}_{2})$ be idempotents in the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ , then:

(1)

$\epsilon_{1}=\epsilon_{2}\iff c_{1}=c_{2}$ , and there is a bijection between the sets $E(\widehat{\mathcal{C}_{\mathfrak{D}}})$ and $v\mathcal{C}$ ;
(2)

$\epsilon_{1}\mathrel{\leqslant_{r}}\epsilon_{2}\iff\mathfrak{d}_{1}=\mathfrak{d}_{1}\wedge\mathfrak{d}_{2}$ .

In particular, the support map $\Gamma\colon v\mathcal{C}\to\mathfrak{D}$ defined by $c\mapsto R_{\epsilon(c,\mathfrak{d})}$ is an order preserving surjection.

Proof.

(1) follows from the fact that there is a unique idempotent in the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ with a given vertex. To prove (2), first observe that $\mathfrak{D}$ is a semilattice and in $\mathfrak{D}$ , we have $\mathfrak{d}_{1}\sqsubseteq\mathfrak{d}_{2}$ if and only if $\mathfrak{d}_{1}=\mathfrak{d}_{1}\wedge\mathfrak{d}_{2}$ . Also, by Proposition 3.14, we know that $\widehat{\mathcal{C}_{\mathfrak{D}}}/\mathrel{\mathscr{R}}$ is order isomorphic to $\mathfrak{D}$ . So,

\epsilon_{1}\mathrel{\leqslant_{r}}\epsilon_{2}\iff\epsilon_{1}\widehat{\mathcal{C}_{\mathfrak{D}}}\sqsubseteq\epsilon_{2}\widehat{\mathcal{C}_{\mathfrak{D}}}\iff\mathfrak{d}_{1}\sqsubseteq\mathfrak{d}_{2}\iff\mathfrak{d}_{1}=\mathfrak{d}_{1}\wedge\mathfrak{d}_{2}.

To show the last part of this proposition, first observe that the map $\Gamma$ is well-defined by (1) and by definition, it is a surjection. Now if $c_{1}\preceq c_{2}$ , then by Lemma 3.15(1), we have $\epsilon_{1}\mathrel{\leqslant_{\ell}}\epsilon_{2}$ . Using biorder properties [39, Definition 1.1(B21)], we have $\epsilon_{1}\mathrel{\mathscr{L}}\epsilon_{2}\epsilon_{1}\leqslant\epsilon_{2}$ in the $\mathrel{\mathscr{L}}$ -unipotent semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ . As each $\mathrel{\mathscr{L}}$ -class in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ contains a unique idempotent, we have $\epsilon_{1}=\epsilon_{2}\epsilon_{1}$ , i.e., $\epsilon_{1}\mathrel{\leqslant_{r}}\epsilon_{2}$ . So by Lemma 3.15(2), we obtain $\mathfrak{d}_{1}\sqsubseteq\mathfrak{d}_{2}$ and therefore $\Gamma$ is order preserving. ∎

We have seen above how a supported category gives rise to an $\mathrel{\mathscr{L}}$ -unipotent semigroup. The proposition below shows the converse, i.e. every $\mathrel{\mathscr{L}}$ -unipotent semigroup determines a supported category.

Proposition 4.6.

Let $S$ be an $\mathrel{\mathscr{L}}$ -unipotent semigroup $S$ . Then $\mathbb{L}(S)_{\mathfrak{R}}$ is a supported category. The support map $\Gamma_{S}\colon v\mathbb{L}(S)\to\mathfrak{R}$ is given by $Se\mapsto\mathfrak{r}_{e}$ .

Proof.

Given an $\mathrel{\mathscr{L}}$ -unipotent semigroup $S$ , by Proposition 4.2, it is left reductive. As in Section 3.3, we can show that the category $\mathbb{L}(S)$ is normal and the semigroup $\widehat{\mathbb{L}(S)}$ is regular. Now, we define the order ideal $\mathfrak{R}\subseteq\widehat{\mathbb{L}(S)}/\mathrel{\mathscr{R}}$ as:

\mathfrak{R}:=\{\mathfrak{r}_{e}:e\in E(S)\}.

We know that $S/\mathrel{\mathscr{R}}$ is order isomorphic to $\mathfrak{R}$ via $R_{e}\mapsto\mathfrak{r}_{e}:=R_{r^{e}}$ , and in addition it is a a meet semilattice when $S$ is $\mathrel{\mathscr{L}}$ -unipotent. Hence $(\mathfrak{R},\wedge)$ is a meet semilattice with

\mathfrak{r}_{e}\wedge\mathfrak{r}_{f}=\mathfrak{r}_{ef}.

Also, since each $\mathrel{\mathscr{L}}$ -class in $S$ contains a unique idempotent, each object $Se$ in $v\mathbb{L}(S)$ is connected by a unique $\mathfrak{r}_{e}\in\mathfrak{R}$ . Further, in an $\mathrel{\mathscr{L}}$ -unipotent semigroup, as $e\mathrel{\leqslant_{\ell}}f$ if and only if $e\leqslant f$ , we have:

Se\preceq Sf\iff e\mathrel{\leqslant_{\ell}}f\iff e\leqslant f\implies e\mathrel{\leqslant_{r}}f\iff R_{e}\sqsubseteq R_{f}\iff\mathfrak{r}_{e}\sqsubseteq\mathfrak{r}_{f}.

Hence the support map $\Gamma_{S}\colon Se\mapsto\mathfrak{r}_{e}$ is an order preserving surjection from $v\mathbb{L}(S)$ to $\mathfrak{R}$ . ∎

Specialising Proposition 3.18, we get:

Proposition 4.7.

Given an $\mathrel{\mathscr{L}}$ -unipotent semigroup $S$ , the connection semigroup $\widehat{\mathbb{L}(S)_{\mathfrak{R}}}$ is isomorphic to $S$ .

Further, the discussion in Section 3.4 carries over verbatim to the $\mathrel{\mathscr{L}}$ -unipotent case. It is clear that $\mathrel{\mathscr{L}}$ -unipotent semigroups form a full subcategory of $\mathbf{LRS}$ , say $\mathbf{LUS}$ and supported categories form a full subcategory of $\mathbf{CC}$ , say $\mathbf{SC}$ . We shall refer to the morphisms in the subcategory $\mathbf{SC}$ as SC-morphisms, in the sequel.

Repeating the arguments in Section 3.4 to obtain Theorem 3.26, we can show the following.

Theorem 4.8.

The category $\mathbf{LUS}$ of $\mathrel{\mathscr{L}}$ -unipotent semigroups is equivalent to the category $\mathbf{SC}$ of supported categories.

5. Right regular bands

Now we further specialise the construction in Section 4 to describe right regular bands and, in this case, we obtain an adjunction. We show that right regular bands form a full coreflective subcategory $\mathbf{RRB}$ of the category $\mathbf{LUS}$ . To guide the readers to this end, we recall the following definitions which shall also be needed later in this section.

Definition 5.1.

Let $\mathbf{C}$ and $\mathbf{D}$ be two arbitrary categories. An adjunction $\mathbf{C}\to\mathbf{D}$ is a triple $(\mathtt{F},\mathtt{G},\eta)$ , where $\mathtt{F}\colon\mathbf{C}\to\mathbf{D}$ and $\mathtt{G}\colon\mathbf{D}\to\mathbf{C}$ are functors, and $\eta$ is a natural transformation $1_{\mathbf{C}}\to\mathtt{F}\mathtt{G}$ such that the following condition holds:

•

For every pair of objects $\mathcal{C}\in v\mathbf{C}$ and $\mathcal{D}\in v\mathbf{D}$ , and for every morphism $\phi\colon\mathcal{C}\to\mathtt{G}(\mathcal{D})$ in the category $\mathbf{C}$ , there exists a unique morphism $\bar{\phi}\colon\mathtt{F}(\mathcal{C})\to\mathcal{D}$ in the category $\mathbf{D}$ such that the following diagram commutes:

In this case, $\mathtt{F}$ and $\mathtt{G}$ are called left and right adjoints , respectively, and $\eta$ is the unit of adjunction.

Definition 5.2.

A coreflective subcategory is a full subcategory $\mathcal{B}$ of a category $\mathcal{S}$ whose inclusion functor $\mathtt{J}\colon\mathcal{B}\to\mathcal{S}$ has a right adjoint. We shall say that a category $\mathcal{B}$ is coreflective in $\mathcal{S}$ if $\mathcal{B}$ is equivalent to a coreflective subcategory of $\mathcal{S}$ .

We refer the reader to [18, Proposition 1.3] for several equivalent characterisations of the (dual of) second definition above. Now, it is routine to verify the following lemma.

Lemma 5.1.

The category $\mathbf{RRB}$ of right regular bands is a coreflective subcategory of the category $\mathbf{LUS}$ of $\mathrel{\mathscr{L}}$ -unipotent semigroups.

Here the coreflector functor $\mathtt{B}\colon\mathbf{LUS}\to\mathbf{RRB}$ maps an $\mathrel{\mathscr{L}}$ -unipotent semigroup $S$ to its right regular band $E(S)$ of idempotents. Combining Lemma 5.1 and Theorem 4.8, one can construct the adjunction between the categories $\mathbf{RRB}$ and $\mathbf{SC}$ , but we shall briefly exposit this adjunction in a direct manner. Putting together Propositions 4.4 and 4.1(4), we are led to the next lemma.

Lemma 5.2.

Let $\mathcal{C}_{\mathfrak{D}}$ be a supported category. Then the set $E(\widehat{\mathcal{C}_{\mathfrak{D}}})$ of all idempotent cones in the connection semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$

E(\widehat{\mathcal{C}_{\mathfrak{D}}}):=\{\>\epsilon(c,\mathfrak{d}):c\text{ is connected by }\mathfrak{d}\>\}

forms a right regular band.

Notice that by Proposition 4.5(1), the band $E(\widehat{\mathcal{C}_{\mathfrak{D}}})$ is in bijection with the set $v\mathcal{C}$ . Further, given an SC-morphism $m:=(F,G)$ from $\mathcal{C}_{\mathfrak{D}}$ to $\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}$ , as shown in Lemmas 3.21 and 3.22 we can prove that $\phi_{m}\colon E(\mathcal{C}_{\mathfrak{D}})\to E(\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}})$ given by:

\phi_{m}\colon\epsilon(c,\mathfrak{d})\mapsto\epsilon(F(c),G(\mathfrak{d}))

is a semigroup homomorphism of right regular bands. Hence we obtain a functor $\mathtt{E}\colon\mathbf{SC}\to\mathbf{RRB}$ as follows:

\mathcal{C}_{\mathfrak{D}}\mapsto E(\widehat{\mathcal{C}_{\mathfrak{D}}})\>\text{ and }\>m\mapsto\phi_{m}

Conversely starting from a right regular band $S$ , since it is also $\mathrel{\mathscr{L}}$ -unipotent, we can easily see that (as shown in Section 4) the category $\mathbb{L}(S)_{\mathfrak{R}}$ constitutes a supported category. This correspondence $S\mapsto\mathbb{L}(S)_{\mathfrak{R}}$ is given by the functor $\mathtt{C}\colon\mathbf{RRB}\to\mathbf{SC}$ . The functor $\mathtt{C}$ is precisely the restriction of the functor defined in the Section 3.4 to the category $\mathbf{RRB}\subseteq\mathbf{LUS}\subseteq\mathbf{LRS}$ . Further, we will show that the functor $\mathtt{C}$ is a left adjoint to the functor $\mathtt{E}$ .

Theorem 5.3.

There is an adjunction from the category $\mathbf{RRB}$ of right regular bands to the category $\mathbf{SC}$ of supported categories. In particular, the category $\mathbf{RRB}$ is coreflective in the category $\mathbf{SC}$ .

Proof.

To begin with, observe that given a right regular band $B$ , it is left reductive and by Proposition 3.18, the connection semigroup $\widehat{\mathbb{L}(B)_{\mathfrak{R}}}$ is isomorphic to $B$ via the map $\bar{\rho}(B)$ . Moreover, $E(\widehat{\mathbb{L}(B)_{\mathfrak{R}}})=\widehat{\mathbb{L}(B)_{\mathfrak{R}}}$ . Hence the assignment $B\mapsto\bar{\rho}(B)$ is a natural isomorphism from the functor $1_{\mathbf{RRB}}$ to the functor $\mathtt{C}\mathtt{E}$ .

Now, given an object $B\in v\mathbf{RRB}$ , by Proposition 4.6, we can see that $\mathtt{C}(B)=\mathbb{L}(B)_{\mathfrak{R}}$ is a supported category. Let $\mathcal{C}_{\mathfrak{D}}\in v\mathbf{SC}$ so that $\mathtt{E}({\mathcal{C}_{\mathfrak{D}}})=E(\widehat{\mathcal{C}_{\mathfrak{D}}})$ . Given a semigroup homomorphism $\phi\colon B\to\mathtt{E}({\mathcal{C}_{\mathfrak{D}}})$ in $\mathbf{RRB}$ , by Lemma 3.19 and Proposition 3.14, we see that $m_{\phi}:=(F_{\phi},G_{\phi})$ is the unique SC-morphism from $\mathbb{L}(B)_{\mathfrak{R}}$ to $\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}\subseteq\mathcal{C}_{\mathfrak{D}}$ , where $\mathcal{C^{\prime}}_{\mathfrak{D^{\prime}}}:=\mathtt{C}(\mathtt{E}(\widehat{\mathcal{C}_{\mathfrak{D}}}))$ . Next, we may routinely verify that the following diagram commutes:

Hence $(\mathtt{C},\mathtt{E},\bar{\rho})$ constitutes an adjunction from the category $\mathbf{RRB}$ to $\mathbf{SC}$ . The last part of the theorem follows directly from the fact that the left adjoint $\mathtt{C}$ is fully-faithful (see [18, Proposition 1.3]). To conclude one can check that $\mathbf{RRB}$ is equivalent to the category $\mathtt{C}(\mathbf{RRB})$ , and the latter is a coreflective subcategory of $\mathbf{SC}$ . ∎

6. Right reductive regular semigroups

In this section, we describe some straightforward applications of our results from the previous sections to their dual classes of semigroups. We begin with the class of right reductive semigroups and also consider their subclasses of $\mathrel{\mathscr{R}}$ -unipotent semigroups and of left regular bands.

In Section 2 (see equation (4)), we introduced the normal category $\mathbb{R}(S)$ of principal right ideals of a regular semigroup $S$ . Observe that given an arbitrary element $a\in S$ , the principal cone $l^{a}$ is the cone in $\mathbb{R}(S)$ with vertex $fS$ given by, for each $eS\in v\mathbb{R}(S)$

l^{a}(eS):=l(e,ae,f)\text{ where }f\in E(R_{a}).

Recall from [13, Section 1.3] that the anti-regular representation of $S$ is the anti-homomorphism $\lambda\colon S\to S_{\lambda}$ given by $a\mapsto\lambda_{a}$ , and $S$ is said to be right reductive if $\lambda$ is injective. We may then show the following dual statement of Theorem 2.6.

Theorem 6.1.

Let $S$ be a regular semigroup. There is an anti-homomorphism $\bar{\lambda}\colon S\to\widehat{\mathbb{R}(S)}$ given by $a\mapsto l^{a}$ and the semigroup $S$ is isomorphic to a subsemigroup of $\widehat{\mathbb{R}(S)}^{\text{op}}$ if and only if $S$ is right reductive.

From Propositions 3.8 and 3.10, given a connected category $\mathcal{C}_{\mathfrak{D}}$ , the connection semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is left reductive. Thus, the opposite semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}^{\text{op}}}:=(\widehat{\mathcal{C}_{\mathfrak{D}}},\circ)$ where, for any cones $\gamma,\delta\in\widehat{\mathcal{C}_{\mathfrak{D}}}$ ,

(12)

\gamma\circ\delta:=\delta\ast(\gamma(z_{\delta}))^{\circ},

is right reductive and regular. In the sequel, we shall refer to this semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}^{\text{op}}}$ as the dual connection semigroup.

Now by [41, Remark III.6], we may see that for the opposite semigroup $S^{\text{op}}$ of a regular semigroup $S$ , we get

(13)

\mathbb{R}(S^{\text{op}})=\mathbb{L}(S)\>\text{ and }\>\mathbb{L}(S^{\text{op}})=\mathbb{R}(S).

Moreover, the following relationships between its regular posets hold:

(14)

S^{\text{op}}/\mathrel{\mathscr{R}}=S/\mathrel{\mathscr{L}}\>\text{ and }\>S^{\text{op}}/\mathrel{\mathscr{L}}=S/\mathrel{\mathscr{R}}.

Therefore, $\mathbb{R}(\widehat{\mathcal{C}_{\mathfrak{D}}^{\text{op}}})=\mathbb{L}(\widehat{\mathcal{C}_{\mathfrak{D}}})$ and using Proposition 3.14, we get $\mathbb{L}(\widehat{\mathcal{C}_{\mathfrak{D}}})$ isomorphic to $\mathcal{C}$ . Hence we may state the next proposition.

Proposition 6.2.

Let $\mathcal{C}_{\mathfrak{D}}$ be a connected category. The semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}^{\text{op}}}$ is a right reductive semigroup. The right category $\mathbb{R}(\widehat{\mathcal{C}_{\mathfrak{D}}^{\text{op}}})$ is a normal category isomorphic to the category $\mathcal{C}$ and the regular poset $\widehat{\mathcal{C}_{\mathfrak{D}}^{\text{op}}}/\mathrel{\mathscr{L}}$ is isomorphic to $\mathfrak{D}$ .

At this point recall that by Theorem 6.1, given a right reductive semigroup $S$ , we have $S\xhookrightarrow{}\widehat{\mathbb{R}(S)}^{\text{op}}$ and so, as in Section 3.3, we may isolate the $\mathrel{\mathscr{L}}$ -classes in $\widehat{\mathbb{R}(S)}^{\text{op}}$ of the form $\mathfrak{l}_{a}$ . Observe that $\mathfrak{l}_{a}$ is the $\mathrel{\mathscr{L}}$ -class in $\widehat{\mathbb{R}(S)}^{\text{op}}$ containing the principal cone $l^{a}$ , for $a\in S$ . now, define

\mathfrak{L}:=\{L_{l^{e}}:e\in E(S)\}=\{\mathfrak{l}_{e}:e\in E(S)\}.

Since $\widehat{\mathbb{R}(S)}^{\text{op}}/\mathrel{\mathscr{L}}=\widehat{\mathbb{R}(S)}/\mathrel{\mathscr{R}}$ , we have $\mathfrak{L}\subseteq\widehat{\mathbb{R}(S)}/\mathrel{\mathscr{R}}$ and can verify the result below.

Proposition 6.3.

Let $S$ be a right reductive semigroup. The normal category $\mathbb{R}(S)$ is connected by the regular poset $\mathfrak{L}$ , and so $\mathbb{R}(S)_{\mathfrak{L}}$ is a connected category. Moreover, the dual connection semigroup $\widehat{\mathbb{R}(S)_{\mathfrak{L}}^{\text{op}}}$ is isomorphic to the semigroup $S$ .

The next step is to extend this to a category equivalence. We denote the category of right reductive semigroups by $\mathbf{RRS}$ . Given a semigroup homomorphism $\phi\colon S\to S^{\prime}$ in $\mathbf{RRB}$ , as in Lemma 3.19, we obtain a CC-morphism $m_{\phi}:=(F_{\phi},G_{\phi})$ from $\mathbb{R}(S)_{\mathfrak{L}}$ to $\mathbb{R}(S^{\prime})_{\mathfrak{L}}$ . Notice that $F_{\phi}$ remains a covariant functor. Hence, we obtain a functor $\mathtt{C}\colon\mathbf{RRS}\to\mathbf{CC}$ as follows:

S\mapsto\mathbb{R}(S)_{\mathfrak{L}}\text{ and }\phi\mapsto m_{\phi}.

Conversely, given a connected category $\mathcal{C}_{\mathfrak{D}}\in v\mathbf{CC}$ , the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}^{\text{op}}}$ is a right reductive, and as in Proposition 3.23, we find a functor $\mathtt{S}\colon\mathbf{CC}\to\mathbf{RRS}$ . Imitating the proofs of Lemmas 3.24 and 3.25, we conclude the required equivalence.

Theorem 6.4.

The category $\mathbf{RRS}$ of right reductive semigroups is equivalent to the category $\mathbf{CC}$ of connected categories.

Naturally, specialising our discussion on right reductive semigroups to $\mathrel{\mathscr{R}}$ -unipotent semigroups and left regular bands, we may emulating Sections 4 and 5.

Theorem 6.5.

The category $\mathbf{RUS}$ of $\mathrel{\mathscr{R}}$ -unipotent semigroups is equivalent to the category $\mathbf{SC}$ of supported categories. The category $\mathbf{LRB}$ of left regular bands is coreflective in the category $\mathbf{SC}$ of supported categories.

7. Inverse semigroups

In this section, we look at a class of regular semigroups, which are both left and right reductive, namely inverse semigroups. Inverse semigroups arguably form the most important class of regular semigroups, mainly due to their ability to capture partial symmetry [32]. Inverse semigroups are in fact $\mathrel{\mathscr{L}}$ -unipotent semigroups, which are left-right symmetrical.

In the joint work [8], the first author described a category equivalence between inverse semigroups and inversive categories. That construction used Nambooripad’s normal categories and admittedly, the description did not reflect the symmetrical nature of inverse semigroups. In contrast, when we employ supported categories, the symmetry of the semigroups gets manifested by the categories ‘supporting’ themselves; so we dub these self-supported categories.

Before continuing we recall some characterisations of inverse semigroups:

Proposition 7.1 ([24, Theorem II.2.6]).

Let $S$ be a regular semigroup. The following are equivalent:

(1)

$S$ is an inverse semigroup;
(2)

every element in $S$ has a unique inverse element;
(3)

$(E(S),\leqslant)$ is a semilattice;
(4)

there is a unique idempotent in each $\mathscr{L}$ -class and each $\mathscr{R}$ -class of $S$ .

As mentioned inverse semigroups form one of the most ‘symmetrical’ classes of semigroups. This symmetry is a reflection of the uniqueness of the inverse, which in turn, defines a natural involution on the semigroup given by $a\mapsto a^{-1}$ . This leads to the proposition below.

Proposition 7.2.

Let $S$ be an inverse semigroup. Then the left category $\mathbb{L}(S)$ is normal category isomorphic to the right category $\mathbb{R}(S)$ . In particular, the semilattice $(v\mathbb{L}(S),\preceq)$ is order isomorphic to $(v\mathbb{R}(S),\sqsubseteq)$ .

Proof.

Define a functor $F\colon\mathbb{L}(S)\to\mathbb{R}(S)$ by, for any inverse semigroups $S$ ,

(15)

vF(Se):=eS\text{ and }F(r(e,u,f)):=l(e,u^{-1},f)

Since inverse semigroups have a unique idempotent in every $\mathrel{\mathscr{L}}$ -class and in every $\mathrel{\mathscr{R}}$ - class, it is easy to see that the map $vF$ is a well-defined bijection. Now, by Proposition 7.1(3), the quasi-orders $\mathrel{\leqslant_{\ell}}$ and $\mathrel{\leqslant_{r}}$ on the idempotents of an inverse semigroups $S$ coincide with the natural partial order $\leqslant$ , and so

Se\preceq Sf\iff e\mathrel{\leqslant_{\ell}}f\iff e\leqslant f\iff e\mathrel{\leqslant_{r}}f\iff eS\sqsubseteq fS.

Hence $vF$ is an order isomorphism between the semilattices $v\mathbb{L}(S)$ and $v\mathbb{R}(S)$ . Also, observe that $vF$ is order isomorphic to the set $E(S)$ of idempotents of $S$ .

Given $u\in eSf$ such that $r(e,u,f)$ is a morphism in $\mathbb{L}(S)$ from $Se$ to $Sf$ , we can see that $u^{-1}\in f^{-1}Se^{-1}=fSe$ so that $l(e,u^{-1},f)$ is a morphism in $\mathbb{R}(S)$ from $eS$ to $fS$ . Then using Lemma 4.3 and Proposition 7.1(2), we verify that the map $F$ is well defined. Now given two composable morphism $r(e,u,f)$ and $r(g,v,h)$ in the category $\mathbb{L}(S)$ such that $f\mathrel{\mathscr{L}}g$ , we know that $f=g$ and

F(r(e,u,f)r(g,v,h))=F(r(e,uv,h))=l(e,{(uv)}^{-1},h)=l(e,v^{-1}u^{-1},h).

On the other hand,

F(r(e,u,f))F(r(g,v,h))=l(e,u^{-1},f)l(g,v^{-1},h)=l(e,v^{-1}u^{-1},h).

Hence $F$ is a covariant functor from $\mathbb{L}(S)$ to $\mathbb{R}(S)$ . Applying Lemma 4.3, one sees that $F$ is inclusion preserving and fully-faithful. Therefore, $F$ is a normal category isomorphism. ∎

From Proposition 4.6, the support map $\Gamma\colon v\mathbb{L}(S)\to\mathfrak{R}$ of an $\mathrel{\mathscr{L}}$ -unipotent semigroup $S$ is given by $Se\mapsto\mathfrak{r}_{e}$ . In the case of inverse semigroups, the next corollary reflects the left-right symmetry of these semigroups.

Corollary 7.3.

Let $S$ be an inverse semigroup. The support map $\Gamma\colon v\mathbb{L}(S)\to\mathfrak{R}$ is an order isomorphism.

Proof.

The map $vF$ in the previous proposition may be interpreted as an order isomorphism from $v\mathbb{L}(S)$ to $S/\mathrel{\mathscr{R}}$ given by $Se\mapsto R_{e}$ . As discussed in Section 3.3, we know that the map $R_{e}\mapsto\mathfrak{r}_{e}$ is an order isomorphism from the poset $S/\mathrel{\mathscr{R}}$ to $\mathfrak{R}$ . Hence $v\mathbb{L}(S)$ is order isomorphic to $\mathfrak{R}$ via the map $Se\mapsto\mathfrak{r}_{e}$ . ∎

Remark 7.1.

When $S$ is an inverse semigroup, we see that $v\mathbb{L}(S)$ is order isomorphic to $\mathfrak{R}$ . In other words, the normal category $\mathbb{L}(S)$ is supported by a partially ordered set which is isomorphic to $v\mathbb{L}(S)$ . Or by abuse of terminology, we can just say that the normal category $\mathbb{L}(S)$ is supported by $v\mathbb{L}(S)$ , i.e., $\mathbb{L}(S)$ is self-supported.

Definition 7.1.

A supported category is said to be self-supported if the support map is an order isomorphism.

Remark 7.2.

In the above definition, we do not need to explicitly specify the supporting semilattice $\mathfrak{D}$ as we know that $\mathfrak{D}$ is the semilattice $\Gamma(v\mathcal{C})$ .

Proposition 7.4.

Given a self-supported category $\mathcal{C}$ with $\mathfrak{D}:=\Gamma(v\mathcal{C})$ , the connection semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is inverse.

Proof.

Since a self-supported category is supported, by Proposition 4.4 the connection semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is $\mathrel{\mathscr{L}}$ -unipotent. Now let $\epsilon_{1}=\epsilon(c_{1},\mathfrak{d})$ and $\epsilon_{2}=\epsilon(c_{2},\mathfrak{d})$ be $\mathrel{\mathscr{R}}$ -related idempotents of $\widehat{\mathcal{C}_{\mathfrak{D}}}$ . Then the objects $c_{1}$ and $c_{2}$ are both connected by $\mathfrak{d}$ . As the support map $\Gamma\colon v\mathcal{C}\to\mathfrak{D}$ is injective, we get $c_{1}=c_{2}$ . Hence $\epsilon_{1}=\epsilon_{2}$ and each $\mathrel{\mathscr{R}}$ -class in $\widehat{\mathcal{C}_{\mathfrak{D}}}$ contains a unique idempotent. By Proposition 7.1(2), the semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$ is inverse. ∎

Using Corollary 7.3 and specialising Proposition 4.6, we conclude the next statement.

Proposition 7.5.

Let $S$ be an inverse semigroup. Then the category $\mathbb{L}(S)$ is self-supported and such that $\mathfrak{R}:=\Gamma_{S}(\mathbb{L}(S))$ is order isomorphic to $v\mathbb{L}(S)$ .

Finally, Theorem 4.8 allows us to obtain the following equivalence theorem for inverse semigroups.

Theorem 7.6.

The category of inverse semigroups is equivalent to the category of self-supported categories.

8. Totally left reductive semigroups and regular monoids

In this section we aim to study regular monoids which also happen to be both left and right reductive. To this end, we identify another special class of left reductive semigroups, which we call totally left reductive. In the process, we prove a couple of interesting results regarding the semigroup of all cones from an arbitrary normal category $\mathcal{C}$ (Proposition 8.1 and Corollary 8.3). We introduce morphisms between arbitrary normal categories and thereby define the category of normal categories. The discussion in this section tells us that if a regular semigroup is totally left reductive (in particular, if it is a monoid), the cross-connection analysis is expendable and connected categories suffice to describe such semigroups completely.

8.1. Totally left reductive semigroups

Recall that, given a regular semigroup $S$ , we have the homomorphism $\bar{\rho}\colon S\to\widehat{\mathbb{L}(S)}$ , $a\mapsto r^{a}$ , and by Theorem 2.6, this is injective when $S$ is left reductive. The question of the surjectivity of $\bar{\rho}$ leads to the following definition.

Definition 8.1.

A regular semigroup $S$ is said to be totally left reductive if $S$ is isomorphic to the semigroup $\widehat{\mathbb{L}(S)}$ of cones in the category $\mathbb{L}(S)$ .

It is not difficult to see that regular monoids are totally left reductive semigroups (see Proposition 8.5), but this class contains several non-monoids, too. It has been showed in [5, Theorem 3.1] and in [2, Theorem 2], respectively, that singular transformation semigroups and singular linear transformation semigroups are totally left reductive. The semigroups of order preserving mappings on finite chains [45] and the Clifford inverse semigroups [3] also fall into this class. All the above mentioned papers were written within the framework of Nambooripad’s cross-connection theory. Our next aim is to employ connected categories to look at these classes and as the reader may see, our analysis will take us to easier and stronger characterisations of each of these classes.

From Section 3, and by definition, a connected category $\mathcal{C}_{\mathfrak{D}}$ connects a normal category $\mathcal{C}$ with an order ideal $\mathfrak{D}\subseteq\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ . In particular, taking $\mathfrak{D}=\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}$ , we obtain $\widehat{\mathcal{C}_{\mathfrak{D}}}=\widehat{\mathcal{C}}$ . Observe that here we are treating an arbitrary normal category as a connected category. For the remaining part of the paper, we shall treat normal categories in this manner without further comment. By Proposition 3.10, we know that the semigroup $\widehat{\mathcal{C}}$ of all cones in a normal category $\mathcal{C}$ , is indeed left reductive. Then applying Proposition 3.14, we come to the next result, which we believe to be of more general interest.

Proposition 8.1.

The semigroup $\widehat{\mathcal{C}}$ of all cones in a normal category $\mathcal{C}$ is left reductive. Moreover, the left category $\mathbb{L}(\widehat{\mathcal{C}})$ is normal and isomorphic to $\mathcal{C}$ .

The latter half of above proposition was already known and one can find an alternate proof in [41, Section III.3.3] where the functor is defined from $\mathcal{C}$ to $\mathbb{L}(\widehat{\mathcal{C}})$ .

The following lemma is of a routine verification.

Lemma 8.2.

Let $\mathcal{C}$ and $\mathcal{C^{\prime}}$ be isomorphic normal categories. Then the regular semigroup $\widehat{\mathcal{C}}$ is isomorphic to the semigroup $\widehat{\mathcal{C^{\prime}}}$ .

Further, combining Proposition 8.1 and Lemma 8.2, we conclude that the semigroup $\widehat{\mathcal{C}}$ is isomorphic to the semigroup $\widehat{\mathbb{L}(\widehat{\mathcal{C}})}$ of cones in the left category $\mathbb{L}(\widehat{\mathcal{C}})$ . Hence, we get the next statement.

Corollary 8.3.

Let $\mathcal{C}$ be a normal category. The semigroup $\widehat{\mathcal{C}}$ of all cones in $\mathcal{C}$ is a totally left reductive semigroup.

Summarising the above discussion: A normal category $\mathcal{C}$ can be realised as a connected category $\mathcal{C}_{\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}}$ such that the connection semigroup $\widehat{\mathcal{C}}=\widehat{\mathcal{C}_{\widehat{\mathcal{C}}/\mathrel{\mathscr{R}}}}$ is totally left reductive, and the left category $\mathbb{L}(\widehat{\mathcal{C}})$ is isomorphic to $\mathcal{C}$ . Conversely, given a totally left reductive semigroup $S$ , the category $\mathbb{L}(S)$ is normal, and the semigroup $\widehat{\mathbb{L}(S)}$ is isomorphic to $S$ . Thus, the regular poset $\widehat{\mathbb{L}(S)}/\mathrel{\mathscr{R}}$ is isomorphic to $S/\mathrel{\mathscr{R}}$ , whence the normal category $\mathbb{L}(S)$ may be realised as the connected category $\mathbb{L}(S)_{S/\mathrel{\mathscr{R}}}$ and the connection semigroup $\widehat{\mathbb{L}(S)_{S/\mathrel{\mathscr{R}}}}=\widehat{\mathbb{L}(S)}$ is isomorphic to $S$ .

Now, normal categories (with CC-morphisms) form a full subcategory of the category $\mathbf{CC}$ , and totally left reductive semigroups (with regular semigroup homomorphisms) form a full subcategory of the category $\mathbf{LRS}$ . Therefore, specialising Theorem 3.26, we obtain:

Theorem 8.4.

The category of normal categories is equivalent to the category of totally left reductive semigroups.

8.2. Regular monoids

Now we look at the most important class of totally left reductive semigroups, namely that of regular monoids. Although the construction we present in this special case, does not seem very insightful regarding the ideal structure of the monoids (as a monoid itself forms a two sided ideal, and consequently, the left and the right actions on this ideal determines the entire actions), our analysis does provide a category equivalence of regular monoids with small categories which we believe it may prove to be quite useful.

To begin with, let $S$ be a regular monoid with identity $1$ . The left ideal category $\mathbb{L}(S)$ of $S$ has a largest object, namely $S1=S$ . Hence we have the corollary to Theorem 2.6 below.

Proposition 8.5 ([41, Corollary III.17]).

Let $S$ be a regular monoid. The normal category $\mathbb{L}(S)$ has a largest object, and $S$ is isomorphic to the semigroup $\widehat{\mathbb{L}(S)}$ of all cones in $\mathbb{L}(S)$ .

The above proposition leads us to identify certain special normal categories.

Definition 8.2.

A normal category $\mathcal{C}$ is said to be bounded above if there exists an object $\mathfrak{d}\in v\mathcal{C}$ such that $c\preceq d$ , for every $c\in v\mathcal{C}$ .

Lemma 8.6.

Let $\mathcal{C}$ be a normal category which is bounded above. Then the semigroup $\widehat{\mathcal{C}}$ of all cones in $\mathcal{C}$ is a regular monoid.

Proof.

By Lemma 2.2, we know that $\widehat{\mathcal{C}}$ is a regular semigroup. Since $\mathcal{C}$ is bounded above, it has a largest object, say $k$ . Let $\epsilon_{k}$ be an idempotent cone in $\mathcal{C}$ with vertex $k$ . Now, for $c\in v\mathcal{C}$ , we have $c\preceq k$ , and given an arbitrary cone $\gamma\in\widehat{\mathcal{C}}$ , by Definition 2.3(1) we get $\gamma(c)=j(c,k)\gamma(k)$ . In particular, $\epsilon_{k}(c)=j(c,k)\epsilon_{k}(k)=j(c,k)1_{k}=j(c,k)$ . Observe that $\gamma(k)$ will always be an epimorphism. Then using equation (2), we see that

\gamma\>\epsilon_{k}=\gamma\ast(\epsilon_{k}(z_{\gamma}))^{\circ}=\gamma\ast(j(z_{\gamma},k))^{\circ}=\gamma\ast 1_{z_{\gamma}}=\gamma.

Also for an arbitrary $c\in v\mathcal{C}$ ,

\epsilon_{k}\gamma(c)=\epsilon_{k}(c)(\gamma(k))^{\circ}=j(c,k)\gamma(k)=\gamma(c).

Hence $\epsilon_{k}\gamma=\gamma=\gamma\epsilon_{k}$ , for any $\gamma\in\widehat{\mathcal{C}}$ . Thus the semigroup $\widehat{\mathcal{C}}$ is a regular monoid with identity $\epsilon_{k}$ . ∎

Since regular monoids form a full subcategory of the category of totally left reductive semigroups, and bounded above categories form a full subcategory of normal categories, specialising Theorem 8.4, we get:

Theorem 8.7.

The category of regular monoids is equivalent to the category of bounded above normal categories (with CC-morphisms).

8.3. Transformation semigroups

The full transformation monoid $\mathscr{T}_{n}$ is regular and it contains the symmetric group $\mathscr{S}_{n}$ as its subgroup of units. From Example 3.1, the full powerset category ${\mathbb{P}}_{\Pi}$ of all subsets of $\mathbf{n}$ is connected and normal with largest object $\mathbf{n}$ , hence ${\mathbb{P}}_{\Pi}$ is bounded above. We denote ${\mathbb{P}}_{\Pi}$ simply by ${\mathbb{P}}$ .

In the light of Theorem 8.7, the next proposition leads to a full description of the regular monoids $\mathscr{T}_{n}$ , in terms of categories.

Proposition 8.8.

The normal category $\mathbb{L}(\mathscr{T}_{n})$ is isomorphic to the full powerset category ${\mathbb{P}}$ , as bounded above categories.

Proof.

From Lemma 3.4(1) the left ideals in the monoid $\mathscr{T}_{n}$ are determined by images. Hence, given $\alpha,\beta\in E(\mathscr{T}_{n})$ and $\theta\in\alpha\mathscr{T}_{n}\beta$ , define a functor $F\colon\mathbb{L}(\mathscr{T}_{n})\to{\mathbb{P}}$ as:

vF(\mathscr{T}_{n}\alpha):=\mathbf{n}\alpha\text{ and }F(r(\alpha,\theta,\beta)):=\theta_{|{\mathbf{n}\alpha}}.

It may be routinely verified that $F$ is a normal category isomorphism from $\mathbb{L}(\mathscr{T}_{n})$ to ${\mathbb{P}}$ .

Notice that the largest object in $\mathbb{L}(\mathscr{T}_{n})$ is $\mathscr{T}_{n}$ and so, we can define a map $G\colon\widehat{\mathbb{L}(\mathscr{T}_{n})}/\mathrel{\mathscr{R}}\to\Pi$ as $R_{\gamma}\mapsto\pi_{\gamma(\mathscr{T}_{n})}$ . Since $\widehat{\mathbb{L}(\mathscr{T}_{n})}$ is isomorphic to $\mathscr{T}_{n}$ and using Lemma 3.4(2), we may verify that $G$ is an order isomorphism. Further, we may prove that the pair $(F,G)$ also satisfies the condition (10), and hence the category $\mathbb{L}(\mathscr{T}_{n})$ is isomorphic to ${\mathbb{P}}$ as bounded above categories. ∎

The above proposition tells that the full transformation monoid $\mathscr{T}_{n}$ is equivalent to the bounded above category ${\mathbb{P}}$ . Hence, any question regarding the monoid $\mathscr{T}_{n}$ may be translated to an equivalent question regarding the connected category ${\mathbb{P}}_{\Pi}$ . In particular, we can obtain the exact same descriptions of the biordered set and the sandwich sets of $\mathscr{T}_{n}$ in terms of subsets and partitions (see [1, Section 5.1]) using the connected category description rather than cross-connections. Observe that the cross-connection description of $\mathscr{T}_{n}$ previously known, [5], involved the rather cumbersome category of partitions[43], but our approach bypasses this by just using the poset $\Pi$ .

Having settled the full monoid case $\mathscr{T}_{n}$ , we move onto one of its most important regular subsemigroups, namely that of singular transformations $\mathscr{T}_{n}\backslash\mathscr{S}_{n}$ . See [14] for a good overview regarding this semigroup. The ideal structure of $\mathscr{T}_{n}\backslash\mathscr{S}_{n}$ was studied in detail inside the cross-connection theory in [5, 42, 43]. Naturally, our next objective is to use our theory of connected categories to realise the semigroup $\mathscr{T}_{n}\backslash\mathscr{S}_{n}$ as a normal category.

It is easy to see that the set of proper subsets of the set $\mathbf{n}$ forms a small category $\mathbb{PS}$ with mappings as morphisms. We shall refer to $\mathbb{PS}$ as the powerset category. Observe that $\mathbb{PS}$ is a full subcategory of the full powerset category ${\mathbb{P}}$ (see Example 3.1).

At this point, we call upon some known results.

Lemma 8.9 ([5, Theorem 3.1][42, Theorem 3]).

Let $T:=\mathscr{T}_{n}\backslash\mathscr{S}_{n}$ be the singular transformation semigroup. The category $\mathbb{PS}$ is normal and isomorphic to the left category $\mathbb{L}(T)$ . The semigroup $\widehat{\mathbb{PS}}$ of all cones in the category $\mathbb{PS}$ is isomorphic to $T$ , and so $T$ is a totally left reductive.

Recall also that, since $\mathscr{T}_{n}\backslash\mathscr{S}_{n}$ is a regular subsemigroup of $\mathscr{T}_{n}$ , the Green relations in $\mathscr{T}_{n}\backslash\mathscr{S}_{n}$ get inherited from $\mathscr{T}_{n}$ . Using Lemma 3.4 and the fact that $\mathscr{T}_{n}\backslash\mathscr{S}_{n}$ is totally left reductive, we see that the poset of right ideals of the semigroup $\widehat{\mathbb{L}(\mathscr{T}_{n}\backslash\mathscr{S}_{n})}$ may be characterised using non-identity partitions of $\mathbf{n}$ .

Next, observe that the non-identity partitions of the set $\mathbf{n}$ form an order ideal, say $\text{N}\Pi$ , of the poset $(\Pi,\supseteq)$ . Also, given an element $\alpha\in\mathscr{T}_{n}$ , we have a principal cone $r^{\alpha}$ in the normal category $\mathbb{L}(\mathscr{T}_{n}\backslash\mathscr{S}_{n})$ . This discussion allows us to verify that the map $R_{r^{\alpha}}\mapsto\pi_{\alpha}$ gives an order isomorphism from the poset of $\mathrel{\mathscr{R}}$ -classes in the semigroup $\widehat{\mathbb{L}(\mathscr{T}_{n}\backslash\mathscr{S}_{n})}$ to the poset $\text{N}\Pi$ . In fact, emulating the proof of Proposition 8.8, we get a description of the semigroup of singular transformations as a normal category.

Proposition 8.10.

The normal category $\mathbb{L}(\mathscr{T}_{n}\backslash\mathscr{S}_{n})$ of the singular transformation semigroup is isomorphic to the powerset category $\mathbb{PS}$ .

8.4. Linear transformation semigroups

Continuing our list of applications, we move onto a brief discussion on the linear transformation semigroups, which are quite analogous to the transformation semigroups. Given a finite dimensional vector space $V$ over a field $K$ , the linear transformations on $V$ form a regular monoid $\mathscr{L\mspace{-5.0mu}T}_{V}$ . The group $\mathscr{G\mspace{-5.0mu}L}_{V}$ of invertible linear transformations on $V$ forms the subgroup of units of $\mathscr{L\mspace{-5.0mu}T}_{V}$ . In [2], the cross-connections of $\mathscr{L\mspace{-5.0mu}T}_{V}$ , its singular part $\mathscr{L\mspace{-5.0mu}T}_{V}\backslash\mathscr{G\mspace{-5.0mu}L}_{V}$ and the variants of $\mathscr{L\mspace{-5.0mu}T}_{V}$ , were discussed. We refer the readers to [2] for a detailed discussion on the normal categories involved. It may be worth mentioning that the right ideal structure of $\mathscr{L\mspace{-5.0mu}T}_{V}$ was described using the annihilator category in [2]. However, employing our approach of connected categories, we can describe the right structure of $\mathscr{L\mspace{-5.0mu}T}_{V}$ with just the null spaces of $V$ .

Given a finite dimensional vector space $V$ , the subspaces of $V$ form a small category $\mathbb{SV}$ with linear transformations as morphisms. This category $\mathbb{SV}$ has a largest object and the Green relations $\mathrel{\mathscr{L}}$ and $\mathrel{\mathscr{R}}$ in $\mathscr{L\mspace{-5.0mu}T}_{V}$ are determined by subspaces and null spaces, respectively [13, Section 2.2]. Let $\mathfrak{N}$ denote the poset of null spaces of $V$ under reverse inclusion. Imitating the proofs in the case of transformation semigroups, we obtain the next results.

Proposition 8.11.

Let $V$ be a finite dimensional vector space over a field $K$ . The category $\mathbb{SV}$ is normal and bounded above. In particular, the poset of $\mathrel{\mathscr{R}}$ -classes in the semigroup $\widehat{\mathbb{SV}}$ is isomophic to the poset $\mathfrak{N}$ , and so $\mathbb{SV}_{\mathfrak{N}}$ is a connected category. The normal category $\mathbb{L}(\mathscr{L\mspace{-5.0mu}T}_{V})$ is isomorphic to $\mathbb{SV}$ , as bounded above categories.

Next, let $\mathscr{L\mspace{-5.0mu}T}_{V}\backslash\mathscr{G\mspace{-5.0mu}L}_{V}$ be the singular linear transformation semigroup and let $\mathbb{PV}$ be the category of proper subspaces of $V$ . Clearly, $\mathbb{PV}$ is a full subcategory of $\mathbb{SV}$ . Using [2, Theorem 2] and the discussion on the semigroup of singular transformations, we can also deal with the singular linear case.

Proposition 8.12.

Let $T:=\mathscr{L\mspace{-5.0mu}T}_{V}\backslash\mathscr{G\mspace{-5.0mu}L}_{V}$ be the singular linear transformation semigroup. The category $\mathbb{PV}$ is normal and isomorphic to $\mathbb{L}(T)$ , as normal categories. Moreover, the semigroup $\widehat{\mathbb{PV}}$ is isomorphic to the semigroup $T$ .

8.5. Symmetric inverse monoids

We conclude the paper with a quick discussion on arguably the most important inverse semigroup: the monoid of all the partial bijections on an arbitrary set $X$ , denoted $\mathscr{I}_{X}$ . We shall realise $\mathscr{I}_{X}$ as a self-supported category $\mathbb{X}$ which is also bounded above.

To begin with, recall that the Green relations $\mathrel{\mathscr{L}}$ and $\mathrel{\mathscr{R}}$ in the semigroup $\mathscr{I}_{X}$ are determined by the images and domains of the partial mappings, respectively [29, Exercise 5.11.2]. Let $\mathbb{X}$ be the category of all subsets of $X$ with partial bijections as morphisms. Clearly, $\mathbb{X}$ is normal and the set $X$ is the largest object in $\mathbb{X}$ , wence $\mathbb{X}$ is bounded above. Now, define a functor $F\colon\mathbb{L}(\mathscr{I}_{X})\to\mathbb{X}$ as follows: for idempotents $\alpha,\beta\in\mathscr{I}_{X}$ and $\theta\in\alpha\mathscr{I}_{X}\beta$ ,

vF(\mathscr{I}_{X}\alpha):=X\alpha\text{ and }F(r(\alpha,\theta,\beta)):=\theta.

We may check that $F$ is a normal category isomorphism. Hence the semigroup $\widehat{\mathbb{X}}$ of cones in $\mathbb{X}$ is isomorphic to the semigroup $\mathscr{I}_{X}$ . Moreover, since the $\mathrel{\mathscr{R}}$ -classes of the semigroup $\mathscr{I}_{X}$ are determined by the domains, the poset of $\mathrel{\mathscr{R}}$ -classes of the semigroup $\widehat{\mathbb{X}}$ is order isomorphic to $v\mathbb{X}$ . Hence the category $\mathbb{X}$ is self-supported too. At last, we can verify that the isomorphism $F$ may be extended to an isomorphism of self-supported categories. We collect the above discussion in the final proposition, which describes how the symmetric inverse monoid may be realised as a category, just as in the ESN Theorem.

Proposition 8.13.

Let $X$ be a set. The category $\mathbb{X}$ is normal and bounded above. The semigroup $\widehat{\mathbb{X}}$ of cones in $\mathbb{X}$ is isomorphic to the semigroup $\mathscr{I}_{X}$ . The normal category $\mathbb{L}(\mathscr{I}_{X})$ and the category $\mathbb{X}$ are isomorphic as self-supported categories.

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Left reductive regular semigroups

Abstract.

Key words and phrases:

1. Introduction

Definition 1.1.

Definition 1.2.

2. Preliminaries

2.1. Semigroups and categories

2.2. Regular semigroups and normal categories

Definition 2.1.

Definition 2.2.

Lemma 2.1 ([41, Corollary II.4, Proposition II.5 and II.7]).

Remark 2.1.

Definition 2.3.

Definition 2.4.

Lemma 2.2 ([41, Theorem I.2]).

Lemma 2.3.

Lemma 2.4.

Theorem 2.5 ([41, Corollary III.20]).

Theorem 2.6 ([41, Theorem III.16]).

3. Left reductive regular semigroups

3.1. Connected categories

Definition 3.1.

Remark 3.1.

Remark 3.2.

Example 3.1.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4 ([13, Section 2.2]).

Lemma 3.5.

Proposition 3.6.

Proof.

Remark 3.3.

3.2. The connection semigroup 𝒞𝔇^\widehat{\mathcal{C}_{\mathfrak{D}}}

Lemma 3.7.

Proof.

Remark 3.4.

Proposition 3.8.

Proof.

Lemma 3.9.

Proof.

Proposition 3.10.

Proof.

Lemma 3.11.

Lemma 3.12.

Proof.

Lemma 3.13.

Proof.

Proposition 3.14.

Remark 3.5.

Lemma 3.15.

3.3. Structure of left reductive regular semigroups

Lemma 3.16.

Proof.

Proposition 3.17.

Proposition 3.18.

Proof.

3.4. Category equivalence

Definition 3.2.

Remark 3.6.

Remark 3.7.

Lemma 3.19.

Proof.

Proposition 3.20.

Lemma 3.21.

Proof.

Lemma 3.22.

Proof.

Proposition 3.23.

Lemma 3.24.

Proof.

Lemma 3.25.

Proof.

Theorem 3.26.

4. ℒ\mathrel{\mathscr{L}}-unipotent semigroups

Proposition 4.1 ([48, Theorem 1][16, Corollary 1.2, 1.3]).

Proposition 4.2.

Proof.

3.2. The connection semigroup $\widehat{\mathcal{C}_{\mathfrak{D}}}$

4. $\mathrel{\mathscr{L}}$ -unipotent semigroups