Journal of Mathematics

Volume 2019, Article ID 8027391, 9 pages

https://doi.org/10.1155/2019/8027391

## Flows on Classes of Regular Semigroups and Cauchy Categories

Department of Mathematics, Science Faculty, King Abdulaziz University, Girls Campus, 21589 Jeddah, Saudi Arabia

Correspondence should be addressed to Suha Ahmed Wazzan; as.ude.uak@nazzaws

Received 4 February 2019; Accepted 3 April 2019; Published 15 April 2019

Academic Editor: Radomír Halaš

Copyright © 2019 Suha Ahmed Wazzan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the structure of the flow monoid for some classes of regular semigroups (which are special case of flows on categories) and for Cauchy categories. In detail, we characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups and also describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. In fact, we obtain a general structure for the flow monoids on Cauchy categories.

#### 1. Introduction and Preliminaries

The term flow monoid first arose in unpublished typescript of Chase; in this manuscript, Chase determines the structure of the flow monoid (named it in his paper* incidence* monoid) and its group of units in general category theory which has categories with fixed vertices set . In [1], Gilbert adopted the term* flow* monoid which connects the concepts with vector fields. Gilbert was solely interested in studying flow monoid in the theory of regular semigroup. He depicted the structure of the flow monoid in accordance to the Green relations on the semigroup. The aim of this paper is to describe flow monoids for some classes of regular semigroups. We characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups. We describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. We find the general structure of the flow monoids on Cauchy categories.

A directed graph consists of two sets: a vertex set and an arrow set and two mappings: and the codomain operation . There are many possible formulations of the definition of* categories* (see, for instance, [2–4]). As in [5], we shall regard a* category* as a directed graph with extra structure in the mean of abstract algebra. We start with a directed graph with the term* objects *used for the vertices, and the term* arrows *used for directed edges, and which has supplementary operations:(C1)Identity, which appoints to each object an arrow ;(C2)Composition, which designates each two arrows and satisfying with an arrow . Composition satisfies two conditions:(i)Associativity: for objects and arrows such that , we have the equality (ii)Identity: for each arrow , we have the composition with identity arrows and and .

In general, a category contains objects set and arrows set between objects; arrows can sometimes be composed, whenever they fit together with* source* and* target* maps , and an identity map such that . Whenever , every object has an identity arrow and so the composition of arrows is associative.

According to [1], a flow is a set-theoratic section to the source map. A flow on a category with vertex set is a function that is a section to the source map: that is, for all , let denote the set of all flows on ; then is a monoid, with composition defined by and identity .

In Section 3, we characterize the structure of the flow monoids for some regular semigroups such as full transformation semigroups, Rees matrix semigroups, and rectangular band.

Cauchy category of a semigroup is a small category whose vertices set is the set of idempotents of . An arrow from to is a triple where . Composition of arrows is given by

The identity at is by [6]. Moreover, again by the same reference, the Cauchy category known as* Karoubi envlope * (or Cauchy completion or idempotent splitting) is a full subcategory of which is named as the* Schützenberger category* of whose vertices are elements of the semigroup . The* Schützenberger category * of a semigroup is defined as follows. The vertices set of is . Also an arrow from to is a triple such that . The composition of arrows is given by where and the identity at is .

In Section 4, we describe Cauchy categories for some small regular semigroups such as rectangular bands, Brandt semigroups, and completely simple semigroups and then find the general structure for the flow monoid on Cauchy categories. We actually prove that, in the Cauchy category , we have exactly one arrow from to (for all if and only if is a rectangular band.

#### 2. Flows on Regular Semigroups

Let be a* regular* semigroup, so for all there exists such that and . We call an* inverse* for . An element may have many inverses, and the set of inverses of is denoted by . At the same time, an* idempotent* in is an element such that . If , then and are idempotents. Let be the set of idempotents. In regular semigroup, need not be a subsemigroup since the product of two idempotents need not be an idempotent element. If is regular and is a subsemigroup of , then is called* orthodox* (cf. [7]).

A* flow * on is a pair of functions with the propertiesand

Due to [1], by letting as the set of all flows on , one can compose flows in the following way: if and are flows on , then a pair of mappings is defined as follows:

Thus we have the following first lemma of this section.

Lemma 1. *The pair of mappings is a flow on .*

*Proof. *For all and , we have to prove that Now Similarly, we obtain . Additionally, we also have Hence the pair of mappings is a flow on .

Define the flow to be the product of the flows and in , and write . Thus we get the following proposition.

Proposition 2. * is a monoid which is called the flow monoid of .*

*Proof. *To show that is a semigroup, we first need to prove the operation is associative. Let be flows on , and so each of those is a pair of mapping from to . To prove , let and . We then have pairs of maps and again from to . So we have to show and . Now Since is a flow, then , and so in we have which implies thatwhereas . ThusTherefore (13) and (14) together imply that . Similarly, we get , and hence is an associative operation. Now the trivial flow on is the pair of mappings , where is the identity map on . So, for all , we have , where . Then is the identity element in since As the next step, for all , we have to show , where is a flow on . In other words, and . Now, from the equality , we get Hence we get which implies that . With a similar calculation, from the equation , we obtain which gives .

Therefore is the identity element in , and so it becomes a monoid.

#### 3. The Structure of Flow Monoids on Classes of Regular Semigroups

In this section, by describing the construction of the flow monoid on a regular semigroup (which is a special case of a flow on a category), we will investigate the special classes of regular semigroups and flow monoids more precisely.

In the following example, by taking into account a regular semigroup where (in fact, by [8], is the full transformation semigroup on , we will compute the number of elements in the flow monoid .

*Example 1. *Recall that the full transformation semigroup on a set consists of all mappings from to itself under the operation composition of mappings. It is known that when .

By taking any element , we will first show that is actually a regular semigroup. To do that we need to find a such that and . Since is a map from to itself, we can define it as where and . Also define by Then which implies . Similarly we can get . Hence is a regular semigroup.

Now we compute the number of elements in the flow monoid . In fact, is defined by Recall that . In Table 1, we find all inverses.

It is clear from the table that , , and . Now, let We can get a (bad) estimate of the size of here: number of maps without any restriction is much too big. Let us look in term at the possibilities for for which element of .

For , we have two possibilities or .

For , we have six possibilities for choosing two elements in whose product is : For , as for , we can only choose Therefore, we get eight flows on as shown in Table 2.

A sample of the composition can be considered as , , and . Then satisfies For the cases , we have and (same as ) , we have and , we have andTo understand the full structure, we would need to complete the table for the composition in . Note that we have the trivial flow , , and for all .