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The term flow monoid first arose in unpublished typescript of Chase; in this manuscript, Chase determines the structure of the flow monoid (named it in his paper incidence monoid) and its group of units in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 .
In [1, Proposition 3.3], Gilbert characterized that a simplicity groupoid arises from the Nambooripad construction. In the following example, we will investigate the flow monoid on a rectangular band.
Example 2. Recall that a semigroup is a band if every element of is an idempotent and a rectangular band if for all in . Define . Then is regular and for all , , and . For flow on , we have which is trivially true. So, for a flow is now a pair of maps such that , let us suppose So which gives . Therefore where . So we get a flow whenever we choose any pair of mappings and the flow is given by and , or could combine into single mappings and so a mapping . Now suppose pairs of functions determine the flow a pair of functions , for , such that and determines the flow a pair of functions , for , such that Then the corresponding functions are defined by Similarly, we obtain . Hence is determined by a single function: with the rule
Let be a semigroup, and nonempty sets, and a matrix indexed by and with entries taken from . Then the Rees matrix semigroup ([7]) is the set together with the multiplication In the next example we describe flows on Rees matrix semigroups.
Example 3. The Rees matrix semigroup is regular since, for all elements in this semigroup, there exist such thatandFrom (34), we get and from (35), we have So . Hence once we have selected and , is uniquely determined.
For any element is an idempotent if So . Therefore we can think of as being identified (once we are given with .
For each , a flow determines a pair of inverse elements (in which ) such that Once is determined, we have free choice for and is determined, and so is determined. Hence flow is a function . We can now think of a flow as two functions: We determine the composition of functions that corresponds to the composition of flows. Take two flows and with corresponding functions We have Now , where and where . So The functions and corresponding to are Thus , and so, for any flow monoid on any category , we get a monoid homomorphism such that elements occur to reflect rule for multiplication in an -class of .
Let be a Rees matrix group. Consider the subset of . In fact is a subgroup of which is isomorphic to . Consider
Thus we have the following lemma and result.
Lemma 4. Let be a group, and fix an element . Define a new binary operation on as follows: for all , . Then is a group isomorphic to .
Corollary 5. contains many copies of the group , all isomorphic to one copy for each .
4. Flows on Cauchy Categories
In this section we will investigate another category that can be built from a semigroup. No special conditions are required: any semigroup gives us its Cauchy category .
The Cauchy category is defined as follows: the set of vertices (or objects) is . Given the set of arrows in from to is as in Figure 1(a).
(a)
(b)
(c)
(d)
The identity arrow at is and the composition of arrows is shown in Figure 1(b).
Lemma 6. In the Cauchy category , if and only if . Hence is always nonempty (so that is a connected category).
Proof. Suppose that ; then . Hence , where .
Conversely, if , then where . We want to show that . Since Hence ; therefore .
Since , and if , the identity arrow at is such that . Therefore is always nonempty.
If we thought of as a category with one vertex, then we have the following. If is a group then is equal to . We have one object (vertex ) which is not an element of and one arrow for each element of , with the arrow being the identity arrow at . Hence we have one arrow for each element of ; i.e., . See Figure 1(c).
Lemma 7. Let be a monoid with identity ; then is a submonoid of and isomorphic to . Hence is a submonoid of .
Proof. Let be a submonoid of . It is easy to check that the operation is associative since associativity will hold automatically in a category. Also is the identity arrow. To show that , suppose defined by again it is easy to check that is one-one correspondence and it is a morphism. In order to prove that is a submonoid of , given , define a flow on as follows: Then is isomorphic to ; this gives us a copy of inside . Hence is a submonoid of .
In the following we describe the Cauchy categories of some small semigroups given in [7].
(i) Cauchy category of a rectangular band : we write using for all . If then so for some and . But it follows that
The set of vertices (or objects) of is , and arrow from idempotent to idempotent is defined by element such that where so and so . Hence is the only arrow connecting .
If we have in the Cauchy category of a regular semigroup which has exactly one arrow from any two idempotents vertices, assume that for all in the set of idempotents and . If we put then . Hence is a subsemigroup of .
Now since is regular let and ; then and ; we want to prove that every element of is an idempotent. Since is an arbitrary element of , and
Hence every element of is an idempotent; therefore is a band; our assumption on tells us that for all , , if , then . Therefore is a rectangular band.
We prove the coming result.
Proposition 8. Let be a regular semigroup. Then in the Cauchy category we have exactly one arrow from to (for all if and only if is a rectangular band.
Take the semigroup with (for example, infinite monogenic semigroup). Define , and for all ; then and satisfies our assumption: . Since we must have , is not regular (0 is the only regular element). Can we find a semigroup that does not have a zero, is not regular, and satisfies for all such that ? Consider (which is a monogenic subsemigroup) is not a band since and not regular since is not a regular element, but satisfies for all since Define with multiplication , which is a semigroup such that no element acts as (if behave like a zero then if moreover is not regular for any we compute for all . But satisfies the low since (remember that in ). Hence if satisfies as a low, then satisfies same low. As a semigroup, if is a rectangular band, then so is .(i)We describe Cauchy categories where is completely simple semigroup which is the Rees matrix semigroup where . The vertex set is . The arrows between the vertices and are where such that . See Figure 1(d). Check the composition of arrows: So let and , as drawn in Figure 2(a). If and , then and Identity at is .(ii)We describe Cauchy categories where is a Brandt semigroup . The general construction of Brandt semigroups proceeds as follows. Let be a finite set with elements. The Brandt semigroup is the set with multiplication given by so has elements, and idempotents: Arrows from idempotent to idempotent are a triple such that such that , if and if . So if then , then but if , then , and so we get two arrows from to as shown in Figure 2(b).Moreover, and is a vertex of . Then so and then .
(a)
(b)
Now has idempotents. If there are at least two nonzero idempotents and ; then has two arrows. If , has only two elements which is certainly a band; but since it is not a rectangular band. See Figure 3.
Our task here is to describe the flow monoid on a Cauchy category. We describe it in terms of functions defined on . A flow on assigns to each an arrow where and . So we can first select the function , and then having chosen we have a second function say, with satisfying for all . The pair then determines the flow . If and are two flows determined by the pairs and , then we compute the composition of flows as
For future work from the point of semigroup theory, it is quite natural to consider partial flows that are only defined on a subset of . This is similar idea of considering the partial transformation semigroup of a set instead of the full transformation semigroup . Partial flow then is a function from a subset assigning to each an arrow starting at ; composition of defined on and defined on is defined on the subset ends at an element of and then ; we need to allow the empty flow which acts as a zero; comes equipped with a monoid homomorphism (partial transformation semigroup) such that . We will try to set up the theory of partial flows on category, generalizing [1, Sections 2 and 4]. Another future work for flows monoid is to present a Gröbner-Shirshov basis for each example in this paper. We may refer to [9–13] for more details.
Data Availability
The [flows monoid concept] used to support the findings of this study has been deposited in “Gilbert ND. Flows on Regular Semigroups. Applied Categorical Structures 2003. (11) 147-155. DOI 10.1007/s00233-008-9065-5 https://rdcu.be/bwQNG” which is included within the reference of the article. The [Cauchy Categories concept] used to support the findings of this study has been deposited in “Costa, A. & Steinberg, B. Semigroup Forum (2015) 91: 543. https://doi-org.sdl.idm.oclc.org/10.1007/s00233-014-9657-1” within the reference of this article. The unpublished manuscript “S. U. Chase, On representations of small categories and some constructions in algebra and combinatorics. Preprint (1977)” supports the findings of this study which are available from the corresponding author upon request.
Conflicts of Interest
The author declares that they have no conflicts of interest.