# On Classification of SemigroupN by Green’s Theorem

Authors: Suha Wazzan; Nurten Urlu Ozalan
Journal: Journal of Mathematics
 (2022)

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## Abstract

In previous papers, it has been recently defined a new class of semigroups based on both Rees matrix and completely 0-simple semigroups. For this new structure, it has been introduced with certain basic properties and finiteness conditions. The main goal of the paper is to prove Green’s Theorem forN by indicating the existence of Green’s lemma. We also study generalized Green’s relation over N and present some new findings. In particular, we classify our semigroup N associated with generalized Green’s relation by constructing good homomorphism.


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## Body

## 1. Introduction and Preliminaries

It is good to keep in mind the definition of the semigroupN given in [1], in order to grasp the importance of this paper. Since this paper is continuation of our work in [1], let us recall the basic facts regarding the semigroup N.The notationMR represents the Rees matrix semigroup M0S0;I,J;P, and the notation MC represents completely 0-simple semigroup M0G0;I,J;P′.Let us consider the mappingγ:MR×MC∗MR×MC⟶MR×MC(1)x,y,z,a,b,c∗k,l,m,r,t,q=x,ypzkl,m,0,ifpzk≠0andpcr′=0,0,a,bpcr′t,q,ifpzk=0andpcr′≠0,x,ypzkl,m,a,bpcr′t,q,ifpzk≠0andpcr′≠0,0R,0C,ifpzk=0andpcr′=0,for the elements x,y,z,k,l,m∈MR, and a,b,c,r,t,q∈MC. By the operation given in (14), the set MR×MC defines a semigroup M0S0,G0;MR,MC;P,P′. We denote this new semigroup shortly by N. The details of the element of N are as follows.(2)N=Xi,Xk=Xi=0,Xk≠01≤k≤morXi≠0,Xk=01≤i≤norXi≠0,Xk≠01≤k≤mand1≤j≤morXi=0,Xk=0.Indeed, everyXi is an element of MR, and Xk is an element of MC. Moreover, the semigroup N is composed of the semigroup S and the group G. Furthermore, we can see the diagram in Figure 1 for N.Figure 1
The diagram for the semigroupN.Then, we obtainedN which satisfies important homological properties and proved that the spined product of the (C)-inversive semigroup and the idempotent semigroup of N is isomorphic to the strictly inverse semigroup N in [1]. Furthermore, we gave some consequences of the results to make a detailed classification over N in the same reference. Despite the progress, several important issues remain to be addressed, which will be focused on them.We now discuss a very helpful tool for the study of monoids/semigroups called Green’s relations. Fundamental equivalence relations,ℛ,ℒ,ℋ,D-relations, were first introduced and studied by Green in 1951. Then, using these equivalences, Green’s lemma and theorem were proved. These relations have played a main role in the development of the semigroup theory. Green’s relation is a subject in its own right, with a substantial body of literature, see for instance [2]. Especially, Green’s relations may be used to depict the structure of regular semigroups (for example, see Theorem 2.1 in [3]). On the other hand, in [4, 5], the authors studied some Green’s relations for semigroup and monoid structures. The Green’s lemma and Green’s theorem are the natural next steps in these relations. In [6], the author studied an application for Γ-semigroups techniques via the Green’s theorem. In the present paper, our aim is to prove Green’s lemma and Green’s theorem for N.On the other hand, in [7], the authors introduced generalized Green’s relation over a semigroup, namely, the generalized Green’s relation. Because of this reason, replacing Green’s relations upon a given semigroup is the most efficient method to study the generalized regular semigroups.Generalized Green’s relation, which is another equivalence class, is crucial for the classification of the semigroup class such as the abundant and adequate semigroup. One can regard the abundant semigroup as another kind of generalized regular semigroups. Obviously, an abundant semigroup is not necessarily a regular semigroup. In abundant semigroups, a homomorphic image of an abundant semigroup must not be abundant, and the notions of good homomorphism for abundant semigroups are introduced in [8, 9]. Another of our aim is to focus on the generalizations of Green’s relations on a semigroup N and think of their applications such as classification of semigroups and good homomorphism in this paper.The paper is structured as follows. We reveal important results related to Green’s relations which are Green’s lemma and Green’s theorem forN (Proposition 1 and Theorem 1) in Section 2. In Section 3, by considering generalizations of Green’s relations on a semigroup N as mentioned by Green’s ∗relations (Lemma 5 and Theorem 2). Furthermore, we make new classifications for N associated with the abundant semigroup (Corollaries 1 and 2). Finally, we present other results which build a structure of abundant and adequate semigroups in terms of good homomorphism (Corollary 3) in the same section.


## 2. Green’s Lemma and Green’s Theorem forN

A natural question in the theory of semigroups can be stated as follows: what kind of new material can be obtained for a semigroupS by considering the results on Green’s relation of the same semigroup S? It is well known that certain equivalence relations ℒ and ℛ for an arbitrary semigroup S were first introduced by Green [10], in which those are defined in a very influential method to make a classification for a new semigroup. As a result of this fact, in [1], we studied ℒ and ℛ-Green’s relations for N. On the other hand, we will prove Green’s lemma and Green’s theorem by developing the results which are obtained in [1].Some basic facts about the behavior of these relations are summarized in the following lemma.Lemma 1 (see [1]).
Forα,β,γ,x,y,z,α′,β′,γ′,x′,y′,z′∈N(i)
γ=γ′,z=z′,βℒβ′ and yℒy′⇔α,β,γ,x,y,zℒα′,β′,γ′,x′,y′,z′.(ii)
α=α′,x=x′,βℛβ′ and yℛy′⇔α,β,γ,x,y,zℛα′,β′,γ′,x′,y′,z′.ForN1,N2∈N, we denote by N1ℒ and N2ℒ the ℒ-classes of N containing the elements N1 and N2, respectively. Furthermore, for s1,s2∈N, let us assume that N1∗s1=N2 and N2∗s2=N1.Lemma 2.
Letx1∈N1ℒ. Then, x1∗s1,N1∗s1∈ℒ and x1∗s1∈N2ℒ.Proof.
Let us choose an elementN1=a,b,c,d,e,f such that x1∈N1ℒ. This clearly implies that x1,N1∈ℒ⇒x1∗s1,N1∗s1∈ℒ⇒x1∗s1,N2∈ℒ⇒x1∗s1∈N2ℒ.
Nevertheless, we can present this short proof in the following alternative way:
Forx1=a,b∗,c,d,e∗,f, we have b∗ℒb and e∗ℒe by Lemma 1 via by the definition of Green’s relation [2], and we certainly have αb∗=b,βb=b∗ and α′e∗=b,β′e=e∗ such that α,α′,β,β′∈N. Thus,(3)x1∗s1=a,b∗,c,d,e∗,f∗a′,b′,c,d′,e′,f=a,b∗pca′b′,c,d,e∗pfd′′e′,f.
Sinceb∗pca′b′=βbpca′b′=βyℒy and e∗pfd′′e′=β′epfd′′e′=βlℒl, we finally have x1∗s1∈N2ℒ.Similarly as in Lemma2, by choosing an element N2=a,y,c,d,l,f such that x2∈N2ℒ, we also obtain the following preresult.Lemma 3.
Letx2∈N2ℒ. Then, x2∗s2,N2∗s2∈ℒ and x2∗s2∈N1ℒ.Proposition 1.
(Green lemma forN) Let N1=a,b,c,d,e,f and N2=a,y,c,d,l,f be ℛ-equivalent elements of the semigroup N, and let s1,s2∈N such that N1∗s1=N2,N2∗s2=N1. Then, the mappings x1↦x1s1x1∈N1ℒ and x2↦x2s2x2∈N2ℒ are mutually inverse and one-to-one mappings of N1ℒ upon N2ℒ and N2ℒ upon N1ℒ, respectively.Proof.
By Lemmas2 and 3, since x1∗s1∈N2ℒ ve x2∗s2∈N1ℒ, one can obtain the following well-defined mappings:(4)Γ:N1ℒ⟶N2ℒ,Γ′:N2ℒ⟶N1ℒ,x1↦x1∗s1,x2↦x2∗s2.
Now, let us show these two mappings are mutually inverse. To do that let us take the elementsD∈N1ℒ and D′∈N2L. In fact D∈N1ℒ implies D=N1, or there exists α3∈N such that α3∗N1=D, and similarly, D′∈N2L implies D′=N2, or there exists t∗∈N such that t∗∗N2=D′. By considering the compositions(5)Γ′∘Γ:N1ℒ⟶N2ℒ,Γ∘Γ′:N2ℒ⟶N1ℒ,x1↦x1∗s1∗s2,x2↦x2∗s2∗s1,of mappings Γ and Γ′,(i)
for the caseD=N1, we have Γ′∘ΓD=Γ′∘ΓN1=Γ′ΓN1=Γ′N1⋆s1=Γ′N2=N2∗s2=N1=D, and(ii)
for the other case, we obtainΓ′∘ΓD=Γ′∘Γα3∗N1=Γ′Γα3∗N1=Γ′α3∗N1⋆s1=Γ′α3∗N2=α3∗N2∗s2=α3∗N1=D.
ThereforeΓ′∘Γ is the identity mapping on N1L. In fact, a quite similar process can be applied on Γ∘Γ′ to get the identity mapping. Hence, Γ and Γ′ are mutually inverse mappings (in other words, ℛ-class preserving). Hence, Γ is one-to-one mapping of N1L onto N2L, and Γ′ is 1−1 mapping of N2L onto N1L.Theorem 1.
(Green theorem forN) Let N1 and N3 be ℛ∘ℒ equivalent elements of a semigroup N. Then, there exists N2∈N such that N1ℛN2 and N2ℒN3, and hence, N1s1=N2, N2s2=N1, tN2=N3, t′N3=N2 for some s,s′,α1,α2∈N. The mappings(6)δ:N1ℛ∩N1ℒ⟶N3ℛ∩N3ℒδ′:N3ℛ∩N3ℒ⟶N1ℛ∩N1ℒ,x1↦α1∗x1∗s1x1↦α2∗x1∗s2,are mutually inverse, one-to-one mappings of N1ℛ∩N1ℒ and N3ℛ∩N3ℒ upon each other.Proof.
For an elementx1∈N1ℛ∩N1ℒ, clearly, we have x1∈N1ℛ and x1∈N1ℒ. Therefore,(7)x1∈N1ℛ⇒x1,N1∈ℛ⇒N1∗z=x1andx1∗z′=N1,for somez,z′∈N,x1∈N1ℒ⇒x1,N1∈ℒ⇒w∗N1=x1andw′∗x1=N1,for somev,v′∈N.
After that we can indicate the elementN3 as(8)N3=α1∗N2=α1∗N1∗s1=α1∗x1∗z′∗s1=α1∗v∗N1∗z′∗s1=α1∗v∗N2∗s2∗z′∗s1=α1∗v∗N1∗s1∗s2∗z′∗s1=α1∗v∗N1∗s1∗s2∗z′∗s1=α1∗x1∗s1∗s2∗z′∗s1,such that(9)α1∗x1∗s1=α1∗N1∗z∗s1=α1∗N2∗s2∗z∗s1=α1∗N2∗s2∗z∗s1=N3∗s2∗z∗s1.
Sinces2∗z′∗s1 and s2∗z∗s1 in elements N, it is satisfied that N3=α1∗x1∗s1∗s2∗z′∗s1 and α1∗x1∗s1=N3∗s2∗z∗s1, and we have α1∗x1∗s1,N3∈ℛ, so α1∗x1∗s1∈N3ℛ.
We can also indicate the elementN3 as(10)N3=α1∗N2=α1∗N1∗s1=α1∗v′∗x1∗s1=α1∗v′∗N1∗z∗s1=α1∗v′∗N2∗s2∗z∗s1=α1∗v′∗α2∗N3∗s2∗z∗s1=α1∗v′∗α2∗α1∗N2∗s2∗z∗s1=α1∗v′∗α2∗α1∗N2∗s2∗z∗s1=α1∗v′∗α2∗α1∗N1∗z∗s1=α1∗v′∗α2∗α1∗N1∗z∗s1=α1∗v′∗α2∗α1∗x1∗s1,such that(11)α1∗x1∗s1=α1∗v∗N1∗s1=α1∗v∗N1∗s1=α1∗v∗N2=α1∗v∗α2∗N3=α1∗v∗α2∗N3.
Sinceα1∗v′∗α2 and α1∗v∗α2 in elements N, it is satisfied that α1∗v′∗α2∗α1∗x1∗s1=N3 and α1∗v∗α2∗N3=α1∗x1∗s1, and we have α1∗x1∗s1,N3∈ℒ, so α1∗x1∗s1∈N3ℒ. Hence, we obtain α1∗x1∗s1∈N3ℛ∪N3ℒ, and the whole process mentioned above implies that the mapping δ is well defined.
By considering an arbitrary elementx1∈N3ℛ∩N3ℒ and applying a similar approach as above, one can also obtain the mapping δ′ which is well defined.
Now, suppose thatD is an element of N1ℛ∩N1ℒ which gives D∈N1ℛ and D∈N1ℒ.(12)D∈N1ℛ⇒D,N1∈ℛ⇒N1∗u=DandD∗u′=N1,for someu,u′∈N,D∈N1ℒ⇒D,N1∈ℒ⇒v∗N1=Dandv′∗D=N1,for somev,v′∈N.
Then, we have(13)δ′∘δD=δ′δD=δ′α1∗D∗s1=α2∗α1∗D∗s1∗s2=α2∗α1∗v∗N1∗s1∗s2=α2∗α1∗v∗N1∗s1∗s2=α2∗α1∗v∗N2∗s2=α2∗α1∗v∗N1=α2∗α1∗D=α2∗α1∗N1∗u=α2∗α1∗N2∗s2∗u=α2∗α1∗N2∗s2∗u=α2∗N3∗s2∗u=α2∗N3∗s2∗u=N2∗s2∗u=N2∗s2∗u=N1∗u=D.
With a similar approach as above, one can further getδ∘δ′D=D by considering an element D∈N3ℛ∪N3ℒ.
The mappingδ is one-to-one mapping of N1ℛ∩N1ℒ onto N3R∩N3L and the mapping δ′ is a one-to-one mapping of N3R∩N3L onto N1ℛ∩N1ℒ so δ′ and δ are mutually inverse.
The proof is now complete.


## 3. Generalized Green’s Relation forN

Let us consider the semigroupS. ES is the set of all idempotents. The element e∈S is called a regular element if there exists an element x∈S such that exe=e and xex=x. S is said to be the regular semigroup if each element of S is regular. Green’s relation may be used to describe the structure of regular semigroups [3]. In fact, the result related to regularity over N is given in [1]. However, in here, we will focus on the generalizations of Green’s relations on a semigroup N and study their applications such as adequate and abundant semigroups. Obviously, these types of semigroups are natural generalizations of regular semigroups, and so, it would be appropriate to study N being abundant and adequate semigroups. We note that the index sets (used to expose the semigroup N) are composed of one element in this section.In [11, 12], Fountain observed that Green’s ∗ relation (i.e., generalized Green’s relation) may be studied to abundant semigroups, namely, superabundant semigroups. Many papers have been dedicated to generalizations and improvements of Green’s ∗ relation in various contexts (see, for instance, [7, 13]).In the following context, we will remind some fundamentals which are related to the terminology generalized Green’s relations over a semigroup.We consider the semigroupS to be a semigroup, and α,β∈S. By [12], it is known that(14)αℒ∗β⇔u,v∈S1αu=αv⇔βu=βv,(15)αℛ∗β⇔u,v∈S1uα=vα⇔uβ=vβ.S is called the abundant semigroup if each ℒ∗-class and ℛ∗-class include an idempotent element. Moreover, if each ℋ∗-class ℋ∗=ℒ∗∩ℛ∗ of S includes an idempotent element, then S is called superabundant. If S is a superabundant semigroup and ES (idempotent semigroup of S) forms a subsemigroup of S, then S is called a cyber-group.On the other hand, in [14], Fountain introduced that the most important subclasses of abundant semigroups are perhaps the classes of adequate semigroups. An abundant semigroup whose idempotents are commutative is considered as adequate semigroup. Therefore, an adequate semigroup in the class of abundant semigroups can be regarded as a generalization of the inverse semigroup in the class of regular semigroups (cf. [15]).If for anyx,y∈S, xℒ∗y implies θxℒ∗θy and xℛ∗y implies θxℛ∗θy, then a semigroup homomorphism θ:S⟶T is called good [9].Definition 1 (see [2]).
LetS be a semigroup. An element x in S is called left cancellative if xy=xz implies y=z for all y and z in S and right cancellative if yx=zx implies y=z for all y and z in S. If every element in S is left cancellative, then S is called a left cancellative semigroup. Similarly, S is called a right cancellative semigroup if every element in S is right cancellative.
Now, we will considerℒ∗ and ℛ∗ equivalences for the semigroup N and so with the operation ∗ defined in (14). Before, let us recall the following lemma, which was proved in ([1], Lemma 3).
Unless stated otherwise, in this whole section,S will denote the semigroup placed in the Rees matrix semigroup MR which was constructed for N as presented in the introduction of this paper.Lemma 4 (see [1]).
The elementx,y,z,k,l,m∈N is an idempotent ⇔S∪0, which is a rectangular band, and pmk′=l−1.Lemma 5.
For the semigroupS,(i)
if it is a right cancellative semigroup, then every element ofEN is ℛ∗ equivalence, and(ii)
if it is a left cancellative semigroup, then every element ofEN is ℒ∗ equivalence.Proof.
The proof will be given for justi since the ℒ∗ equivalence case can be obtained quite similarly as ℛ∗ equivalence.
Since index sets are composed of one element, the structure of two arbitrary elements inEN are the form of a,b,c,a,d,c and a,s,c,a,g,c (see [1], Lemma 3). So, we have(16)a,b′,c,a,d′,c⋆a,b,c,a,d,c=a,b″,c,a,d″,c∗a,b,c,a,d,c⇒a,b′pcab,c,a,d′pca′d,c=a,b″pcab,c,a,d″pca′d,c.
Sincea,b,c,a,d,c is in EN, by Lemma 4, we will have(17)pca′=d−1,d′pca′d=d″pca′d⇒d′=d″.
In addition, by the assumption, sinceS is a right cancellative semigroup, we have b′pcas=b″pcas, which implies b′=b″. So, by taking into account (15), we can write(18)a,b′,c,a,d′,c∗a,s,c,a,g,c=a,b″,c,a,d″,c∗a,s,c,a,g,c⇒a,b′pcas,c,a,d′pca″g,c=a,b″pcas,c,a,d″pca″g,c.
All of the preceding progress results inEN as ℛ∗ equivalence, as required.
One of the main theorems of this paper is the following.Theorem 2.
LetS be a cancellative semigroup. Then, the set EN is not only an abundant semigroup but also a superabundant semigroup.Proof.
Consider any two idempotent elementsx¯=a,b,c,a,d,c and y¯=a,s,c,a,g,c in EN. By Lemma 4, we have pca′=d−1 and pca″=g−1. In addition, since x¯ and y¯ are idempotents, ℒ∗, ℛ∗ and ℋ∗-classes consist of idempotent elements. According to Lemma 5 and the definition of an abundant semigroup, EN is the abundant semigroup. Furthermore, according to the definition of the superabundant semigroup, EN is the superabundant semigroup since ℋ∗-classes consist of idempotent elements.
The following consequences are immediate.Corollary 1.
IfN is commutative, then EN is an abundant semigroup as well as an adequate semigroup.Proof.
Assume thatN is commutative. Thus, the elements of EN are commutative which gives the meaning of an adequate semigroup since an abundant semigroup whose idempotents are commutative is actually an adequate semigroup.Corollary 2.
For anyN, every EN is a superabundant semigroup as well as a cyber-group.Proof.
By Theorem2, EN is a superabundant semigroup of N. Furthermore, since EN is a subsemigroup of N, we reach that it is also a cyber-group.
As we mentioned in Corollary1, it is well known that every commutative abundant semigroup N is an adequate semigroup for N. By utilizing this result, we show that the homomorphism of abundant semigroups for N is a good kind of homomorphism. Now, we let Na and NA denote the abundant semigroup and commutative abundant semigroup (adequate) for N, respectively.
Our final result in this paper is as follows.Corollary 3.
Iff:NA⟶Na is a homomorphism, then f is a good homomorphism.Proof.
Letf:NA⟶Na be a homomorphism, then we have fA,fB∈Na for A,B∈NA. Since it is known that A,B∈NA by using the definition of adequate semigroup, we have Aℒ∗B andAℛ∗B. Furthermore, since fA,fB∈Na, we have fAℒ∗fB and fAℛ∗fB, and so, f is a kind of good homomorphism.


## 4. Summary and Conclusions

Green’s relations are one of the interested areas in the semigroup theory. There are different types of it also, which are used for Green’s theorem. In this paper, we prove Green’s theorem overN by using Green’s relations of the semigroup N. Furthermore, we present the classification of the semigroup N considering generalized Green’s relation. Finally, we obtain a new result about good homomorphism.As future work, we are going to study some other properties and classification of the special semigroupN over semigroups.



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