# Regularity of Semigroups of Transformations Whose Characters Form the Semigroup of aΔ-Structure

Authors: Jittisak Rakbud; Malinee Chaiya
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

In this paper, we make use of the notion of the character of a transformation on a fixed setX, provided by Purisang and Rakbud in 2016, and the notion of a Δ-structure on X, provided by Magill Jr. and Subbiah in 1974, to define a sub-semigroup of the full-transformation semigroup TX. We also define a sub-semigroup of that semigroup. The regularity of those two semigroups is also studied.


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## Body

## 1. Introduction and Preliminaries

For any semigroupS, we call an element a of S a regular element of S if there exists an element b of S such that aba=a. A semigroup S is said to be regular if every element of S is regular. The semigroup of transformations on a nonempty set X, denoted by TX, is a well-known regular semigroup. In [1], Nenthein et al. studied the regularity of the following two sub-semigroups of TX:(1)TX,Y≔α∈TX:Ranα⊆Y,T¯X,Y≔α∈TX:αY⊆Y,where Y is a fixed nonempty subset of X and Ranα denotes the range of α for all α∈TX. The authors obtained that the two semigroups are regular if and only Y=1 or Y=X. The regularity of some other sub-semigroups of TX has been studied by many people (see [2–6] for some recent works).In this paper, by apartition of a nonempty set X, we mean a family ℱ=Yi:i∈I of nonempty subsets of X such that X=∪i∈IYi and Yi∩Yj=∅ for all i,j∈I with i≠j. For a given partition ℱ=Yi:i∈I of a nonempty set X, let(2)TℱX=α∈TX:∀i∈I∃j∈I,αYi⊆Yj.Note thatTℱX is exactly the semigroup of all transformations on X preserving the equivalence relation induced by partition ℱ. There have been several works involving transformations preserving an equivalence relation (see [3–5, 7–12] for some references).Throughout this paper, letX be a nonempty set, and let ℱ=Yi:i∈I be a partition of X, which are arbitrarily fixed. From the definition of a partition, Purisang and Rakbud [11] defined a function χα:I⟶I associated with each α∈TℱX called the character of α, by(3)∀i,j∈I,χαi=j⟺αYi⊆Yj.They also defined and studied the regularity of the following sub-semigroup ofTX:(4)TℱJX≔α∈TX:∀i∈I∃j∈J,αYi⊆Yj=α∈TX:χα∈TI,J,where J is an arbitrarily fixed nonempty subset of the index set I. We summarize some of their results as follows.Lemma 1 (see [11], Lemma 2.3).
For everyα,β∈TℱJX, χαβ=χαχβ.By making use of the notion of the character, the following two equivalence relationsχ and χ˜ on TℱJX were defined:(5)∀α,β∈TℱJX,α,β∈χ⟺χα=χβ,∀α,β∈TℱJX,α,β∈χ˜⟺χα|J=χβ|J.Note that, by Lemma1, they both are congruence relations. The authors studied the regularity of the quotient semigroups TℱJX/χ and TℱJX/χ˜. The following are what they obtained.Theorem 1 (see [11], Theorem 2.4).
The following statements hold:(1)
TℱJX/χ≅TI,Jby the isomorphismα↦χα;(2)
TℱJX/χ˜≅TJby the isomorphismα˜↦χα|J,where for eachα∈TℱJX, αandα˜denote the equivalence classes ofαunder the equivalence relationsχandχ˜, respectively.Corollary 1 (see [11], Corollary 2.5).
The following statements hold.(1)
The three statements below are all equivalent:(a)
The quotient semigroupTℱJX/χis regular;(b)
The semigroupTI,Jis regular;(c)
J=IorJ=1.(2)
The quotient semigroupTℱX/χis regular.(3)
The quotient semigroupTℱJX/χ˜is regular.The regularity of the semigroupTℱJX was obtained as follows.Theorem 2 (see [11], Theorem 2.6).
The semigroupTℱJXis regular if and only ifTℱJX=1orTℱJX=TX.It is easy to see that for eachα∈TℱX, the equivalence class α of α under the equivalence relation χ is a sub-semigroup of TℱX if and only if χα is an idempotent element of TI. In [11], the authors studied the regularity of the semigroup α when α is an idempotent element of TI. They also defined some further sub-semigroups of TℱX by making use the notion of the character as follows: let IℱX, SℱX, and BℱX be the sets of all elements of TℱX whose characters are injective, surjective, and bijective, respectively. Note that, by Lemma 1, the sets IℱX, SℱX, and BℱX are sub-semigroups of TX. The regularity of each of these three semigroups was also studied.It was observed by Rakbud [12] that the semigroups TℱJX, α when χα is idempotent, IℱX, SℱX, and BℱX can simultaneously be generalized by making use of the notion of the character as follows: for every sub-semigroup S of TI, let(6)TℱSX=α∈TℱX:χα∈S.Note that, for eachγ∈TI, the function α:X⟶X defined on each Yi by αYi=zi, where zi is a fixed element of Yγi for all i∈I, is an element of TℱX whose character is exactly γ. Hence, TℱSX≠∅, and by Lemma 1, it is a sub-semigroup of TℱX.LetS be a sub-semigroup of TI. Then, by considering the congruence relation χ on TℱX restricted to TℱSX, we have the quotient semigroup TℱSX/χ. Obviously, TℱSX/χ=α:α∈TℱSX and TℱSX/χ is a sub-semigroup of TℱX/χ. Analogously to Theorem 1, the following result was established.Theorem 3 (see [12], Theorem 1.14).
TℱSX/χ≅S by the isomorphism defined byα↦χα.Immediately from Theorem3, the following corollary was obtained.Corollary 2 (see [12], Corollary 1.15).
The quotient semigroupTℱSX/χis regular if and only if the semigroupSis regular.Besides the above results, in [12], the author also used the notion of the character to define the notion of a weakly regular transformation and study the regularity of a semigroup of weakly regular transformations in that sense. However, the regularity of TℱSX has not been studied in general yet. This will be in our attention here when S has a certain property. We are mentioning that S is “the semigroup of a Δ-structure”on I.We now refer to the definition of aΔ-structure on a set and some other related ones from [13] by Magill and Subbiah. Let A be a family of nonempty subsets of X such that X∈A. And, let ℳ=HomA,B:A,B∈A, where HomA,B is a nonempty set of functions from A into B for all A,B∈A, with the following properties:(Δ1)
EndX:=HomX,X is a monoid;(Δ2)
Ranα∈A for all α∈EndX;(Δ3)
For allB∈A, α∈EndX and β∈HomRanα,B, βα∈EndX;(Δ4)
For allα,β∈EndX and A,B∈A with αB⊆A and βA⊆B, if αβ|A=idA and βα|B=idB, then β|A∈HomA,B and α|B∈HomB,A, where idZ denotes the identity function on Z for any nonempty set Z.The pairA,ℳ is called a Δ-structure on X, and the monoid EndX is called the semigroup of theΔ-structureA,ℳ. A subset A of X is called a Δ-retract of X if there is an idempotent element ρ of EndX such that A=Ranρ. For any A,B∈A, an element λ of HomA,B is called a Δ-isomorphism (from A onto B) if there is σ∈HomB,A such that σλ=idA and λσ=idB. It is clear for any A,B∈A that an element λ of HomA,B is Δ-isomorphism if and only if λ is bijective and λ−1∈HomB,A.In [13], the authors gave some characterizations of regular elements of the semigroup EndX as follows.Theorem 4 (see [13], Theorem 2.4).
Letα∈EndX. Then, the following statements are equivalent:(1)
αis regular;(2)
Ranαis aΔ-retract ofX, and there is aΔ-retractAofXsuch thatα|Ais aΔ-isomorphism fromAontoRanα;(3)
Ranαis aΔ-retract ofX, and there isA∈Asuch thatα|Ais aΔ-isomorphism fromAontoRanα.It is clear thatTX=EndX when X is equipped with the Δ-structure A,ℳ, where A is the family of all nonempty subsets of X, and HomA,B is the set of all functions from A into B for all A,B∈A. In this setting, the Δ-retracts of X are exactly the nonempty subsets of X, and the Δ-isomorphisms are exactly the bijective functions. More interesting semigroups of Δ-structures on X were given in [13] as follows:(i)
The semigroupSX of all continuous maps on X is exactly EndX when X is a topological space equipped with the Δ-structure A,ℳ, where A is the family of all nonempty subsets of X and HomA,B is the set of all continuous maps from A into B for all A,B∈A. In this setting, the Δ-retracts of X are exactly the Δ-retracts of the topological space X, and the Δ-isomorphisms are exactly the homeomorphisms.(ii)
The semigroupLX of all linear transformations on X coincides with EndX when X is a vector space equipped with the Δ-structure A,ℳ, where A is the family of all subspaces of X and HomA,B is the set of all linear transformations from A into B for all A,B∈A. In this setting, the Δ-retracts of X are exactly the subspaces of the vector space X, and the Δ-isomorphisms are exactly the vector space isomorphisms.(iii)
The semigroupΓX of all closed maps on X is exactly EndX when X is a T1-space equipped with the Δ-structure A,ℳ, where A is the family of all nonempty closed subsets of X and HomA,B is the set of all closed maps from A into B for all A,B∈A. In this setting, the Δ-retracts of X are exactly the nonempty closed subsets of the topological space X, and the Δ-isomorphisms are exactly the homeomorphisms.The regularity of the semigroupsSX, LX, and ΓX was also deduced via Theorem 4. In addition, we note here that the semigroup TℰX of all transformations on X preserving an equivalence relation ℰ on X can be considered as the semigroup of all continuous maps on X, where X is equipped with the topology having the family of all equivalence classes as a base. This was proved by Huisheng [8] (see Theorem 2.8).The main aim of this paper is to study the regularity of the semigroupTℱSX, introduced in [12], when S is the semigroup of a Δ-structure on the index set I of the partition ℱ. We also define, in this situation, a sub-semigroup of TℱSX whose regularity coincides with that of the semigroup S.


## 2. The Semigroup and Its Regularity

For any nonempty setsZ and W and partitions P=Zi:i∈H and Q=Wi:i∈K of Z and W, respectively, let TP,QZ,W be the set of all functions λ:Z⟶W satisfying the condition that for all i∈H, there is j∈K such that λZi⊆Wj. And, for each λ∈TP,QZ,W, let χP,Qλ:H⟶K be defined by(7)∀i,j∈H×K,χP,Qλi=j⟺λZi⊆Wj.It is easy to verify that the mapλ↦χP,Qλ from the set TP,QZ,W into the set of all functions from H into K is surjective. If we have three nonempty sets Z, W, and U with partitions P=Zi:i∈H, Q=Wi:i∈K, and ℛ=Ui:i∈M of Z, W, and U, respectively, we see for any λ∈TP,QZ,W and ρ∈TQ,ℛW,U that ρλ∈TP,ℛZ,U and χP,ℛρλ=χQ,ℛρχP,Qλ.For allα,β∈TℱX, let(8)Tℱα,β=TP,QRanα,Ranβ,where(9)P=Ranα∩Yi:i∈Ranχα,Q=Ranβ∩Yi:i∈Ranχβ.And, for eachλ∈Tℱα,β, let χα,βλ=χP,Qλ.LetJ,C be a Δ-structure on the index set I of the partition ℱ, and let(10)EndℱX=TℱEndIX≔α∈TℱX:χα∈EndI.By the property (Δ1) of theΔ-structure J,C on I, we have that χidX=idI∈EndI, which yields that idX∈EndℱX. Hence, EndℱX is a submonoid of TℱX.Theorem 5.
There is aΔ-structure onXsuch thatEndX=EndℱX.Proof.
Let(11)A=Ranα:α∈EndℱX.
SinceEndℱX is a monoid, we have X∈A. For any α,β∈EndℱX, let(12)HomRanα,Ranβ=λ∈Tℱα,β:χα,βλ∈HomRanχα,Ranχβ.
Then, by the surjectivity of the mapλ↦χα,βλ from the set Tℱα,β into the set of all functions from Ranχα into Ranχβ, we have that HomRanα,Ranβ≠∅ for all α,β∈EndℱX. Note that for all α,β,λ,ρ∈EndℱX, if Ranα=Ranλ and Ranβ=Ranρ, then Ranχα=Ranχλ and Ranχβ=Ranχρ, which yields that HomRanα,Ranβ=HomRanλ,Ranρ. Let(13)ℳ=HomRanα,Ranβ:α,β∈EndℱX.
Then, from the above explanation, the familyℳ is well defined. We will show that A,ℳ is a Δ-structure on X. It is easy to see that(14)HomX,X=HomRanidX,RanidX=λ∈TℱX:χλ∈EndI=EndℱX.
Thus, the property (Δ1) is satisfied. From the definition of the familyA, we immediately have that the property (Δ2) holds. Next, we will show that the property (Δ3) is satisfied. Let α,β∈EndℱX, and let λ∈HomRanα,Ranβ. We want to show that λα∈EndℱX. Since λ∈HomRanα,Ranβ, we have χα,βλ∈HomRanχα,Ranχβ. Since α∈EndℱX, we have that χα∈EndI. Thus, by the property (Δ3) of the Δ-structure J,C on I, we get χα,βλχα∈EndI. It is clear that λα∈TℱX, and that χλα=χα,βλχα. Hence, by the membership of χα,βλχα in EndI, we obtain that λα∈EndℱX as desired. Finally, we will show that the property (Δ4) holds. Let α,β,ψ,ϕ∈EndℱX be such that αRanϕ⊆Ranψ and βRanψ⊆Ranϕ. Then, χα,χβ,χψ,χϕ∈EndI. Thus, by the property (Δ2) of the Δ-structure J,C on I, we get that Ranχψ and Ranχϕ belong to J. Since Ranαϕ=αRanϕ⊆Ranψ, we have that Ranχαϕ⊆Ranχψ. To see this explicitly, let j∈Ranχαϕ. Then, there is i∈I such that χαϕi=j, which yields that αϕYi⊆Yj. Fix x∈Yi. Since Ranαϕ⊆Ranψ, we get that αϕx∈Ranψ. Therefore, there is z∈X such that αϕx=ψz. Let k∈I be such that z∈Yk, and let ν=χψk. Then, ψYk⊆Yν; so αϕx∈Yν, which implies that Yj=Yν. Hence, j=ν∈Ranχψ. Similarly, by the inclusion βRanψ⊆Ranϕ, we obtain Ranχβψ⊆Ranχϕ. We now have two inclusions Ranχαϕ⊆Ranχψ and Ranχβψ⊆Ranχϕ. From these, we get by Lemma 1 that χαRanχϕ⊆Ranχψ and χβRanχψ⊆Ranχϕ. By the assumptions that αRanϕ⊆Ranψ and that βRanψ⊆Ranϕ, we have α|Ranϕ:Ranϕ⟶Ranψ and β|Ranψ:Ranψ⟶Ranϕ, respectively. And, from the membership of α in TℱX, we get that for all i∈Ranχϕ, there exists j∈Ranχψ such that α|RanϕYi∩Ranϕ⊆Yj. Thus, α|Ranϕ∈Tℱϕ,ψ. By the inclusion χαRanχϕ⊆Ranχψ, we have χα|Ranχϕ:Ranχϕ⟶Ranχψ, which yields that χϕ,ψα|Ranϕ=χα|Ranχϕ. Similarly, we have that β|Ranψ∈Tℱψ,ϕ, and that χψ,ϕβ|Ranψ=χβ|Ranχψ. Suppose that βα|Ranϕ=idRanϕ, and that αβ|Ranψ=idRanψ. We want to show that α|Ranϕ∈HomRanϕ,Ranψ, and that β|Ranψ∈HomRanψ,Ranϕ. By the inclusions χαRanχϕ⊆Ranχψ and χβRanχψ⊆Ranχϕ, we have, respectively, that χβχα|Ranχϕ:Ranχϕ⟶Ranχϕ and χαχβ|Ranχψ:Ranχψ⟶Ranχψ. And, since βα|Ranϕ=idRanϕ and αβ|Ranψ=idRanψ, it follows that χβχα|Ranχϕ=idRanχϕ and χαχβ|Ranχψ=idRanχψ, respectively. Therefore, by the property (Δ4) of the Δ-structure J,C on I, we obtain that χϕ,ψα|Ranϕ=χα|Ranχϕ∈HomRanχϕ,Ranχψ and χψ,ϕβ|Ranψ=χβ|Ranχψ∈HomRanχψ,Ranχϕ. These yield, respectively, that α|Ranϕ∈HomRanϕ,Ranψ and β|Ranψ∈HomRanψ,Ranϕ.From now on, we will consider theΔ-structure A,ℳ on X defined in the proof of Theorem 5, and the semigroup EndℱX of this △-structure will be in our attention. By Theorem 4, the regularity of elements of the semigroup EndℱX can roughly be characterized. To get more precise characterizations, according to Theorem 4, the notions of a Δ-retract and a Δ-isomorphism in the Δ-structure A,ℳ on X should particularly be studied. For that purpose, the following elementary theorem, stating some characterizations of idempotent elements in a transformation semigroup, is needed.Theorem 6.
LetZ be a nonempty set and α∈TZ. Then, the following statements are equivalent:(1)
α is idempotent;(2)
α|Ranα=idRanα;(3)
There is a partitionZj:j∈E of Z and a subset zj:j∈E of Z such that zj∈Zj for all j∈E and αZj=zj for all j∈E.
In this situation, the partitionZj:j∈E and the subset Zj:j∈E of Z are unique determined by α.For each nonempty subsetA of X, we see that there is a unique subset of I, denoted by JA, such that the family A∩Yi:i∈JA is a partition of A. In particular, we have that JA is exactly Ranχα if A=Ranα for some α∈TℱX.Proposition 1.
Letγ∈TI, and letA⊆X. Ifγis idempotent andJA=Ranγ, then there is an idempotent elementαofTℱXsuch thatχα=γandA=Ranα.Proof.
Suppose thatγ is idempotent, and that JA=Ranγ. Then, by Theorem 6, there is a partition Ij:j∈E of I and a subset ij:j∈E of I such that ij∈Ij and γIj=ij for all j∈E. From this, we have that JA=Ranγ=ij:j∈E. For each j∈E, let αj∈Tℱj∪i∈IjYi, where ℱj=Yi:i∈Ij, be idempotent such that χαji=ij for all i∈Ij and Ranαj=A∩Yij. And, finally, let α:X⟶X be defined by α|∪i∈IjYi=αj for all j∈E. Since ∪i∈IjYi:j∈E is a partition of X, it follows that α is well defined. It is clear by the way of defining α that α∈TℱX with χα=γ. It is also clear that α is idempotent with Ranα=A.From Proposition1, the following corollary is easily obtained.Corollary 3.
LetA⊆X. Then,Ais aΔ-retract ofXif and only ifJAis aΔ-retract ofI.Proof.
Suppose thatA is a Δ-retract of X. Then, there is an idempotent element α of EndℱX such that A=Ranα, which yields that JA=Ranχα. Since α is idempotent, we have by Lemma 1 that χα is an idempotent element of EndI. Hence, JA is a Δ-retract of I. Conversely, suppose that JA is a Δ-retract of I. Then, there is an idempotent element γ of EndI such that JA=Ranγ. Thus, by Proposition 1, the set A is a Δ-retract of X.Proposition 2.
Letα,β∈EndℱX, and letλ∈HomRanα,Ranβ. Ifλis aΔ-isomorphism, thenχα,βλis aΔ-isomorphism.Proof.
Letλ∈HomRanα,Ranβ. Then, χα,βλ∈HomRanχα,Ranχβ. Suppose that λ is a Δ-isomorphism. Then, there exists ψ∈HomRanβ,Ranα such that λψ=idRanβ and ψλ=idRanα. By the membership of ψ in HomRanβ,Ranα, we have χβ,αψ∈HomRanχβ,Ranχα. Since λψ=idRanβ, we get that χα,βλχβ,αψ=χβ,βλψ=χβ,βidRanβ=idRanχβ. Similarly, since ψλ=idRanα, we get χβ,αψχα,βλ=idRanχα. Thus, χα,βλ is a Δ-isomorphism.In the following theorem, we provide a characterization of the regularity of elements ofEndℱX in terms of the Δ-retract and the Δ-isomorphism of I.Theorem 7.
Letα∈EndℱX. Then,αis regular if and only ifRanχαis aΔ-retract ofI, and there isA⊆Xsuch that each of the following statements holds true:(i)
JAisΔ-retract ofI;(ii)
χα|JAis aΔ-isomorphism fromJAontoRanχα;(iii)
α|A∩Yiis a bijective function fromA∩YiontoRanα∩Yχαifor alli∈JA.Proof.
Suppose thatRanχα is a Δ-retract of I, and that there exists a subset A of X such that each of the following statements holds true:(i)
JA is Δ-retract of I;(ii)
χα|JA is a Δ-isomorphism from JA onto Ranχα;(iii)
α|A∩Yi is a bijective function from A∩Yi onto Ranα∩Yχαi for all i∈JA.
SinceRanχα is a Δ-retract of I, we have by Corollary 3 that Ranα is a Δ-retract of X. And, since JA is a Δ-retract of I, there is an idempotent element γ of EndI such that JA=Ranγ. Thus, by Proposition 1, there is an idempotent element β of TℱX such that χβ=γ and A=Ranβ. This yields that A is a Δ-retract of X. By condition (ii), we have that χα|JA is a bijective function from JA onto Ranχα. Thus, by condition iii, we get that α|A, which is an element of Tℱβ,α, is bijective. By condition (ii) once again, we have χβ,αα|A=χα|JA∈HomJA,Ranχα and χα,βα|A−1=χα|JA−1∈HomRanχα,JA. It follows that α|A is a Δ-isomorphism from A onto Ranα. Therefore, by Theorem 4, we obtain that α is regular. Conversely, suppose that α is regular. Then, by Theorem 4, we get that Ranα is a Δ-retract of X, and that there is a Δ-retract A of X such that α|A is a Δ-isomorphism from A onto Ranα. Since Ranα and A are Δ-retracts of X, we have by Corollary 3 that Ranχα and JA are Δ-retracts of I, respectively. Since α|A is a Δ-isomorphism from A onto Ranα, we have that (iii) holds. And, by Proposition 2, we get that (ii) holds.IfEndI=TI, then EndℱX becomes TℱX. Hence, by Theorem 7, the following corollary is immediately obtained.Corollary 4.
Letα∈TℱX. Then,αis regular if and only if there isA⊆Xsuch that each of the following statements holds true:(i)
χα|JAis a bijective function fromJAontoRanχα;(ii)
α|A∩Yiis a bijective function fromA∩YiontoRanα∩Yχαifor alli∈JA.Note that by consideringTℱX as the semigroup of all continuous maps on X, where X is equipped with the topology having the family of all equivalence classes as a base, the regularity of α∈TℱX can be deduced from Theorem 4 as well. This was provided by Huisheng [9]. The author obtained for any α∈TℱX that α is regular if and only if for each i∈I, there is j∈I such that Yi∩Ranα⊆αYj. Here, we get another characterization of the regularity of elements of TℱX in terms of the character.The following three corollaries are immediately obtained from Theorem7 as well.Corollary 5.
Suppose thatIis a topological space, and letEndI=SI. Letα∈EndℱX. Then,αis regular if and only if there isA⊆Xsuch that each of the following statements holds true:(i)
χα|JAis a homeomorphism fromJAontoRanχα;(ii)
α|A∩Yiis a bijective function fromA∩YiontoRanα∩Yχαifor alli∈JA.Corollary 6.
Suppose thatIis aT1-space, and letEndI=ΓI. Letα∈EndℱX. Then,αis regular if and only if there isA⊆Xsuch that each of the following statements holds true:(i)
JAis a closed subset of I;(ii)
χα|JAis a homeomorphism fromJAontoRanχα;(iii)
α|A∩Yiis a bijective function fromA∩YiontoRanα∩Yχαifor alli∈JA.Corollary 7.
Suppose thatIis a vector space, and letEndI=LI. Letα∈EndℱX. Then,αis regular if and only if there isA⊆Xsuch that each of the following statements holds true:(i)
JAis a subspace of I;(ii)
χα|JAis an isomorphism fromJAontoRanχα;(iii)
α|A∩Yiis a bijective function fromA∩YiontoRanα∩Yχαifor alli∈JA.We end this section with a discussion on the regularity of the quotient semigroupEndℱX/χ, where χ is the congruence relation on EndℱX defined by(15)∀α,β∈EndℱX,αχβ⟺χα=χβ.By virtue of Corollary2, we immediately get that the semigroup EndℱX/χ is regular if and only if the semigroup EndI is regular.


## 3. A Subsemigroup ofEndℱX and Its Regularity

In this section, we define a sub-semigroup ofEndℱX and study the regularity of that semigroup. Let κ be a cardinal number, and let(16)MℱκX=α∈TℱX:∀i∈Ranχα,|Yi∩Ranα|≤κ.Lemma 2.
Letσ,ρ∈MℱκXandγ∈Tℱσ,ρ. Then,γσ∈MℱκX.Proof.
Letβ=γσ. We now want to show that β∈MℱκX. It is clear that β∈TℱX. Next, let i∈Ranχβ. Since χβ=χσ,ργχσ, we have Ranχβ=χσ,ργRanχσ⊆Ranχσ,ργ⊆Ranχρ. Thus, i∈Ranχρ, and there is j∈Ranχσ such that χσ,ργj=i, which yields that γYj∩Ranσ⊆Yi∩Ranρ. Since j∈Ranχσ, there is k∈I such that χσk=j, which implies that σYk⊆Yj. Thus, βYk=γσYk=γσYk∩Ranσ⊆γYj∩Ranσ⊆Yi∩Ranρ. From this, we obtain that Yi∩Ranβ≠∅. Since i∈Ranχρ, we have that Yi∩Ranρ≤κ. And, since Ranβ⊆Ranγ⊆Ranρ, it follows that 0<Yi∩Ranβ≤Yi∩Ranρ≤κ. Thus, β∈MℱκX.Corollary 8.
The setMℱκXis a sub-semigroup ofTℱX.Proof.
Letα,β∈MℱκX. Then, α|Ranβ∈Tℱβ,α. Thus, by Lemma 2, we get that αβ=α|Ranββ∈MℱκX.Let(17)EℱκX=EndℱX∩MℱκX.Note that for eachγ∈TI, there is α∈MℱκX such that χα=γ. Such a function α can easily be defined as follows: for each j∈Ranγ, let zj be a fixed element of Yj and let α:X⟶X be defined by α∪i∈γ−1jYi=zj for all j∈Ranγ. Thus, by the note, we have that EℱκX≠∅. Furthermore, it is a sub-semigroup of EndℱX.Theorem 8.
Letα∈Eℱ1X. Then, the following statements are equivalent:(1)
αis a regular element ofEℱ1X;(2)
αis a regular element ofEndℱX;(3)
χαis a regular element ofEndI.Proof.
3⇒2. Suppose that χα is a regular element of EndI. Then, by the regularity of χα in EndI, we get by Theorem 4 that Ranχα is a Δ-retract of I, and that there is an idempotent element γ of EndI such that χα|Ranγ is a Δ-isomorphism from Ranγ onto Ranχα. Since γ is an idempotent element of EndI, we have by Theorem 6 that there is a partition Ij:j∈E of I and a subset ij:j∈E of I with ij∈Ij for all j∈E such that γIj=ij for all j∈E. So, Ranγ=ij:j∈E. Since χα|Ranγ is a Δ-isomorphism from Ranγ onto Ranχα, we have that χα|Ranγ∈HomRanγ,Ranχα is bijective, and that χα|Ranγ−1∈HomRanχα,Ranγ. So, Ranχα=χαij:j∈E with Ranχα=E. Since α∈Mℱ1X, we have that Ranα∩Yi=1 for all i∈Ranχα. Let Ranα∩Yχαij=aj for all j∈E. Then, Ranα=aj:j∈E. For each j∈E, fix bj∈α−1aj∩Yij. Let β:X⟶X be defined by β∪i∈IjYi=bj for all j∈E. Since the family ∪i∈IjYi:j∈E is a partition of X, we get that β is well defined. It is clear that β is an idempotent element of TℱX, and that χβ=γ∈EndI. Thus, β is an idempotent element of EndℱX yields that Ranβ is a Δ-retract of X. Moreover, we have α|Ranβ∩Yi is a bijective function from Ranβ∩Yi onto Ranα∩Yχαi for all i∈Ranχβ. Therefore, by Theorem 7, we obtain that α is regular.
2⇒1. Suppose that α is a regular element of EndℱX. Then, by Theorem 4 and Proposition 1, there exists an idempotent element σ of EndℱX such that χσ=χα and Ranα=Ranσ. Since α∈Eℱ1X, it follows that σ∈Eℱ1X. By Theorem 7, there exists an idempotent element ρ of EndℱX such that(a)
χα|Ranχρ is a △-isomorphism from Ranχρ onto Ranχσ;(b)
α|Ranρ∩Yi is a bijective function from Ranρ∩Yi onto Ranσ∩Yχσi for all i∈Ranχρ.
From (a) and (b), we have thatα|Ranρ−1∈HomRanσ,Ranρ and ρ∈Eℱ1X, respectively. Let β=α|Ranρ−1σ. Then, by the property (Δ3), we have that β∈EndℱX. And, by Lemma 2, we immediately obtain that β∈Mℱ1X. Thus, β∈Eℱ1X. Finally, we will show that αβα=α. Let x∈X. Then, αx∈Ranσ. Hence, there is z∈X such that αx=σz. And, since σ is idempotent, it follows that αβαx=σαx=σσz=σz=αx. Therefore, αβα=α, and hence α is a regular element of Eℱ1X.
1⇒3. It follows directly from Lemma 1.Corollary 9.
The semigroupEℱ1Xis regular if and only ifEndIis regular.


## 4. Conclusions

The semigroupTℱSX, where S is a sub-semigroup of TI, was first defined by Rakbud [12] in 2018 via the notion of the character introduced by Purisang and Rakbud [11] in 2016. Here, we focus on studying the regularity of the semigroup TℱSX when S is the semigroup of a Δ-structure on I, which is written as S=EndI. In our study, we obtain that TℱSX, which is denoted by EndℱX, is the semigroup of a Δ-structure on X. From this, the regularity of elements of EndℱX can generally be explained via Theorem 4 established by Magill and Subbiah [13] in 1974. We also obtain a characterization of regular elements of EndℱX in terms of the Δ-structure on I (see Theorem 7). From this result, we deduce the regularity of EndℱX when EndI is one of the following semigroups: the transformation semigroup TI, the semigroup SI of continuous maps on I when I is a topological space, the semigroup ΓI of closed maps on I when I is a T1-space, and the semigroup LI of linear transformations on I when I is a vector space (see Corollaries 4–7). Apart from the regularity of EndℱX, we provide a sub-semigroup of EndℱX, namely, the semigroup Eℱ1X, whose regularity coincides with that of EndI. In [13], Magill and Subbiah also generally gave some characterizations of Green’s relations for regular elements of the semigroup of a Δ-structure. Since our semigroup EndℱX is the semigroup of a Δ-structure on X, some rough characterizations of Green’s relations for regular elements of EndℱX can immediately be deduced from the results of Magill Jr. and Subbiah.We end this paper with some interesting questions:(1)
Can Green’s relations for regular elements ofEndℱX be characterized more deeply in terms of the Δ-structure on I?(2)
Can other notions such as the ideal, the rank, the left regularity, and the right regularity in the semigroupEndℱX be explained in terms of those in the semigroup EndI?



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