Regularity of Semigroups of Transformations Whose Characters Form the Semigroup of a Δ-Structure

where Y is a fixed nonempty subset of X and Ran(α) denotes the range of α for all α ∈ T(X). (e authors obtained that the two semigroups are regular if and only |Y| � 1 or Y � X. (e regularity of some other sub-semigroups of T(X) has been studied by many people (see [2–6] for some recent works). In this paper, by a partition 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 partitionF � Yi: i ∈ I 􏼈 􏼉 of a nonempty set X, let


Introduction and Preliminaries
For any semigroup S, 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. e semigroup of transformations on a nonempty set X, denoted by T(X), is a well-known regular semigroup. In [1], Nenthein et al. studied the regularity of the following two sub-semigroups of T(X): where Y is a fixed nonempty subset of X and Ran(α) denotes the range of α for all α ∈ T(X). e authors obtained that the two semigroups are regular if and only |Y| � 1 or Y � X. e regularity of some other sub-semigroups of T(X) has been studied by many people (see [2][3][4][5][6] for some recent works). In this paper, by a partition of a nonempty set X, we mean a family F � Y i : i ∈ I of nonempty subsets of X such that X � ∪ i∈I Y i and Y i ∩ Y j � ∅ for all i, j ∈ I with i ≠ j. For a given partition F � Y i : i ∈ I of a nonempty set X, let T F (X) � α ∈ T(X): ∀i ∈ I∃j ∈ I, α Y i ⊆Y j . (2) Note that T F (X) is exactly the semigroup of all transformations on X preserving the equivalence relation induced by partition F.
e following statements hold.
(1) e three statements below are all equivalent: (2) e quotient semigroup T F (X)/χ is regular.
e regularity of the semigroup T (J) F (X) was obtained as follows.
It is easy to see that for each α ∈ T F (X), the equivalence class [α] of α under the equivalence relation χ is a subsemigroup of T F (X) if and only if χ (α) is an idempotent element of T(I). In [11], the authors studied the regularity of the semigroup [α] when α is an idempotent element of T(I).
ey also defined some further sub-semigroups of T F (X) by making use the notion of the character as follows: let I F (X), S F (X), and B F (X) be the sets of all elements of T F (X) whose characters are injective, surjective, and bijective, respectively. Note that, by Lemma 1, the sets I F (X), S F (X), and B F (X) are sub-semigroups of T(X). e regularity of each of these three semigroups was also studied.
It was observed by Rakbud [12] that the semigroups T (J) F (X), [α] when χ (α) is idempotent, I F (X), S F (X), and B F (X) can simultaneously be generalized by making use of the notion of the character as follows: for every sub-semigroup S of T(I), let Note that, for each c ∈ T(I), the function α: X ⟶ X defined on each Y i by α(Y i ) � z i , where z i is a fixed element of Y c(i) for all i ∈ I, is an element of T F (X) whose character is exactly c. Hence, T (S) F (X) ≠ ∅, and by Lemma 1, it is a sub-semigroup of T F (X).
Let S be a sub-semigroup of T(I). en, by considering the congruence relation χ on T F (X) restricted to T (S) F (X), we have the quotient semigroup T (S) F (X)/χ. Obviously, F (X) and T (S) F (X)/χ is a subsemigroup of T F (X)/χ. Analogously to eorem 1, the following result was established.
Immediately from eorem 3, the following corollary was obtained.
e quotient semigroup T (S) F (X)/χ is regular if and only if the semigroup S is 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 T (S) F (X) has not been studied in general yet. is 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 a Δ-structure on a set and some other related ones from [13]  In [13], the authors gave some characterizations of regular elements of the semigroup End(X) as follows.

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International Journal of Mathematics and Mathematical Sciences Theorem 4 (see [13], eorem 2.4). Let α ∈ End(X). en, the following statements are equivalent: (1) α is regular; (2) Ran(α) is a Δ-retract of X, and there is a Δ-retract A of X such that α| A is a Δ-isomorphism from A onto Ran(α); It is clear that T(X) � End(X) when X is equipped with the Δ-structure (A, M), where A is the family of all nonempty subsets of X, and Hom(A, 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: (iii) e semigroup Γ(X) of all closed maps on X is exactly End(X) when X is a T 1 -space equipped with the Δ-structure (A, M), where A is the family of all nonempty closed subsets of X and Hom(A, 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.
e regularity of the semigroups S(X), L(X), and Γ(X) was also deduced via eorem 4. In addition, we note here that the semigroup T E (X) 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. is was proved by Huisheng [8] (see eorem 2.8).
e main aim of this paper is to study the regularity of the semigroup T (S) F (X), 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 T (S) F (X) whose regularity coincides with that of the semigroup S.

The Semigroup and Its Regularity
For any nonempty sets Z and W and partitions P � Z i : i ∈ H and Q � W i : i ∈ K of Z and W, respectively, let T P,Q (Z, W) be the set of all functions λ: Z ⟶ W satisfying the condition that for all It is easy to verify that the map λ↦χ (λ) P,Q from the set T P,Q (Z, 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 where And, for each λ ∈ T F (α, β), let χ (λ) α,β � χ (λ) P,Q . Let (J, C) be a Δ-structure on the index set I of the partition F, and let By the property (Δ1) of the Δ-structure (J, C) on I, we have that χ (id X ) � id I ∈ End(I), which yields that id X ∈ End F (X). Hence, End F (X) is a submonoid of T F (X).

Theorem 5.
ere is a Δ-structure on X such that End(X) � End F (X).
□ From now on, we will consider the Δ-structure (A, M) on X defined in the proof of eorem 5, and the semigroup End F (X) of this △-structure will be in our attention. By eorem 4, the regularity of elements of the semigroup End F (X) can roughly be characterized. To get more precise characterizations, according to eorem 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. Let Z be a nonempty set and α ∈ T(Z). en, the following statements are equivalent:
(1) α is idempotent; (2) α| Ran(α) � id Ran(α) ; (3) ere is a partition Z j : j ∈ E of Z and a subset z j : j ∈ E of Z such that z j ∈ Z j for all j ∈ E and α(Z j ) � z j for all j ∈ E.
In this situation, the partition Z j : j ∈ E and the subset Z j : j ∈ E of Z are unique determined by α.
For each nonempty subset A of X, we see that there is a unique subset of I, denoted by J A , such that the family A ∩ Y i : i ∈ J A is a partition of A. In particular, we have that J A is exactly Ran(χ (α) ) if A � Ran(α) for some α ∈ T F (X).

Proposition 1. Let c ∈ T(I), and let A⊆X. If c is idempotent and J A � Ran(c), then there is an idempotent element α of T F (X) such that χ (α) � c and A � Ran(α).
Proof. Suppose that c is idempotent, and that J A � Ran(c). en, by eorem 6, there is a partition I j : j ∈ E of I and a subset i j : j ∈ E of I such that i j ∈ I j and c(I j ) � i j for all j ∈ E. From this, we have that J A � Ran(c) � i j : j ∈ E . For each j ∈ E, let α j ∈ T F j ( ∪ i∈I j Y i ), where F j � Y i : i ∈ I j , be idempotent such that χ (α j ) (i) � i j for all i ∈ I j and Ran(α j ) � A ∩ Y i j . And, finally, let α: X ⟶ X be defined by α| ∪ i∈I j Y i � α j for all j ∈ E. Since ∪ i∈I j Y i : j ∈ E is a partition of X, it follows that α is well defined. It is clear by the way 4 International Journal of Mathematics and Mathematical Sciences of defining α that α ∈ T F (X) with χ (α) � c. It is also clear that α is idempotent with Ran(α) � A.
□ From Proposition 1, the following corollary is easily obtained.

Corollary 3. Let A⊆X. en, A is a Δ-retract of X if and only if J A is a Δ-retract of I.
Proof. Suppose that A is a Δ-retract of X. en, there is an idempotent element α of End F (X) such that A � Ran(α), which yields that J A � Ran(χ (α) ). Since α is idempotent, we have by Lemma 1 that χ (α) is an idempotent element of End(I). Hence, J A is a Δ-retract of I. Conversely, suppose that J A is a Δ-retract of I. en, there is an idempotent element c of End(I) such that J A � Ran(c).
us, by Proposition 1, the set A is a Δ-retract of X.
en, there exists ψ ∈ Hom (Ran(β), Ran(α)) such that λψ � id Ran(β) and ψλ � id Ran(α) . By the membership of ψ in Hom(Ran(β), Ran(α)), we have χ (ψ) (β,α) ∈ Hom(Ran (χ (β) ), Ran(χ (α) )). Since λψ � id Ran(β) , we get that In the following theorem, we provide a characterization of the regularity of elements of End F (X) in terms of the Δ-retract and the Δ-isomorphism of I. Theorem 7. Let α ∈ End F (X). en, α is regular if and only if Ran(χ (α) ) is a Δ-retract of I, and there is A⊆X such that each of the following statements holds true: Proof. Suppose that Ran(χ (α) ) is a Δ-retract of I, and that there exists a subset A of X such that each of the following statements holds true: Since Ran(χ (α) ) is a Δ-retract of I, we have by Corollary 3 that Ran(α) is a Δ-retract of X. And, since J A is a Δ-retract of I, there is an idempotent element c of End(I) such that J A � Ran(c). us, by Proposition 1, there is an idempotent element β of T F (X) such that χ (β) � c and A � Ran(β). is yields that A is a Δ-retract of X. By condition (ii), we have that χ (α) | J A is a bijective function from J A onto Ran(χ (α) ).
us, by condition (iii), we get that α| A , which is an element of T F (β, α), is bijective. By condition (ii) once again, we have χ Hom(Ran(χ (α) ), J A ). It follows that α| A is a Δ-isomorphism from A onto Ran(α). erefore, by eorem 4, we obtain that α is regular. Conversely, suppose that α is regular. en, by eorem 4, we get that Ran(α) 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 Ran(α). Since Ran(α) and A are Δ-retracts of X, we have by Corollary 3 that Ran(χ (α) ) and J A are Δ-retracts of I, respectively. Since α| A is a Δ-isomorphism from A onto Ran(α), we have that (iii) holds. And, by Proposition 2, we get that (ii) holds.

□
If End(I) � T(I), then End F (X) becomes T F (X). Hence, by eorem 7, the following corollary is immediately obtained.

Corollary 4. Let α ∈ T F (X). en, α is regular if and only if
there is A⊆X such that each of the following statements holds true: Note that by considering T F (X) 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 ∈ T F (X) can be deduced from eorem 4 as well.
is was provided by Huisheng [9]. e author obtained for any α ∈ T F (X) that α is regular if and only if for each i ∈ I, there is j ∈ I such that Y i ∩ Ran(α)⊆α(Y j ). Here, we get another characterization of the regularity of elements of T F (X) in terms of the character. e following three corollaries are immediately obtained from eorem 7 as well.

Corollary 5. Suppose that I is a topological space, and let End(I) � S(I).
Let α ∈ End F (X). en, α is regular if and only if there is A⊆X such that each of the following statements holds true:

Corollary 6.
Suppose that I is a T 1 -space, and let End(I) � Γ(I). Let α ∈ End F (X). en, α is regular if and only if there is A⊆X such that each of the following statements holds true: International Journal of Mathematics and Mathematical Sciences

Corollary 7. Suppose that I is a vector space, and let End(I) � L(I).
Let α ∈ End F (X). en, α is regular if and only if there is A⊆X such that each of the following statements holds true: We end this section with a discussion on the regularity of the quotient semigroup End F (X)/χ, where χ is the congruence relation on End F (X) defined by By virtue of Corollary 2, we immediately get that the semigroup End F (X)/χ is regular if and only if the semigroup End(I) is regular.

A Subsemigroup of End F (X) and Its Regularity
In this section, we define a sub-semigroup of End F (X) and study the regularity of that semigroup. Let κ be a cardinal number, and let Lemma 2. Let σ, ρ ∈ M (κ) F (X) and c ∈ T F (σ, ρ). en, cσ ∈ M (κ) F (X).
Since the family ∪ i∈I j Y i : j ∈ E is a partition of X, we get that β is well defined. It is clear that β is an idempotent element of T F (X), and that χ (β) � c ∈ End(I). us, β is an idempotent element of End F (X) yields that Ran(β) is a Δ-retract of X. Moreover, we have α| Ran(β)∩Y i is a bijective function from Ran(β) ∩ Y i onto Ran(α) ∩ Y χ (α) (i) for all i ∈ Ran(χ (β) ). erefore, by eorem 7, we obtain that α is regular. ((2)⇒(1)). Suppose that α is a regular element of End F (X).

Corollary 9.
e semigroup E (1) F (X) is regular if and only if End(I) is regular.

Conclusions e semigroup T (S)
F (X), where S is a sub-semigroup of T(I), 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 T (S) F (X) when S is the semigroup of a Δ-structure on I, which is written as S � End(I). In our study, we obtain that T (S) F (X), which is denoted by End F (X), is the semigroup of a Δ-structure on X. From this, the regularity of elements of End F (X) can generally be explained via eorem 4 established by Magill and Subbiah [13] in 1974. We also obtain a characterization of regular elements of End F (X) in terms of the Δ-structure on I (see eorem 7). From this result, we deduce the regularity of End F (X) when End(I) is one of the following semigroups: the transformation semigroup T(I), the semigroup S(I) of continuous maps on I when I is a topological space, the semigroup Γ(I) of closed maps on I when I is a T 1 -space, and the semigroup L(I) of linear transformations on I when I is a vector space (see . Apart from the regularity of End F (X), we provide a sub-semigroup of End F (X), namely, the semigroup E (1) F (X), whose regularity coincides with that of End(I). 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 End F (X) is the semigroup of a Δ-structure on X, some rough characterizations of Green's relations for regular elements of End F (X) 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 of End F (X) 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 semigroup End F (X) be explained in terms of those in the semigroup End(I)?

Data Availability
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Conflicts of Interest
e authors declare that they have no conflicts of interest.