For any semigroup , we call an element a of a regular element of if there exists an element b of such that aba = a. It is well-known that an element a of a semigroup is regular if and only if there is such that aca = a and cac = c. We denote the set of all regular elements of a semigroup by . A semigroup is said to be regular if every element of is regular, that is, if .

The notion of regularity plays an important role in semigroup theory. Over the years, there have been many people studying the regularity of subsemigroups of the regular semigroup T(X) of functions on a nonempty set X under the composition, called a full transformation semigroup. The following are two simple subsemigroups of T(X) which have widely been investigated or used as bases for building up some other subsemigroups of T(X):

and

where Y is a fixed nonempty subset of X (see [2, 3, 7, 9, 10, 11] for some references). Here are some results on the regularity of T(X, Y ) and T̄(X, Y ) provided in [7] by Nenthein, Youngkhong and Kemprasit, in [3] by Honyam and Sanwong, and in [10] by Sanwong and Sommanee.

Theorem 1.1.([7, Theorem 2.1])

Let X be a nonempty set, and let Y be a nonempty subset of X. Then for any α ∈ T(X, Y ), the following statements are equivalent:

• α ∈ R(T(X, Y ));

• Xα = Y α;

• Y ∩ (xα)α⁻¹ ≠ ∅︀ for all x ∈ X;

• Y ∩ xα⁻¹ ≠ ∅︀ for all x ∈ Xα.

By Thorem 1.1, the following corollary was deduced.

Corollary 1.2.([7, Corollary 2.2])

Let X be a nonempty set, and let Y be a nonempty subset of X. Then T(X, Y ) is regular if and only if Y = X or |Y | = 1.

Remark 1.3

In the proof of the implication (4) ⇒ (1) in Theorem 1.1, the authors defined a function β in T(X, Y ) under the assumption that Y ∩ xα⁻¹ ≠ ∅︀ for all x ∈ Xα to make α regular by xβ = y_x if x ∈ Xα and xβ = c otherwise, where c is a fixed element of Y and for each x ∈ Xα, y_x is a fixed element of Y ∩ xα⁻¹.

Theorem 1.4. ([7, Theorem 2.3])

Let X be a nonempty set, and let Y be a nonempty subset of X. Then for any α ∈ T̄(X, Y ), the following statements are equivalent:

• α ∈ R(T̄(X, Y ));

• Xα ∩ Y = Y α;

• Y ∩ (xα)α⁻¹ ≠ ∅︀ for all x ∈ Y α⁻¹;

• Y ∩ xα⁻¹ ≠ ∅︀ for all x ∈ Xα ∩ Y.

By Thorem 1.4, the following corollary was obtained.

Corollary 1.5. ([7, Corollary 2.4])

Let X be a nonempty set, and let Y be a nonempty subset of X. Then T̄(X, Y ) is regular if and only if Y = X or |Y | = 1.

Theorem 1.6. ([10, Theorem 2.4])

Let X be a nonempty set, and let Y be a nonempty subset of X. Then R(T(X, Y )) is the largest regular subsemigroup of T(X, Y ).

Theorem 1.7. ([3, Lemma 1])

Let X be a nonempty set, and let Y be a nonempty subset of X. Then R(T̄(X, Y )) is a subsemigroup of T̄(X, Y ) if and only if Y = X or |Y | = 1. In this trivial situation, R(T̄(X, Y )) = T̄(X, Y ) is regular.

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. Each of the two partitions {X} and {{x} : x ∈ X} is called a trivial partition of X.

Throughout the remainder of this paper, let X be a nonempty set, and let ℱ = {Y_i : i ∈ I} be a partition of X, which are arbitrarily fixed. Let

It is clear that T_ℱ(X) is a subsemigroup of the full transformation semigroup T(X). Note that T_ℱ(X) is exactly the semigroup of transformations preserving the equivalence ℰ induced by the partition ℱ (see [4] for more detials). There have been several works on the semigroup of transformations preserving an equivalence (see [1, 5, 6] for some references). For each α ∈ T_ℱ(X), let χ^(α) : I → I be defined by iχ^(α) = j if and only if Y_iα ⊆ Y_j. By the defintion of a partition, we see that χ^(α) is well-defined, that is, χ^(α) ∈ T(I). For each α ∈ T_ℱ(X), we call the function χ^(α) the character of α with respect to ℱ. In addition to the set X and the partition ℱ of X, let J be an arbitrarily fixed nonempty subset of I. Let

It is clear that

The set TF(J)(X), which is indeed a subsemigroup of T_ℱ(X), as well as the notion of character were first introduced in [8] by Purisang and Rakbud. In that paper, the authors studied the regularity of the semigroup TF(J)(X) and some other semigroups defined via the notion of character. We summarize some of their results as follows.

Proposition 1.8. ([8, Proposition 2.2])

Let Y = ∪_j∈J Y_j. Then the following statementshold:

• TF(J)(X)=T(X,Y)if and only if |J| = 1 or ℱ = {{x} : x ∈ X}.

• TF(J)(X)=T(X)if and only if J = I or ℱ is trivial.

Lemma 1.9. ([8, Lemma 2.3])

For every α, β∈TF(J)(X), χ^(αβ) = χ^(α)χ^(β).

By using the notion of character, the authors defined two congruence relations χ and χ̃ on TF(J)(X) as follows:

And then they studied the regularity of the quotient semigroups TF(J)(X)/χ and TF(J)(X)/χ˜. The following are what they obtained.

Theorem 1.10. ([8, Theorem 2.4])

For eachα∈TF(J)(X), let [α] and[α]˜be the equivalence classes of α under the equivalence relations χ and χ̃ respectively. Then the following statements hold:

• TF(J)(X)/χ≅T(I,J)by the isomorphism [α] ↦ χ^(α).

• TF(J)(X)/χ˜≅T(J)by the isomorphism[α]˜↦χ(α)∣J.

By Corollary 1.2 and Theorem 1.10, the following corollary was obtained.

Corollary 1.11. ([8, Corollary 2.5])

The following statements hold:

• The three statements (a), (b) and (c) below are all equivalent:

  • the quotient semigroupTF(J)(X)/χis regular;

  • the semigroup T(I, J) is regular;

  • J = I or |J| = 1.

• The quotient semigroup T_ℱ(X)/χ, which is exactlyTF(I)(X)/χ, is regular.

• The quotient semigroupTF(J)(X)/χ˜is regular.

In [8], the regularity of the semigroup TF(J)(X) was obtained as follows.

Theorem 1.12. ([8, Theorem 2.6])

The semigroupTF(J)(X)is regular if and only if∣TF(J)(X)∣=1orTF(J)(X)=T(X).

Note that, from Theorem 1.12, we immediately have that T_ℱ(X) is regular if and only if ℱ is trivial. This can also be deduced from Proposition 2.4 of Huisheng [5].

It is clear 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 the full transformation semigroup T(I). The regularity of the semigroup [α], in the case where α is an idempotent element of T(I), was also studied in [8]. In [8] as well, some other subsemigroups of T_ℱ(X) were defined by using the notion of 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. The regularity of each of these three semigroups was also studied.

Observe that the semigroups TF(J)(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 character as follows: For every subsemigroup of T(I), let

By Lemma 1.9, we see that TF(S)(X) is a subsemigroup of T_ℱ(X). And, furthermore, Lemma 1.9 also implies that for every subsemigroup ℋ of T_ℱ(X), ℋ is necessarily of the form TF(S)(X) for some subsemigroup of T(I), in fact, . We state this pleasant result in the following theorem.

Theorem 1.13

For every ℋ ⊆ T_ℱ(X), ℋ is a subsemigroup of T_ℱ(X) if and only if there is a subsemigroupof T(I) such thatH=TF(S)(X). In this situation, .

Let be a subsemigroup of T(I). Then by considering the congruence relation χ on T_ℱ(X) restricted to TF(S)(X), we have the quotient semigroup TF(S)(X)/χ. It is clear that TF(S)(X)/χ={[α]:α∈TF(S)(X)}, and that TF(S)(X)/χ is a subsemigroup of T_ℱ(X)/χ. Analogously to Theorem 1.10(1), the following result is obtained.

Theorem 1.14

TF(S)(X)/χ≅Sby the isomorpism [α] ↦ χ^(α).

Immediately from Theorem 1.14, we have the following corollary.

Corollary 1.15

The quotient semigroupTF(S)(X)/χis regular if and only if the semigroupis regular.

We note here that the regularity of may not imply TF(S)(X) to be regular. For example, when is exactly the regular semigroup T(I), we have that TF(S)(X)=TF(X) which is regular only when ℱ is trivial. By making use of the notion of character, we can define a subset of TF(S)(X) as follows: Let

By Lemma 1.9, we have that R (TF(S)(X))⊆Rw (TF(S)(X)). And obviously, if TF(S)(X) is regular, then R (TF(S)(X))=TF(S)(X)=Rw (TF(S)(X)). It is easy to see that the set Rw (TF(S)(X)) is a subsemigroup of TF(S)(X) if and only if is a subsemigroup of . And in this situation, we have Rw (TF(S)(X))=TF(R(S))(X).

From the inclusion R (TF(S)(X))⊆Rw (TF(S)(X)) and the defintion of the set Rw (TF(S)(X)), it makes perfect sense to call every α∈Rw (TF(S)(X)) a weakly regular transformation with respect to, or simply an -weakly-regular transformation. By Theorem 1.14, we have for any α∈TF(S)(X) that α is an -weakly-regular transformation if and only if the equivalence class [α] of α under the congruence relation χ is a regular element of the quotient semigroup TF(S)(X)/χ. This yileds, in the case where is a subsemigroup of , that Rw (TF(S)(X))/χ=R (TF(S)(X)/χ). Note that for any semigroup and a subsemigroup of , if , then . From this elementary fact, we immediately obtain that if is a subsemigroup of , then R (Rw (TF(S)(X)))=R (TF(S)(X)).

The aim of this paper is to investigate the regularity of Rw (TF(S)(X)) for a certain subsemigroup of T(I) with a subsemigroup of .

By virtue of Theorem 1.6, we have that the set Rw (TF(J)(X)) is a subsemigroup of TF(J)(X). This section is devoted to studying the regualrity of this semigroup. From a result of Huisheng [5], the regularity of elements of the semigroup T_ℱ(X) can immediately be deduced as follows.

Proposition 2.1

An element α of the semigroup T_ℱ(X) is regualr if and only if for every i ∈ I, there exists j ∈ I such that Y_i ∩ Xα ⊆ Y_jα.

The result in Proposition 2.1 can straightforwardly be generalized to the semigroup TF(J)(X) as follows.

Theorem 2.2

For everyα∈TF(J)(X), α is regualr if and only if for every i ∈ I, there exists j ∈ J such that Y_i ∩ Xα ⊆ Y_jα.

Theorem 2.3

R (TF(J)(X))=Rw (TF(J)(X))if and only if |Y_j | = 1 for all j ∈ J.

Corollary 2.4

The semigroupRw (TF(J)(X))is regular if and only if |Y_j | = 1 for all j ∈ J.

By the definition of TF(J)(X):=TF(T(I,J))(X), defined relatively to the semigroup T(I, J), and the regularity of R(T(I, J)), we expect to have that R (TF(J)(X)) is a regular subsemigroup of TF(J)(X). But we find that, in general, the set R (TF(J)(X)) is not a subsemigroup of TF(J)(X). This is affirmed by the following proposition.

Proposition 2.5

If |J| > 1 and there is j ∈ J such that |Y_j | > 1, then the setR (TF(J)(X))is not a subsemigroup ofTF(J)(X).

Corollary 2.6

The setR (TF(J)(X))is a subsemigroup ofTF(J)(X)if and only if |J| = 1 or |Y_j | = 1 for all j ∈ J. In this circumstance, R (TF(J)(X))=R(T(X,Yj))if |J| = 1 with J = {j}, andR (TF(J)(X))=Rw (TF(J)(X))if |Y_j | = 1 for all j ∈ J. Furthermore, we have in each case that the semigroupR (TF(J)(X))is regular.