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):
$$T(X,Y)=\{\alpha \in T(X):X\alpha \subseteq Y\}$$and
$$\overline{T}(X,Y)=\{\alpha \in T(X):Y\alpha \subseteq Y\},$$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
$${T}_{\mathcal{F}}(X)=\{\alpha \in T(X):\forall i\in I\exists j\in I,{Y}_{i}\alpha \subseteq {Y}_{j}\}.$$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_{i}α ⊆ 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
$${T}_{\mathcal{F}}^{(J)}(X)=\left\{\alpha \in T(X):{\chi}^{(\alpha )}\in T(I,J)\right\}.$$It is clear that
$${T}_{\mathcal{F}}^{(J)}(X)=\{\alpha \in T(X):\forall i\in I\exists j\in J,{Y}_{i}\alpha \subseteq {Y}_{j}\}.$$The set ${T}_{\mathcal{F}}^{(J)}(X)$, 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 ${T}_{\mathcal{F}}^{(J)}(X)$ 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:
${T}_{\mathcal{F}}^{(J)}(X)=T(X,Y)$if and only if |J| = 1 or ℱ = {{x} : x ∈ X}.
${T}_{\mathcal{F}}^{(J)}(X)=T(X)$if and only if J = I or ℱ is trivial.
Lemma 1.9. ([8, Lemma 2.3])
For every α, $\beta \in {T}_{\mathcal{F}}^{(J)}(X)$, χ^{(}^{αβ}^{)} = χ^{(}^{α}^{)}χ^{(}^{β}^{)}.
By using the notion of character, the authors defined two congruence relations χ and χ̃ on ${T}_{\mathcal{F}}^{(J)}(X)$ as follows:
$$\begin{array}{c}(\alpha ,\beta )\in \chi \iff {\chi}^{(\alpha )}={\chi}^{(\beta )},\\ (\alpha ,\beta )\in \tilde{\chi}\iff {\chi}^{(\alpha )}{\mid}_{J}={\chi}^{(\beta )}{\mid}_{J}.\end{array}$$And then they studied the regularity of the quotient semigroups ${T}_{\mathcal{F}}^{(J)}(X)/\chi $ and ${T}_{\mathcal{F}}^{(J)}(X)/\tilde{\chi}$. The following are what they obtained.
Theorem 1.10. ([8, Theorem 2.4])
For each$\alpha \in {T}_{\mathcal{F}}^{(J)}(X)$, let [α] and$\tilde{[\alpha ]}$be the equivalence classes of α under the equivalence relations χ and χ̃ respectively. Then the following statements hold:
${T}_{\mathcal{F}}^{(J)}(X)/\chi \cong T(I,J)$by the isomorphism [α] ↦ χ^{(}^{α}^{)}.
${T}_{\mathcal{F}}^{(J)}(X)/\tilde{\chi}\cong T(J)$by the isomorphism$\tilde{[\alpha ]}\mapsto {\chi}^{(\alpha )}{\mid}_{J}$.
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 semigroup${T}_{\mathcal{F}}^{(J)}(X)/\chi $is regular;
the semigroup T(I, J) is regular;
J = I or |J| = 1.
The quotient semigroup T_{ℱ}(X)/χ, which is exactly${T}_{\mathcal{F}}^{(I)}(X)/\chi $, is regular.
The quotient semigroup${T}_{\mathcal{F}}^{(J)}(X)/\tilde{\chi}$is regular.
In [8], the regularity of the semigroup ${T}_{\mathcal{F}}^{(J)}(X)$ was obtained as follows.
Theorem 1.12. ([8, Theorem 2.6])
The semigroup${T}_{\mathcal{F}}^{(J)}(X)$is regular if and only if$\mid {T}_{\mathcal{F}}^{(J)}(X)\mid =1$or${T}_{\mathcal{F}}^{(J)}(X)=T(X)$.
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 ${T}_{\mathcal{F}}^{(J)}(X)$, [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
$${T}_{\mathcal{F}}^{(\mathcal{S})}(X)=\left\{\alpha \in {T}_{\mathcal{F}}(X):{\chi}^{(\alpha )}\in \mathcal{S}\right\}.$$By Lemma 1.9, we see that ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)$ 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 ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)$ 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 that$\mathcal{H}={T}_{\mathcal{F}}^{(\mathcal{S})}(X)$. In this situation, .
Let be a subsemigroup of T(I). Then by considering the congruence relation χ on T_{ℱ}(X) restricted to ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)$, we have the quotient semigroup ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)/\chi $. It is clear that ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)/\chi =\left\{[\alpha ]:\alpha \in {T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right\}$, and that ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)/\chi $ is a subsemigroup of T_{ℱ}(X)/χ. Analogously to Theorem 1.10(1), the following result is obtained.
Theorem 1.14
${T}_{\mathcal{F}}^{(\mathcal{S})}(X)/\chi \cong \mathcal{S}$by the isomorpism [α] ↦ χ^{(}^{α}^{)}.
Immediately from Theorem 1.14, we have the following corollary.
Corollary 1.15
The quotient semigroup${T}_{\mathcal{F}}^{(\mathcal{S})}(X)/\chi $is regular if and only if the semigroupis regular.
We note here that the regularity of may not imply ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)$ to be regular. For example, when is exactly the regular semigroup T(I), we have that ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)={T}_{\mathcal{F}}(X)$ which is regular only when ℱ is trivial. By making use of the notion of character, we can define a subset of ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)$ as follows: Let
$${R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)=\left\{\alpha \in {T}_{\mathcal{F}}^{(\mathcal{S})}(X):{\chi}^{(\alpha )}\in R(\mathcal{S})\right\}.$$By Lemma 1.9, we have that $R\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)\subseteq {R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)$. And obviously, if ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)$ is regular, then $R\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)={T}_{\mathcal{F}}^{(\mathcal{S})}(X)={R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)$. It is easy to see that the set ${R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)$ is a subsemigroup of ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)$ if and only if is a subsemigroup of . And in this situation, we have ${R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)={T}_{\mathcal{F}}^{(R(\mathcal{S}))}(X)$.
From the inclusion $R\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)\subseteq {R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)$ and the defintion of the set ${R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)$, it makes perfect sense to call every $\alpha \in {R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)$ a weakly regular transformation with respect to, or simply an -weakly-regular transformation. By Theorem 1.14, we have for any $\alpha \in {T}_{\mathcal{F}}^{(\mathcal{S})}(X)$ 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 ${T}_{\mathcal{F}}^{(\mathcal{S})}(X)/\chi $. This yileds, in the case where is a subsemigroup of , that ${R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)/\chi =R\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)/\chi \right)$. 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 $R\hspace{0.17em}\left({R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)\right)=R\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)$.
The aim of this paper is to investigate the regularity of ${R}_{w}\hspace{0.17em}\left({T}_{\mathcal{F}}^{(\mathcal{S})}(X)\right)$ for a certain subsemigroup of T(I) with a subsemigroup of .