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

Abstract

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

1. Introduction and Preliminaries

For any semigroup , we call an element of a regular element of if there exists an element of such that . A semigroup is said to be regular if every element of is regular. The semigroup of transformations on a nonempty set , denoted by , is a well-known regular semigroup. In [1], Nenthein et al. studied the regularity of the following two sub-semigroups of :where is a fixed nonempty subset of and denotes the range of for all . The authors obtained that the two semigroups are regular if and only or . The regularity of some other sub-semigroups of has been studied by many people (see [2–6] for some recent works).

In this paper, by a partition of a nonempty set , we mean a family of nonempty subsets of such that and for all with . For a given partition of a nonempty set , let

Note that is exactly the semigroup of all transformations on preserving the equivalence relation induced by partition . There have been several works involving transformations preserving an equivalence relation (see [3–5, 7–12] for some references).

Throughout this paper, let be a nonempty set, and let be a partition of , which are arbitrarily fixed. From the definition of a partition, Purisang and Rakbud [11] defined a function associated with each called the character of , by

They also defined and studied the regularity of the following sub-semigroup of :where is an arbitrarily fixed nonempty subset of the index set . We summarize some of their results as follows.

Lemma 1 (see [11], Lemma 2.3). For every , .

By making use of the notion of the character, the following two equivalence relations and on were defined:

Note that, by Lemma 1, they both are congruence relations. The authors studied the regularity of the quotient semigroups and . The following are what they obtained.

Theorem 1 (see [11], Theorem 2.4). The following statements hold:(1)by the isomorphism;(2)by the isomorphism,where for each, anddenote the equivalence classes ofunder the equivalence relationsand, respectively.

Corollary 1 (see [11], Corollary 2.5). The following statements hold.(1)The three statements below are all equivalent:(a)The quotient semigroupis regular;(b)The semigroupis regular;(c)or.(2)The quotient semigroupis regular.(3)The quotient semigroupis regular.

The regularity of the semigroup was obtained as follows.

Theorem 2 (see [11], Theorem 2.6). The semigroupis regular if and only ifor.

It is easy to see that for each , the equivalence class of under the equivalence relation is a sub-semigroup of if and only if is an idempotent element of . In [11], the authors studied the regularity of the semigroup when is an idempotent element of . They also defined some further sub-semigroups of by making use the notion of the character as follows: let , , and be the sets of all elements of whose characters are injective, surjective, and bijective, respectively. Note that, by Lemma 1, the sets , , and are sub-semigroups of . The regularity of each of these three semigroups was also studied.

It was observed by Rakbud [12] that the semigroups , when is idempotent, , , and can simultaneously be generalized by making use of the notion of the character as follows: for every sub-semigroup of , let

Note that, for each , the function defined on each by , where is a fixed element of for all , is an element of whose character is exactly . Hence, , and by Lemma 1, it is a sub-semigroup of .

Let be a sub-semigroup of . Then, by considering the congruence relation on restricted to , we have the quotient semigroup . Obviously, and is a sub-semigroup of . Analogously to Theorem 1, the following result was established.

Theorem 3 (see [12], Theorem 1.14). by the isomorphism defined by.

Immediately from Theorem 3, the following corollary was obtained.

Corollary 2 (see [12], Corollary 1.15). The quotient semigroupis regular if and only if the semigroupis 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 has not been studied in general yet. This will be in our attention here when has a certain property. We are mentioning that is “the semigroup of a -structure”on .

We now refer to the definition of a -structure on a set and some other related ones from [13] by Magill and Subbiah. Let be a family of nonempty subsets of such that . And, let , where is a nonempty set of functions from into for all , with the following properties:(Δ1) is a monoid;(Δ2) for all ;(Δ3)For all , and , ;(Δ4)For all and with and , if and , then and , where denotes the identity function on for any nonempty set .

The pair is called a -structure on , and the monoid is called the semigroup of the-structure. A subset of is called a -retract of if there is an idempotent element of such that . For any , an element of is called a -isomorphism (from onto ) if there is such that and . It is clear for any that an element of is -isomorphism if and only if is bijective and .

In [13], the authors gave some characterizations of regular elements of the semigroup as follows.

Theorem 4 (see [13], Theorem 2.4). Let . Then, the following statements are equivalent:(1)is regular;(2)is a-retract of, and there is a-retractofsuch thatis a-isomorphism fromonto;(3)is a-retract of, and there issuch thatis a-isomorphism fromonto.

It is clear that when is equipped with the -structure , where is the family of all nonempty subsets of , and is the set of all functions from into for all . In this setting, the -retracts of are exactly the nonempty subsets of , and the -isomorphisms are exactly the bijective functions. More interesting semigroups of -structures on were given in [13] as follows:(i)The semigroup of all continuous maps on is exactly when is a topological space equipped with the -structure , where is the family of all nonempty subsets of and is the set of all continuous maps from into for all . In this setting, the -retracts of are exactly the -retracts of the topological space , and the -isomorphisms are exactly the homeomorphisms.(ii)The semigroup of all linear transformations on coincides with when is a vector space equipped with the -structure , where is the family of all subspaces of and is the set of all linear transformations from into for all . In this setting, the -retracts of are exactly the subspaces of the vector space , and the -isomorphisms are exactly the vector space isomorphisms.(iii)The semigroup of all closed maps on is exactly when is a -space equipped with the -structure , where is the family of all nonempty closed subsets of and is the set of all closed maps from into for all . In this setting, the -retracts of are exactly the nonempty closed subsets of the topological space , and the -isomorphisms are exactly the homeomorphisms.

The regularity of the semigroups , , and was also deduced via Theorem 4. In addition, we note here that the semigroup of all transformations on preserving an equivalence relation on can be considered as the semigroup of all continuous maps on , where 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 semigroup , introduced in [12], when is the semigroup of a -structure on the index set of the partition . We also define, in this situation, a sub-semigroup of whose regularity coincides with that of the semigroup .

2. The Semigroup and Its Regularity

For any nonempty sets and and partitions and of and , respectively, let be the set of all functions satisfying the condition that for all , there is such that . And, for each , let be defined by

It is easy to verify that the map from the set into the set of all functions from into is surjective. If we have three nonempty sets , , and with partitions , , and of , , and , respectively, we see for any and that and .

For all , letwhere

And, for each , let .

Let be a -structure on the index set of the partition , and let

By the property (Δ1) of the -structure on , we have that , which yields that . Hence, is a submonoid of .

Theorem 5. There is a-structure onsuch that.

Proof. LetSince is a monoid, we have . For any , letThen, by the surjectivity of the map from the set into the set of all functions from into , we have that for all . Note that for all , if and , then and , which yields that . LetThen, from the above explanation, the family is well defined. We will show that is a -structure on . It is easy to see thatThus, the property (Δ1) is satisfied. From the definition of the family , we immediately have that the property (Δ2) holds. Next, we will show that the property (Δ3) is satisfied. Let , and let . We want to show that . Since , we have . Since , we have that . Thus, by the property (Δ3) of the -structure on , we get . It is clear that , and that . Hence, by the membership of in , we obtain that as desired. Finally, we will show that the property (Δ4) holds. Let be such that and . Then, . Thus, by the property (Δ2) of the -structure on , we get that and belong to . Since , we have that . To see this explicitly, let . Then, there is such that , which yields that . Fix . Since , we get that . Therefore, there is such that . Let be such that , and let . Then, ; so , which implies that . Hence, . Similarly, by the inclusion , we obtain . We now have two inclusions and . From these, we get by Lemma 1 that and . By the assumptions that and that , we have and , respectively. And, from the membership of in , we get that for all , there exists such that . Thus, . By the inclusion , we have , which yields that . Similarly, we have that , and that . Suppose that , and that . We want to show that , and that . By the inclusions and , we have, respectively, that and . And, since and , it follows that and , respectively. Therefore, by the property (Δ4) of the -structure on , we obtain that and . These yield, respectively, that and .

From now on, we will consider the -structure on defined in the proof of Theorem 5, and the semigroup of this -structure will be in our attention. By Theorem 4, the regularity of elements of the semigroup can roughly be characterized. To get more precise characterizations, according to Theorem 4, the notions of a -retract and a -isomorphism in the -structure on 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 be a nonempty set and . Then, the following statements are equivalent:(1) is idempotent;(2);(3)There is a partition of and a subset of such that for all and for all .In this situation, the partition and the subset of are unique determined by .

For each nonempty subset of , we see that there is a unique subset of , denoted by , such that the family is a partition of . In particular, we have that is exactly if for some .

Proposition 1. Let, and let. Ifis idempotent and, then there is an idempotent elementofsuch thatand.

Proof. Suppose that is idempotent, and that . Then, by Theorem 6, there is a partition of and a subset of such that and for all . From this, we have that . For each , let , where , be idempotent such that for all and . And, finally, let be defined by for all . Since is a partition of , it follows that is well defined. It is clear by the way of defining that with . It is also clear that is idempotent with .

From Proposition 1, the following corollary is easily obtained.

Corollary 3. Let. Then,is a-retract ofif and only ifis a-retract of.

Proof. Suppose that is a -retract of . Then, there is an idempotent element of such that , which yields that . Since is idempotent, we have by Lemma 1 that is an idempotent element of . Hence, is a -retract of . Conversely, suppose that is a -retract of . Then, there is an idempotent element of such that . Thus, by Proposition 1, the set is a -retract of .

Proposition 2. Let, and let. Ifis a-isomorphism, thenis a-isomorphism.

Proof. Let . Then, . Suppose that is a -isomorphism. Then, there exists such that and . By the membership of in , we have . Since , we get that . Similarly, since , we get . Thus, is a -isomorphism.

In the following theorem, we provide a characterization of the regularity of elements of in terms of the -retract and the -isomorphism of .

Theorem 7. Let. Then,is regular if and only ifis a-retract of, and there issuch that each of the following statements holds true:(i)is-retract of;(ii)is a-isomorphism fromonto;(iii)is a bijective function fromontofor all.

Proof. Suppose that is a -retract of , and that there exists a subset of such that each of the following statements holds true:(i) is -retract of ;(ii) is a -isomorphism from onto ;(iii) is a bijective function from onto for all .Since is a -retract of , we have by Corollary 3 that is a -retract of . And, since is a -retract of , there is an idempotent element of such that . Thus, by Proposition 1, there is an idempotent element of such that and . This yields that is a -retract of . By condition (ii), we have that is a bijective function from onto . Thus, by condition , we get that , which is an element of , is bijective. By condition (ii) once again, we have and . It follows that is a -isomorphism from onto . Therefore, by Theorem 4, we obtain that is regular. Conversely, suppose that is regular. Then, by Theorem 4, we get that is a -retract of , and that there is a -retract of such that is a -isomorphism from onto . Since and are -retracts of , we have by Corollary 3 that and are -retracts of , respectively. Since is a -isomorphism from onto , we have that (iii) holds. And, by Proposition 2, we get that (ii) holds.

If , then becomes . Hence, by Theorem 7, the following corollary is immediately obtained.

Corollary 4. Let. Then,is regular if and only if there issuch that each of the following statements holds true:(i)is a bijective function fromonto;(ii)is a bijective function fromontofor all.

Note that by considering as the semigroup of all continuous maps on , where is equipped with the topology having the family of all equivalence classes as a base, the regularity of can be deduced from Theorem 4 as well. This was provided by Huisheng [9]. The author obtained for any that is regular if and only if for each , there is such that . Here, we get another characterization of the regularity of elements of in terms of the character.

The following three corollaries are immediately obtained from Theorem 7 as well.

Corollary 5. Suppose thatis a topological space, and let. Let. Then,is regular if and only if there issuch that each of the following statements holds true:(i)is a homeomorphism fromonto;(ii)is a bijective function fromontofor all.

Corollary 6. Suppose thatis a-space, and let. Let. Then,is regular if and only if there issuch that each of the following statements holds true:(i)is a closed subset of ;(ii)is a homeomorphism fromonto;(iii)is a bijective function fromontofor all.

Corollary 7. Suppose thatis a vector space, and let. Let. Then,is regular if and only if there issuch that each of the following statements holds true:(i)is a subspace of ;(ii)is an isomorphism fromonto;(iii)is a bijective function fromontofor all.

We end this section with a discussion on the regularity of the quotient semigroup , where is the congruence relation on defined by

By virtue of Corollary 2, we immediately get that the semigroup is regular if and only if the semigroup is regular.

3. A Subsemigroup of and Its Regularity

In this section, we define a sub-semigroup of and study the regularity of that semigroup. Let be a cardinal number, and let

Lemma 2. Letand. Then,.

Proof. Let . We now want to show that . It is clear that . Next, let . Since , we have . Thus, , and there is such that , which yields that . Since , there is such that , which implies that . Thus, . From this, we obtain that . Since , we have that . And, since , it follows that . Thus, .

Corollary 8. The setis a sub-semigroup of.

Proof. Let . Then, . Thus, by Lemma 2, we get that .

Let

Note that for each , there is such that . Such a function can easily be defined as follows: for each , let be a fixed element of and let be defined by for all . Thus, by the note, we have that . Furthermore, it is a sub-semigroup of .

Theorem 8. Let. Then, the following statements are equivalent:(1)is a regular element of;(2)is a regular element of;(3)is a regular element of.

Proof. . Suppose that is a regular element of . Then, by the regularity of in , we get by Theorem 4 that is a -retract of , and that there is an idempotent element of such that is a -isomorphism from onto . Since is an idempotent element of , we have by Theorem 6 that there is a partition of and a subset of with for all such that for all . So, . Since is a -isomorphism from onto , we have that is bijective, and that . So, with . Since , we have that for all . Let for all . Then, . For each , fix . Let be defined by for all . Since the family is a partition of , we get that is well defined. It is clear that is an idempotent element of , and that . Thus, is an idempotent element of yields that is a -retract of . Moreover, we have is a bijective function from onto for all . Therefore, by Theorem 7, we obtain that is regular.
. Suppose that is a regular element of . Then, by Theorem 4 and Proposition 1, there exists an idempotent element of such that and . Since , it follows that . By Theorem 7, there exists an idempotent element of such that(a) is a -isomorphism from onto ;(b) is a bijective function from onto for all .From (a) and (b), we have that and , respectively. Let . Then, by the property (Δ3), we have that . And, by Lemma 2, we immediately obtain that . Thus, . Finally, we will show that . Let . Then, . Hence, there is such that . And, since is idempotent, it follows that . Therefore, , and hence is a regular element of .
. It follows directly from Lemma 1.

Corollary 9. The semigroupis regular if and only ifis regular.

4. Conclusions

The semigroup , where is a sub-semigroup of , 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 when is the semigroup of a -structure on , which is written as . In our study, we obtain that , which is denoted by , is the semigroup of a -structure on . From this, the regularity of elements of can generally be explained via Theorem 4 established by Magill and Subbiah [13] in 1974. We also obtain a characterization of regular elements of in terms of the -structure on (see Theorem 7). From this result, we deduce the regularity of when is one of the following semigroups: the transformation semigroup , the semigroup of continuous maps on when is a topological space, the semigroup of closed maps on when is a -space, and the semigroup of linear transformations on when is a vector space (see Corollaries 4–7). Apart from the regularity of , we provide a sub-semigroup of , namely, the semigroup , whose regularity coincides with that of . In [13], Magill and Subbiah also generally gave some characterizations of Green’s relations for regular elements of the semigroup of a -structure. Since our semigroup is the semigroup of a -structure on , some rough characterizations of Green’s relations for regular elements of 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 be characterized more deeply in terms of the -structure on ?(2)Can other notions such as the ideal, the rank, the left regularity, and the right regularity in the semigroup be explained in terms of those in the semigroup ?

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was supported by the Faculty of Science, Silpakorn University, under the grant no. SRF-JRG-2561-06. The authors are grateful to the Faculty of Science, Silpakorn University, for the financial support.