Abstract

Let be the full transformation semigroup on a set . For a fixed nonempty subset of a set , let be the semigroup consisting of all full transformations from into . In a paper published in 2008, Sanwong and Sommanee proved that the set is the largest regular subsemigroup of . In this paper, we describe the maximal inverse subsemigroups of and completely determine all the maximal regular subsemigroups of its ideals.

1. Introduction

Let be the set of all full transformations from a nonempty set into itself. It is well-known that is a regular semigroup under composition of functions; see [1, p. 63]. Moreover, every semigroup can be embedded in for some nonempty set ; see [1, p. 7]. In [2, 3], Schein posed the problem to determine all maximal inverse subsemigroups of ; this problem is still unsolved. In 1976, Nichols [4] characterized one class of maximal inverse subsemigroup of . Later in 1978, Reilly [5] has generalized Nichols construction and obtained a much wider class of maximal inverse subsemigroups of . In 1999, Yang [6] described all of the maximal inverse subsemigroups of the finite symmetric inverse semigroup. Later in 2001, Yang [7] obtained the maximal subsemigroups of the finite singular transformation semigroups. In 2002, You [8] determined all the maximal regular subsemigroups of all ideals of the finite full transformation semigroup. In 2004, H. B. Yang and X. L. Yang [9] completely described the maximal subsemigroups of ideals of the finite full transformation semigroup. In 2014, Zhao et al. [10] showed that any maximal regular subsemigroup of ideals of the finite full transformation semigroup is idempotent generated. After that in 2015, East et al. [11] classified the maximal subsemigroups of when is an infinite set containing certain subgroups of the symmetric group on .

Let be a fixed nonempty subset of and let be the subsemigroup of of all elements with ranges contained in . In 1975, Symons [12] introduced and studied the semigroup . He described all the automorphisms of and also determined the isomorphism theorem for two semigroups of type . In 2005, Nenthein et al. [13] characterized the regular elements of . In general, is not a regular semigroup. In 2008, Sanwong and Sommanee [14] definedand showed that is the largest regular subsemigroup of . Obviously, when . Hence, we may regard as a generalization of . The authors also characterized Green’s relations on and gave one class of maximal inverse subsemigroups of . Later in 2009, Sanwong et al. [15] described all maximal and minimal congruences on . In 2011, Mendes-Gonçalves and Sullivan [16] obtained all the ideals of . In the same year, Sanwong [17] described Green’s relations and ideals and all maximal regular subsemigroups of . The author also proved that every regular semigroup can be embedded in . After that, in 2013, Sommanee and Sanwong [18] computed the rank of when is a finite set. They also obtained the rank and the idempotent rank of its ideals when is finite. In 2014, Fernandes and Sanwong [19] calculated the rank of when is finite. In 2016, L. Sun and J. Sun [20] characterised the natural partial order on . In the same year, Tinpun and Koppitz [21] determined the relative rank of modulo the semigroup of all extensions of the bijections on when is finite.

Let be a chain and the full order-preserving transformation semigroup on . In 2011, Dimitrova and Koppitz [22] characterized the maximal regular subsemigroups of ideals of when is a finite chain. In 2015, Sommanee and Sanwong [23] investigated the regularity and Green’s relations of the order-preserving transformation semigroupThey also proved that is idempotent generated when is finite.

Let be a vector space and let denote the linear transformation semigroup from into . For a fixed subspace of , let be the semigroup consisting of all linear transformations from into . Recently in 2017, Sommanee and Sangkhanan [24] determined all the maximal regular subsemigroups of the semigroupwhen is a finite dimensional subspace of over a finite field.

In this paper, we describe the maximal inverse subsemigroups of and completely determine all the maximal regular subsemigroups of its ideals.

2. Preliminaries and Notations

For all undefined notions, the reader is referred to [1].

For any set , means the cardinality of the set . If is a subset of a semigroup , then denotes the subsemigroup of generated by . An element of a semigroup is called idempotent if . If is a subset of a semigroup , then denotes the set of all idempotents in the set . An element of a semigroup is called regular if there exists such that . The semigroup is called regular if all its elements are regular. If is an element of a semigroup , we say that is an inverse of if and . We denote the set of inverses of an element by . A semigroup is called an inverse semigroup if every element in has a unique inverse in ; it is equivalent to being regular and idempotent elements commute. A proper (regular, inverse) subsemigroup of a semigroup is a maximal (regular, inverse) subsemigroup of if whenever for some a (regular, inverse) subsemigroup of , then or .

The Green’s relations , , , , and on a semigroup are defined as follows. For ,(1) if and only if ;(2) if and only if ;(3) if and only if ;(4) and .

For each , we denote -class, -class, -class, -class, and -class containing by , and , respectively.

Lemma 1 (see [5, Lemma  1]). Let be a maximal inverse subsemigroup of a semigroup . If contains a minimum idempotent , then .

Lemma 2. Let be any inverse semigroup. If is a minimum idempotent in , then for all .

Proof. Assume that is a minimum idempotent in . Then for all ; that is, . Let . Then and .

Throughout the paper we assume that is a nonempty subset of a set . Let , where denotes the image of . DefineIt is known that is the largest regular subsemigroup of .

If and , then the image of under is written as . The set of all inverse images of under is denoted by .

Lemma 3 (see [17, Lemma  1]). Let . Then if and only if for all .

In [17], the author gave a complete description of Green’s relations on as the following lemma.

Lemma 4. Let . Then(1) if and only if ;(2) if and only if , where ;(3) if and only if ;(4). In addition, if and only if and .

Lemma 5. is an inverse semigroup if and only if .

Proof. If , then and so it is inverse. Conversely, assume that is an inverse semigroup. Let and be elements in . Then there exist constant maps and with ranges and , respectively. Thus they are idempotents in such that ; it follows that . Hence, .

Proposition 6. is a maximal inverse subsemigroup of if and only if is a maximal inverse subsemigroup of .

Proof. One direction is clear. Indeed, each inverse subsemigroup of is contained in , the largest regular subsemigroup of . Conversely, assume that is a maximal inverse subsemigroup of . Then is an inverse subsemigroup of . Let be an inverse subsemigroup of such that . Since is the largest regular subsemigroup of , it follows that . We get that or by the maximality of in . If , then is an inverse semigroup and so by Lemma 5; this implies that . Therefore, is a maximal inverse subsemigroup of .

For each , letThe authors [14] showed that is a maximal inverse subsemigroup of . Thus by Proposition 6, we have that is also a maximal inverse subsemigroup of .

3. Maximal Inverse Subsemigroups

In this section, we write for the permutation group on a set .

Let be any idempotent in . In 1978, Reilly defined to be the set of those elements which satisfy the following three conditions:(1)The restriction of to is a permutation of ; that is, .(2)For , if , then .(3).

Lemma 7 (see [5, Theorem  2]). Let be an idempotent in and let be defined as above. Then is a maximal inverse subsemigroup of with minimum idempotent .

Notice that if is a constant map with range , then , wheresee [4, 5] for more details.

Now, let be any idempotent in and defineSince , we obtain and so is a subsemigroup of . We note that if , then and .

Here, we aim to prove that is the maximal inverse subsemigroups of with minimum idempotent .

To prove our main result we need the following five lemmas.

Lemma 8. Let be any idempotent in and . Then if and only if(1),(2)for , if , then ,(3).

Proof. One direction is clear. Conversely, assume that conditions (1), (2), and (3) hold. We prove that . Suppose that and . Since , it follows that for some . By condition (2), we get that . Thus and so .

Lemma 9. Let be any idempotent in and . Then condition (2) in Lemma 8 is equivalent to the statement

Proof. Assume that, for , if , then . We prove . Let for some . Since , there exists such that . Thus . If , then by the assumption. Whence , this is a contradiction. Hence ; that is, . Next, we prove is injective on . Let be such that . Then and . Since , we obtain by our assumption.
Conversely, suppose that and is injective on . Let and . To show that , we let . Then ; this implies that . Since , we obtain . Thus and so since is injective on .

Lemma 10. Let be any element in and let be an idempotent in . If there is such that and , then .

Proof. Assume that the conditions hold. Sinceit follows that ; that is, maps onto . We prove that is injective. Let be such that . ThenHence is bijective.

Lemma 11. Let be an idempotent in . Then is an inverse subsemigroup of with minimum idempotent .

Proof. Let be any element in . Then . Since is an inverse semigroup, there exists such that and . We prove . Since , we obtain ; that is, . To show , we let for some . Since , by Lemma 9. If , then . It follows that ; this is a contradiction. Hence, and so . Thus, . Whence is a regular subsemigroup of , it follows from Lemma 7 that is an inverse subsemigroup of and is the minimum idempotent in .

Lemma 12. Let be an idempotent in . If is any inverse subsemigroup of such that , then .

Proof. Assume that is an inverse subsemigroup of containing . Let be any idempotent in . Since are idempotents in , we get . We prove that . Suppose that . Then , so there exists . We see that and . Now, we have such that . Let be such that and . We define byWe see that and ; we obtain . We prove . Let be such that . Then ; that is, . Thus since is injective on . Since , is surjective on . Hence, . Since for all , whence condition (2) in Lemma 8 holds. To prove . Let . Since and is an idempotent, we get that So, . By Lemma 11, we have as an inverse subsemigroup of . Then there exists and . Since , we obtain . It is clear that is an idempotent in . We see thatSince , this implies that which is a contradiction. Therefore, . Hence and so . It is easy to see that , the identity map on , which implies that . Thus, conditions (1) and (3) in Lemma 8 are satisfied for .
In order to verify condition (2) in Lemma 8 for to be an element of , let be such that . We show that . Let . Then . Suppose that . Assume that . We define byObviously, and it is easy to verify that for all . Hence, and . It is easy to show that is injective, whence . We prove . Let for some . If , then . But if , then . Thus, and so . We prove satisfies the condition in Lemma 9. By the definition of , we have . Let . Then and , which implies that . So, if , then . Hence, is injective on and . Next, we prove is an idempotent. For , . If , then . Thus, . Since and are idempotents in , . However,a contradiction. Hence, . Define bySimilarly, by using the same argument as in the proof above for , we can show that is an idempotent in . Since , we obtain . We see thatand this is a contradiction. Thus, and so condition (2) in Lemma 8 is also satisfied. Whence , therefore, as required.

Now, we are ready to prove our main result (Theorem 13).

Theorem 13. Let be an idempotent in . Then is a maximal inverse subsemigroup of with minimum idempotent .

Proof. From Lemma 11, is an inverse subsemigroup of with minimum idempotent . To prove the maximality, let be any inverse semigroup of such that . Then by Lemma 12, we have , which implies that is the minimum idempotent in . We prove by letting . Since , we obtain and by Lemma 2. And since and are idempotents in , and . Then by Lemma 10, we get that . Thus, conditions (1) and (3) in Lemma 8 are satisfied for to be an element of . Next, we prove that satisfies condition (2) in Lemma 8. Let be such that . Let . Then ; this implies that . Now, we have and . Since , we obtain that by using condition (2) in Lemma 8 for . Hence, and so . Therefore, and is a maximal inverse subsemigroup of .

Corollary 14. Let be an idempotent in . Then is a maximal inverse subsemigroup of .

Proof. It follows directly from Theorem 13 and Proposition 6.

Remark 15. for all distinct idempotents . In fact, if there exist idempotents and in such that , then and such that and are minimum idempotents of and , respectively. Thus, we get and ; it follows that .

Corollary 16. Let be an idempotent in such that . Then is a maximal inverse subsemigroup of .

Proof. Since is a maximal inverse subsemigroup of with the minimum idempotent , we obtain that by Lemma 1. To show that , let . Then and . Since , it follows that . We prove that . Let such that . Then ; this implies that since is injective. Hence . On the other hand, let . Then . Since and is an idempotent, we obtain . Hence and . Then by Lemma 4, we get that . Thus, . Therefore, is the maximal inverse subsemigroup of .

Notice that if is a constant map with range , then andthat is, . Thus, we get the following corollary which appeared in [14, Theorem  4.3].

Corollary 17. If is a constant map with range , then is a maximal inverse subsemigroup of and .

Next, we count the number of elements in . Let be any idempotent in . We know that ; see [17, Section ]. Thus, if , then by Corollary 16. Hence, when . In what follows, we assume that .

Recall that the number of ways that objects can be chosen from distinct objects written as is given byLet and be (possibly empty) sets and let be the set of all partial injective maps from into . We note that contains the empty map. Moreover, if or , then . And if and are finite sets such that and , then we shall write instead of . By a consideration of the cardinality of a domain of partial injective maps, we can verify thatSince , there exists such that . We see that . For each , we defineThen is a disjoint union such that and . Moreover, for some since . LetThen and for all . It is easy to see that

Let be any element in . Then and for some . For each , for some and . If , then . Since , we obtain ; that is, . Since and , we obtain that . If , then by Lemma 9. Since is an idempotent, we get . By condition (2) in Lemma 8, we see that is injective on all elements of which are not mapped into for all . Thus for each , corresponds with an element of ; that is, for ,

We have shown the following. Given . Then is a union of a permutation , a union of partial injections for each , and a union of functions from into for each .

Theorem 18. Let be any idempotent in with . Let and . Then there is a one-to-one correspondence between and .

Proof. For each , we have for some and partial injections for all . We defineby for all . We verify that is injective. Let be such that . Then for some and . Thus, and for all ; it follows that for all . For , for some . We get that . Whence , next, we prove is surjective. Let and . We define which is determined by and for all as follows. For ,We see that, for each , for some . Then and . So, , that is . By the construction of , conditions (1) and (2) in Lemma 8 are satisfied for to be an element of . To prove , we let . If , then , since and is an idempotent. If , then ; it follows that and . Thus, and since . If , then and . And if , then and , since . Hence, . Whence and , therefore, is a bijection.

The following corollary is a straightforward consequence of Theorem 18.

Corollary 19. Let be an idempotent in such that and . Let . For each , we let and . Then

It is known that if , then and . Then by Corollary 19, we have the following corollary.

Corollary 20. Let be an idempotent in such that and . Let . For each , we let and . Then

If is a constant map with range , then we get the following corollary which appeared in [14, Theorem  4.5].

Corollary 21. Let be a finite subset of with and a constant map in with range ; then .

Proof. We note that by Corollary 17. Since , we obtain and there is a unique ; that is, . By Corollary 19, we havewhereTherefore, .

4. Maximal Regular of Ideals

Throughout this section, is a finite set with elements and a nonempty subset of with elements. For convenience, we write for .

For each with , it follows from Lemma 3 that we can writewhere and is a partition of such that for all .

For , we define . Then is a -class of the semigroup . Let , where . Then is an ideal of and it is a regular subsemigroup of ; see [17, Lemma  7].

Recall that the principal factor of is the Rees quotientwhere . It is usually convenient to think of it as , and the product of two elements of is taken to be zero if it falls in . Since is finite and it is not a zero semigroup, we obtain that is a completely 0-simple semigroup; see [18, p. 233] for more details.

We need the following lemmas for proof of our main result.

Lemma 22 (see [1, p. 98]). Let be a completely 0-simple semigroup. If and , then .

Lemma 23 (see [1, Proposition  2.3.7]). Let be elements in -class . Then if and only if contains an idempotent.

Lemma 24. Let be any element in a semigroup and a regular subsemigroup of .(1)If , then contains an idempotent.(2)If , then contains an idempotent.

Proof. (1) Assume that . Then there exists . So, and for some . We get that is an idempotent such that and . Therefore, is an idempotent in .
(2) If there is , then we can write for some and show that is an idempotent in .

Lemma 25 (see [18, Lemma  3.5]). For , .

Lemma 26 (see [18, Lemma  4.2]). If , then can be written as a finite product of idempotents in .

The following fact can be obtained from Lemmas 25 and 26 immediately.

Lemma 27. For , .

Lemma 28. Let and . If , then for some such that .

Proof. Suppose that . Thus ; we writewhere and for all . Since , there is such that for some . We choose for all and defineThen such that and .

Lemma 29. Let and . If , then for some such that .

Proof. Assume that . Thus ; we writewhere and for all . DefineIt is clear that . If , then and . If such , we definewhere . So, such that and .

For the case , the ideal consisting of all constant maps with range . It is easy to verify that is a maximal regular subsemigroup of for all .

Lemma 30. For , is a maximal regular subsemigroup of for all .

Proof. Let be any element in . Let . If , then . If , then in ; that is, by Lemma 22. Thus, and so . Hence, is a subsemigroup of . We prove is regular. Let . If , then is a regular element in . If , we writewhere and for all . Since , there exists such that for some . We choose for all and defineThen , , and . Hence, is a regular semigroup. Next, we prove is maximal. Let be a regular subsemigroup of such that . Then there exists ; that is, . Let be any element in . Then there is such that and by Lemma 28; that is, . Thus, . Hence, and therefore .

Lemma 31. For , is a maximal regular subsemigroup of for all .

Proof. Let be any element in . Let . If , then . If , then in and so by Lemma 22. Thus, . So, . Hence, is a subsemigroup of . We prove is regular. Let . If , then is a regular element in . If , we writewhere and for all . We choose for all and defineThen and . If , then is a regular element in . If , we definewhere since . We obtain and . Hence, is a regular semigroup. For a maximality of , we let be a regular subsemigroup of such that . Then there exists ; that is, . Let be any element in . Then there is such that and by Lemma 29; that is, . Thus, and so . Therefore, .

Theorem 32. Each maximal regular subsemigroup of must be one of the following forms:(1);(2),

where , is the -class containing in , and is the -class containing in .

Proof. By Lemmas 30 and 31, both (1) and (2) are maximal regular subsemigroups of .
On the other hand, let be an arbitrary maximal regular subsemigroup of . It is easy to see that is a regular subsemigroup of such that . Then or by the maximality of . If , then . We obtain by Lemma 25; this implies that , a contradiction. Whence and so , assume that and for all . We first prove that . Let be an idempotent in . Thus, . By Lemma 24, and contain idempotents, say and respectively. So, and . Since contains the idempotent , by Lemma 23. Since and is regular, for some . We see that , which implies that ; that is, in . Then by Lemma 22, we get . Also, we have . Whence , the group -class contains . So, there exists a positive integer such that since . Therefore, . It follows from Lemma 27 that ; hence , a contradiction. Therefore, for some or for some . If for some , thenThus, by the maximality of . If for some , thenThus, by the maximality of .

Notice that the number of -classes in equals and the number of -classes in equals where is the Stirling number of the second kind; see [18, Section ] for details. Therefore, the number of maximal regular subsemigroups of is equal to when .

We shall normally write instead of . For , defineIt is well-known that and is an ideal of for all . If , then , , and . Therefore, we establish the following corollary which first appeared in [8, Theorem  2].

Corollary 33. Each maximal regular subsemigroup of must be one of the following forms:(1);(2), where , is the -class containing in of , and is the -class containing in of .

We note that if , then we immediately obtain that the number of maximal regular subsemigroups of is equal to when .

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

Financial support from the Coordinating Center for Thai Government Science and Technology Scholarship Students (CSTS), National Science and Technology Development Agency (NSTDA), is acknowledged.