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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 489674, 14 pages
http://dx.doi.org/10.1155/2011/489674
Research Article

On Maximal Subsemigroups of Partial Baer-Levi Semigroups

1Department of Mathematics, Chiang Mai University, Chiangmai 50200, Thailand
2Material Science Research Center, Faculty of Science, Chiang Mai University, Chiangmai 50200, Thailand

Received 20 September 2010; Revised 14 January 2011; Accepted 28 February 2011

Academic Editor: Robert Redfield

Copyright © 2011 Boorapa Singha and Jintana Sanwong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Suppose that is an infinite set with and is the symmetric inverse semigroup defined on . In 1984, Levi and Wood determined a class of maximal subsemigroups (using certain subsets of ) of the Baer-Levi semigroup dom and . Later, in 1995, Hotzel showed that there are many other classes of maximal subsemigroups of , but these are far more complicated to describe. It is known that is a subsemigroup of the partial Baer-Levi semigroup . In this paper, we characterize all maximal subsemigroups of when , and we extend to obtain maximal subsemigroups of when .

1. Introduction

Suppose that is a nonempty set, and let denote the semigroup (under composition) of all partial transformations of (i.e., all mappings , where ). For any , we let and (or ) denote the domain and the range of , respectively. We also write and refer to these cardinals as the gap, the defect, and the rank of , respectively. Let denote the symmetric inverse semigroup on : that is, the set of all injective mappings in . If , we write where is the Baer-Levi semigroup of type defined on (see [1, 2]). It is wellknown that this semigroup is right simple, right cancellative, and idempotent-free. On the other hand, in [3] the authors showed that , the partial Baer-Levi semigroup on , does not have these properties but it is right reductive in the sense that for every , if for all , then . Also, they showed that satisfies the dual property, that is, it is left reductive (see [1, 2]). The authors also characterized Green's relations and ideals of and, in [3, Corollary  1], they proved that contains an inverse subsemigroup: namely, the set defined by This set consists, in fact, of all regular elements of , as shown in [3, Theorem  4]. Recently, in [4], the authors studied some properties of Mitsch's natural partial order defined on a semigroup (see [5, Theorem  3]) and some other partial orders defined on . In particular, they described compatibility and the existence of maximal and minimal elements.

For any nonempty subset of such that , let In other words, given , we have if and only if does not contain , or contains and either or . In [6], Levi and Wood showed that is a maximal subsemigroup of . Later, Hotzel [7] showed that there are many other maximal subsemigroups of .

In this paper, we study maximal subsemigroups of . In particular, in Section 3 we describe all maximal subsemigroups of when . We also determine some maximal subsemigroups of a subsemigroup of defined by where . Moreover, we extend to determine maximal subsemigroups of . In Section 4, we determine some maximal subsemigroups of when .

2. Preliminaries

In this paper, means is a disjoint union of sets and . As usual, denotes the empty (one-to-one) mapping which acts as a zero for . For each nonempty , we write for the identity transformation on : these mappings constitute all the idempotents in and belong to precisely when .

We modify the convention introduced in [1, 2]: namely, if is non-zero, then we write and take as understood that the subscript belongs to some (unmentioned) index set , that the abbreviation denotes , and that for each and . To simplify notation, if , we sometimes write in place of .

Let be a semigroup and . Then denotes the subsemigroup of generated by . Recall that a proper subsemigroup of is in if, whenever and is a subsemigroup of , then . Note that this is equivalent to each one of the following:(a) for all ;(b)for any , can be written as a finite product of elements of (note that is not expressible as a product of elements of since ).

Throughout this paper, we will use this fact to show the maximality of subsemigroups of .

3. Maximal Subsemigroups of When

The characterisation of maximal subsemigroups of a given semigroup is a natural topic to consider when studying its structure. Sometimes, it is difficult to describe all of them (see [6, 7], e.g.), but for a semigroup with some special properties, we can easily describe some of its maximal subsemigroups.

Lemma 3.1. Let be a semigroup and suppose that is a disjoint union of a subsemigroup and an ideal of . Then, (a)for any maximal subsemigroup of , is a maximal subsemigroup of ; (b)for any maximal subsemigroup of such that and , the set is a maximal subsemigroup of .

Proof. To see that (a) holds, let be a maximal subsemigroup of . Since is an ideal, we have is a subsemigroup of . Clearly, . If , then and thus . Since contains , we have and so is maximal in as required.
To prove (b), let be a maximal subsemigroup of , where and , and let . Since is maximal in , we have . Thus, for each , for some natural and some for all . Since , we have for some . Moreover, since , we have for all . It follows that , therefore that is, and thus is maximal in .

Let be a cardinal number. The successor of , denoted by , is defined as Note that always exists since the cardinals are wellordered, and when is finite we have .

From [3, p 95], for , is a subsemigroup of . Also, when , the proper ideals of are precisely the sets: where (see [3, Theorem  13]). Thus, for any , it is clear that that is, can be written as a disjoint union of the semigroup and the ideal . Hence, the next result follows directly from Lemma 3.1(a).

Corollary 3.2. Suppose that . If is a maximal subsemigroup of , then is a maximal subsemigroup of .

Lemma 3.3. Let and suppose that is a maximal subsemigroup of . Then, (a) for all ; (b)if there exists with , then for some .

Proof. To show that (a) holds, we first note that is contained in for all . If , then and thus by the maximality of . But where is a subsemigroup of (since is an ideal), so we get a contradiction. Therefore, for all .
To show that (b) holds, suppose there is with . If , then for all . Otherwise, if , then . Hence (b) holds.

For what follows, for any cardinal , we let

Then and . Moreover, if and , then , and so is a subsemigroup of (since it is the intersection of two semigroups). Also, is bisimple and idempotent-free, when and (see [3, Corollary  3]).

From [3, Theorem  5], if , then for each , and by [3, Theorem  6], for each when .

This motivates the following result.

Lemma 3.4. Suppose that . Then for each .

Proof. Let and . Since where and the second intersection on the right has cardinal at most (since ), we have . This means that Since , we have , and so . Hence and therefore .
For the converse, if , then . Since , every element in has rank , so we write Now write and where and (note that this is possible since and ). Define where and are bijections. Then and , that is, and equality follows.

Now we can describe all maximal subsemigroups of when .

Theorem 3.5. Suppose that . Then is a maximal subsemigroup of if and only if equals one of the following sets: (a);(b), where and is a maximal subsemigroup of .

Proof. Let be such that and . Clearly . Then Hence, and this shows that is a subsemigroup of . To show that is maximal in , we let . By Lemma 3.4, for some . Thus, can be written as a finite product of elements of , and hence is maximal in . Also, if and is a maximal subsemigroup of , then is maximal in by Corollary 3.2.
We now suppose that is a maximal subsemigroup of such that . Then there exists with . Thus, Lemma 3.3 implies that and for some , where . Since , Lemma 3.1(b) implies that is maximal in . We also see that where is maximal in by Corollary 3.2. This means that by the maximality of .

By the previous theorem, when , most of the maximal subsemigroups of are induced by maximal subsemigroups of where . Hence we now determine some maximal subsemigroups of .

As mentioned in Section 1, for every nonempty subset of with , is a maximal subsemigroup of . Here we extend the definition of and consider the set defined as that is, in belongs to if and only if(a), or(b) and either , or .

The next result gives more detail on .

Lemma 3.6. Suppose that , and let be a nonempty subset of such that . Then, (a)for any cardinal such that , there exist such that and ;(b)for each , and .

Proof. To show that (a) holds, let , and let be a cardinal such that . We write where and . If , then ; if not, then , and this implies , and so . Fix and let . Then, and . We write where and . Then there exists a bijection and so , . Also, since , we have .
To find with , we consider two cases. First, if , we write where . Fix and define where and are bijections and . On the other hand, if , then . In this case we write and where and . Fix and redefine where and are bijections and . In both cases, we have , , , , and , that is .
To see that (b) holds, suppose that there is , then and . So since is injective. Also, where . Since and by our assumption , we have as required.

In [6, Theorem  1], the authors proved that is a maximal subsemigroup of for every nonempty subset of such that . Using a similar argument, we show that is a subsemigroup of .

Lemma 3.7. Suppose that , and let be a nonempty subset of such that . Then is a proper subsemigroup of .

Proof. Let . If , then . Now we suppose that . Then, and since , we either have , or . If , then and so . Otherwise, we have and hence since is injective. Since , we either have , or . If the latter occurs, then therefore . On the other hand, if , we have . Moreover, , that is, . Therefore , and hence is a subsemigroup of . Finally, this subsemigroup is properly contained in by Lemma 3.6(a).

Remark 3.8. For any cardinal such that , is a proper subsemigroup of but it is not maximal when . To see this, suppose is maximal and choose such that and (possible by Lemma 3.6(a)). Then where . Moreover , and so for some and , , . If or , then and so , a contradiction. Thus, and . Since , it follows that . Moreover, , and this implies, and hence .
Since is maximal in , a subsemigroup of , it is natural to think that is maximal in . But when , by taking , the above observation shows that this claim is false since . Thus, is not always a maximal subsemigroup of .

The proof of the next result follows some ideas from [6, Theorem  1].

Theorem 3.9. Suppose that , and let be a nonempty subset of such that . Then is a maximal subsemigroup of precisely when .

Proof. In Remark 3.8, we have shown that is not maximal in when . It remains to show is maximal in . Let . Then and Lemma 3.6(b) implies that We also have or . In the case that , we have . Thus, there exists , so . Since , we can write where and . Also, since , we have and . Thus, for convenience, write , let be such that for each , and let . Hence, we can write
Now define by Then, , that is, . Also, since , we have and so . Moreover, since and , we have , that is, . Also, since , we can write and define in by Then , that is, . Moreover, since . Finally, we can see that where .
On the other hand, if , then there exists . In this case, we rewrite and where . Like before, we write and where , then Define by Then, , , , and so . Also, since (note that ) and . Moreover, . In other words, we have shown that for every , can be written as a finite product of elements of . Therefore, is maximal in .

We now determine some other classes of maximal subsemigroups of .

Lemma 3.10. Suppose that . Let be a cardinal such that or . Then is a proper subsemigroup of .

Proof. Since , we have . If , then , and this is a subsemigroup of since, for , , and this implies . Now suppose and let be such that . We claim that or . To see this, assume that . Since we have , thus
Note that where the intersection on the right has cardinal at most . Hence, and we have shown that is a subsemigroup of .

Remark 3.11. Observe that, if then is not a semigroup for all . To see this, let and for some subset of such that (possible since ), then since . Moreover, since and , we have . But thus , that is, .

Theorem 3.12. Suppose that . Then the following statements hold: (a) is a maximal subsemigroup of ;(b)if , then for each cardinal such that , is a maximal subsemigroup of .

Proof. By Lemma 3.10, is a subsemigroup of . To see that it is maximal, let . By [3, Theorem  5], , and this implies that for some . Hence is maximal in .
Now suppose that and let . Let . If , then and, by [3, Theorem  6], . If , then (by Lemma 3.4). Therefore, for some , and so is maximal in .

Corollary 3.13. Suppose that and let be a nonempty subset of such that . Then the following sets are maximal subsemigroups of : (a);(b) where or .

Proof. By Theorem 3.9, is maximal in . Then Corollary 3.2 implies that is maximal in . But and so (a) holds. To show that (b) holds, let in Theorem 3.12. Then and thus is maximal in .

Theorem 3.14. Suppose that and equals 0 or . Let be a nonempty subset of such that . Then the two classes of maximal subsemigroups and of are always disjoint.

Proof. By Theorems 3.9 and 3.12, and are maximal subsemigroups of . By Lemma 3.6(a), there exists with . Then but , that is, . Also, by the maximality of and . Therefore, is not equal to .

4. Maximal Subsemigroups of When

We first recall that, when , the empty transformation belongs to since . In this case, the ideals of are precisely the sets: where (see [3, Theorem  14]). Clearly, and is the largest proper ideal. In this case, the complement of each in is not a semigroup. To see this, write where and . Then whereas . Hence, unlike what was done in Section 3, we cannot use Lemma 3.1 to find maximal subsemigroups of when . In this section, we determine some maximal subsemigroups of , for , using a different approach. We first describe some properties of each maximal subsemigroup in this case.

Lemma 4.1. Suppose that and is a maximal subsemigroup of . Then the following statements hold: (a) contains all with ,(b)if , then .

Proof. Suppose that there exists with . Then , and we write in the usual way Also, write and where , and define in by where and are bijections. Then . Also, thus . But is a subsemigroup of and this means that by the maximality of . Since all mappings in have rank at most , it follows that contains all mappings with rank greater than . Therefore and thus , a contradiction.
To show that (b) holds, suppose that . If there exists , then [3, Theorem  5] implies that (note that when ), so , contrary to the maximality of . Thus .

Remark 4.2. If , then every has rank . This contrasts with Lemma 4.1(a). Also, by Corollary 3.13, if and , is a maximal subsemigroup of containing , this contrasts with Lemma 4.1(b).

As in Section 3, for any cardinal , we let

By Lemma 3.10 and Remark 3.11, if , then is a subsemigroup of exactly when or . From Corollary 3.13(b), when , is a maximal subsemigroup of . But when , Lemma 4.1(a) implies that is not maximal since . Moreover, Lemma 4.1(a) implies that every maximal subsemigroup of must contain the largest proper ideal Note that itself is a subsemigroup of , but it is not maximal since (in case , implies ).

Theorem 4.3. Suppose that , and let be a nonempty subset of such that . The following are maximal subsemigroups of :(a);(b);(c).

Proof. If , then , and so (a) holds by Theorem 3.9. Also, by taking in Theorem 3.12(a), we see that (b) holds. To show that (c) holds, take in Lemma 3.10, we have is a subsemigroup of . Moreover, is also a subsemigroup of since is an ideal. To show the maximality of , let . Then . Write in the usual way where , and let where . Then define by where and are bijections. Thus since . Moreover since . It is clear that and therefore is maximal in .

Remark 4.4. When , if is a maximal subsemigroup containing , then by Lemma 4.1(b). Thus, by the maximality of . So we conclude that is the only maximal subsemigroup of containing .

Remark 4.5. As we showed in Section 3, to see all maximal subsemigroups of when , it is necessary to describe all maximal subsemigroups of where . So we leave this as a direction for future research.

Acknowledgments

The authors wish to thank the referees for the careful review and the valuable comments, which helped to improve the readability of this paper. B. Singha thanks the Office of the Higher Education Commission, Thailand, for its support by a grant. He also thanks the Graduate School, Chiang Mai University, Chiangmai, Thailand, for its financial support during the preparation of this paper. J. Sanwong thanks the National Research University Project under the Office of the Higher Education Commission, Thailand, for its financial support.

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