American Journal of Mathematical and Computational Sciences, Vol.1, No.1, Page: 55-61

Equivalent Characterizations and Structure Theorem of Right C-wrpp Semigroups

Deju Zhang1, Xiaomin Zhang2, Enxiao Yuan3

1School of Science, Linyi University, Shandong, P. R. China

2School of Logistic, Linyi University, Shandong, P. R. China

3School of Yishui, Linyi University, Shandong, P. R. China

Email address

(Deju Zhang)
(Xiaomin Zhang)
(Enxiao Yuan)

Citation

Deju Zhang, Xiaomin Zhang, Enxiao Yuan. Equivalent Characterizations and Structure Theorem of Right C-wrpp Semigroups. American Journal of Mathematical and Computational Sciences. Vol. 1, No. 1, 2016, pp. 55-61.

Abstract

The aim of this paper is to study the right dual of left C-wrpp semigroup, that is, the strongly wrpp semigroup whose set of idempotents forms a right regular band and the relation  is a congruence. We call this kind of strongly wrpp semigroups right C-wrpp semigroups. This paper generalizes results of right C-rpp semigroups. Some properties and characterizations of right C-wrpp semigroups are investigated.

Keywords

Right C-wrpp Semigroup, Right Regular Band, Strongly Wrpp Semigroup

1. Introduction

A semigroup  is an rpp semigroup if all of its principal right ideals  regarded as right -systems, are projective (see [3]-[5]). These classes of semigroups and its special subclasses have been studied by Fountain. He also defined a generalized Green’s relation  on a semigroup  by  if and only if the elements  of  are related by the Green’s relation  in some oversemigroup of  in [5]. In particular, if  for some idempotent elements , then . Next, he showed that a monoid  is rpp if and only if each -class of  contains an idempotent. Thus, a semigroup  is an rpp semigroup if each -class of  contains at least one idempotent. An rpp semigroup having all of its idempotents lying in its center is called a C-rpp semigroup. It is well known that a semigroup  is a C-rpp semigroup if and only if  is a strongly semilattice of left cancellative monoids. Thus, C-rpp semigroups are natural generalizations of Clifford semigroups.

For rpp semigroups, the concept of strongly rpp semigroups was first introduced by Guo-Shum-Zhu, that is, an rpp semigroup in which each  contains a unique idempotent  such that , where  is set of idempotents of . Strongly rpp semigroups and their special cases have been extensively studied by many authors (see [6]-[10], [14]-[17]). In particular, we call a strongly rpp semigroup  left C-rpp semigroup if  is a congruence and  for all . For such semigroups, Guo has proved that an rpp semigroup is a left C-rpp semigroup if and only if  is a semilattce of direct products of left cancellative monoids and left zero bands. Thus, a left C-rpp semigroup is clearly a generalized C-rpp semigroup and left C-semigroup.

On the other hand, we call a strongly an rpp semigroup  a right C-rpp semigroup if  is a congruence and  for all . It was observed that the concept of right C-rpp semigroups is not a dual of left C-rpp semigroups by Guo [9]. The structure of right C-rpp semigroups has been recently described by Shum and Ren in [17]. It is noteworthy that many properties of left C-rpp semigroups are not dual to those of right C-rpp semigroups (see [9], [17]).

Tang [18] has introduced a new set of Green’s two stars relations on a semigroup  by modifying Green’s one star relations on semigroups. Let  be a Green’s relation on a semigroup . A relation  on  by: for some , if and only if  for all . In view of this new Green’s two stars relation, he defined the concept of wrpp semigroups, that is, a wrpp semigroup in which each -class contains at least one idempotent. A wrpp semigroup is a C-wrpp semigroup if all the idempotents of  are central. We have known that a C-wrpp semigroup can be expressed as a strongly semilattice of left -cancellative monoids. Recently, C-wrpp semigroups have been extended to left C-wrpp semigroups by Du-Shum [2]. A wrpp semigroups  is called a left C-wrpp semigroup if  satisfies the following conditions: (i)for all  where  is set of idempotents in ; (ii)for all , there exists a unique idempotent  satisfying  and ; (iii) for all , , where  is the smallest left **-ideal of  generated by . if the condition (i) only holds, then  is called quasi strong wrpp semigroup, if conditions (i) and (ii) hold, then  is called strong wrpp semigroup. For left C-wrpp semigroups, Du-Shum [3] have obtained a description of curler structure. In fact, left C-wrpp semigroups are indeed a common generalizations on left C-semigroups and left C-rpp semigroups.

Naturally, one would ask whether we can describe the kind of wrpp semigroup as an analogy of right C-rpp semigroups. In this paper, we generalize right C-rpp semigroups to right C-wrpp semigroups. We first introduce the concept of right C-wrpp semigroups, and some properties on right C-wrpp semigroups are given. We shall show that a right C-wrpp semigroup can be described as a semilattice of the direct product of left- cancellative monoids and left zero bands, but the task is not simple as we need to consider the semilattice congruence on abundant wrpp semigroups. Anyway, our results clearly further generalizes both results of Shum-Ren on right C-rpp semigroups and Guo on a notes on right dual of left C-rpp semigroups. Last, the characterizations of a C-wrpp semigroup are given.

For the notations and terminology not given in this paper, the reader is referred to [2], [13], [18].

2. Basic Definitions and Results

We first introduce some definitions and results that are useful in the sequel.

In order to describe wrpp semigroups, Du-Shum introduced the following (+)-Green’s relations. For elements  in a semigroup , we define:

                                 (1)

                          (2)

                           (3)

                           (4)

where  is the smallest ideal of  containing  such that  is the union of some -classes or some -classes. For the sake of simplicity, we denote the -class (resp.,-class,-class,-class and -class) of  containing  by (resp., and ). The Green’s "egg box" diagram for Green’s relation still holds for these Green’s (+) relations. We have

Lemma 1 [2] The equalities  and  hold on a semigroup.

On the other hand, the Green’s (+) relations on  are also similar to the Green’s relations on , for instance, we have . Moreover, we have the following lemma:

Lemma 2 [2] Let  be a semigroup, . Then  if and only if there exists  with  and  such that  for all .

A -class may contain more than one regular -class. This is because if  for some idempotent  and , then the relation  need not always hold. For example, if  is a Rees matrix semigroup over a group  with , then it is not difficult to see that  is an -simple semigroup containing  regular -classes. But if  is a quasi strongly wrpp semigroup, then we have

Lemma 3 Let  be a quasi strongly wrpp semigroup and . Then  if and only if .

Proof Necessity. We only need verify that . If , then . By the quasi strong wrpp property of , we have , so . Thus .

Sufficiency. It is clear and we omit the proof.

Lemma 4 Let  be a quasi strongly wrpp semigroup. Then each -class contains at most one regular -class.

Proof According to Lemma 1, we have . Let  If  then there exists  such that . By regularity of , we obtain that  is a regular element of . According to Lemma 3, we know that . Hence , so .

Definiton 1 A wrpp semigroup  is called a right C-wrpp semigroup, if  satisfies the following conditions:

(1).  is a quasi strong wrpp semigroup;

(2).  is a congruence on ;

(3). .

We call a band a right regular band if it satisfies the identity . We now cite the following lemma:

Lemma 5 [15] The following statements are equivalent on a band :

(1).  is a right regular band;

(2).  is a congruence;

(3).  is a semilattice of right zero bands.

An immediate result of this lemma is:

Corollary 1 If  is a right regular band, then each -class of  contains precisely one idempotent.

3. Characterizations of Right C-wrpp Semigroups

In this section, we shall describe some characterizations of right C-wrpp semigroups and hence generalize the main results of right C-rpp semigroups obtained by Guo in [9]. The results obtained in [9] will be amplified and strengthened.

Lemma 6 Let  be a right C-wrpp semigroup. Then the following hold:

(1).   is a right regular band;

(2).  Reg is a right C-semigroup.

Proof (1) Let . Since , then there exists  such that , so . Hence , it implies that  is a band and a right regular band.

(2) According to (1),  is a band, so Reg is a regular subsemigroup of . And for all , (RegRegReg), consequently, Reg is a right C-semigroup (see [20]).

Theorem 1 The following statements are equivalent:

(1).  is a right C-wrpp semigroup;

(2).  is a strong wrpp semigroup such that  is a semilattice congruence, and ;

(3).  is a semilattice of -simple strong wrpp semigroups, and ;

(4).  is a semilattice of  for , where  is a left cancellative monoid,  is a right zero band.

Proof (1)(2). Let  be a right wrpp semigroup. Then  is a congruence of . Let , and . Then clearly . But  is a congruence, we have , so . Notice that  is a right regular band, it leads to   Consequently,  is a semilattice congruence.

According to Lemma 6, Reg is a right C-semigroup. Therefore, Reg/ is a semilattice, and  by Lemma 6, We easily prove that . By quasi wrpp property of , and Lemma 4, we know that each -class exactly contains one regular -class, and each -class exactly contains one regular -class, it means that each -class in each -class contains a unique idempotent which is a left identity of this -class. Again,  for all , so this unique idempotent is also a right identity of above -class. Hence  is a strongly wrpp semigroup.

(2)(3). Let  be a semilattice decomposition corresponding to the semilattice congruence . Obviously, for an arbitrary subsemigroup  of , we have . Hence the elements of  having  relation in  also have  relation in . By , and Lemma 4, it implies that each  only contains one regular -class. Therefore, the elements of  having  relation in  and also have  relation in , so each -class  is -simple, and is a strongly wrpp semigroup.

(3)(4). Let  be a semilattice decomposition , where  is a -simple strong wrpp semigroup. Let . According to , we know that each -class  of  contains a unique idempotent , and  is a right zero band. Next we shall verify that . Let . Then , so . Since , it leads to . Hence , so . Conversely, if  (), then , and , it is easily observed that , it means . Thus . By strong wrpp property of , we have , which is a monoid with identity element  for all . We claim that  is left- cancellative. In fact, for all , if , notice that , then . Now define a mapping:

                          (5)

for any fixed . Then we deduce that . Thus,  is a semigroup homomorphism.

We now show that  is a semigroup isomorphism. By virtue of the strongly wrpp property of , for all , there exists  such that . By the definition of , this means that , and hence  is an epimorphism. To prove  is a monomorphism, we assume that . Then we have  Since  is a right zero band, we have . This implies that  for all . Invoking the strongly wrpp property of , we obtain that  This shows that  is a monomorphism as well. Thus . The proof is completed.

1em (4)(1). Let  is a semilattice of  for , where  is a left- cancellative monoid,  is a right zero band. Then , where  is unique identity of left- monoid . We now show that  is a -class. Let , and . Then there exists  such that . Since  if and only if , so . Hence we have  for any , it implies that . This means that . Similarly, we can verify that . Hence we conclude that . Because  is just a -class of ,  must be a semilattice congruence on .

1emNext, we need verify that . Let  and  with . Then . In fact,  , then , so

that is, Clearly, , by using above analogous methods, we obtain that   for any . Hence, we have

This means that  is a right regular band. Now let , then

This verifies that .

Summing up the above results, then  is a right C-wrpp semigroup.

Corollary 2 Let  is a right C-wrpp semigroup. Then .

Proof Because , we only need to prove that . Suppose that . Then . By Lemma 2, there exists  with  and  such that  for all Since  and  is a congruence, we have. By Theorem 1, we know that  is a semilattice congruence. We denote semilattice  by . Index -class in virtue of the elements  in the semilattice , and let . We are not difficult to see that . Similarly, . Hence .

Lemma 7 Let  be a strongly wrpp semigroup whose set of idempotents is a semilattice . Then  for all .

Lemma 8 Let  be a strongly wrpp semigroup whose set of idempotents is a semilattice , and . If , then the following statements are hold:

(1).  is a left- cancellative monoid;

(2). If , and , then the mapping :

                            (6)

is a semigroup homomorphism. Moreover, with respect to the following multiplication "":

                           (7)

 form a C-wrpp semigroup, where  is the product in ;

(3). , where  is the product of  and  in .

Proof (1) Let . Notice that there is exactly one idempotent in , we have . By the fact that  being a strongly wrpp semigroup, we have . Since  is a right congruence, we know that . Hence  is a subsemigroup of . Notice that , it follows that  is the identity of . Now put , and . Then . Thus , that is,  is a left- cancellative monoid.

(2) Let . Since  is a right congruence, we have , that is, . By (1), we know that  is a left- cancellative monoid with identity . Consequently, for all , we have

                    (8)

Thus  is a semigroup homomorphism. It is not difficult to verify that  is a strongly semilattice structure homomorphism on . Therefore,  is a C-wrpp semigroup.

(3) Since  is a right congruence, we have . By Lemma 7, we obtain that . It means that . Because  is the identity of , we have

                     (9)

Lemma 9 Let  be a semigroup satisfying the conditions in Lemma 8. Then every regular element of  is completely regular, that is, a regular element is -related to an idempotent element.

Proof Let  be a regular element of . Then there exists  such that , so . Hence . As it is argued in Lemma 3.5, . Hence  is a regular element of  and  is an idempotent of . But there is only one idempotent in , so . Thus , that is,  is a completely regular element.

Lemma 10 Let  be a strongly semigroup whose set of idempotents is a band. Then every regular element of  is a completely regular element

Proof Since  is a band, Reg is a orthodox semigroups. Since  is a strongly wrpp semigroup, we can easily see that Reg is a strongly wrpp semigroup. Hence Reg  is a strongly wrpp semigroup, where  is the smallest inverse semigroup congruence on Reg. According to Lemma 9, we can follow that Reg  is a Clofford semigroup. Let . Then there exists  such that . It follows that . By Reg being a Clifford semigroup, . On the other hand, since , we have  and hence . Therefore , that is,  is a completely regular element.

As an application of above results, we now give some conditions which lead to a C-wrpp semigroup  for some congruence  defined on a right C-wrpp semigroup . In fact, all we need to find a congruence  on  so that  preserves the -classes of .

For convenience, we denote the rectangular band  by  if the idempotent  is in . Also, we write  if .

We now characterize right C-wrpp semigroups.

Theorem 2 The following conditions are equivalent for a strongly wrpp semigroup :

(1).  is a right C-wrpp semigroup;

(2).  is a right regular band and  is a semilattice congruence on ;

(3). The relation  is a congruence on  such that  is a C-wrpp semigroup.

Proof (1)(2). This part is an immediate consequence of Lemma 6 and Theorem 1.

(2)(1). Let  be a right regular band and suppose that  is a congruence on . To show that  is a right C-wrpp semigroup, we only need to verify that . By Lemma 5, we have . On the other hand, for all  with , by Lemma 10, there exists  such that . Clearly, , and then there exists some  such that . Again by Lemma 10, there exists  such that . Thus, , that is, . Consequently,  and . Thus, we have proved that . The proof is completed.

(2)(3). We can assume that  is a right C-wrpp semigroup. Then we have  for every  and because  is a strongly wrpp semigroup,  for . Thus, . This means that  is reflexive. To see that  is symmetric, we let . Then, by the definition of , we have  for . Since , we also have . Consequently, we get  and . Thus . From , we immediately get . This shows that  is symmetric. To see that  is transitive, we let  and . Then there exists  such that . By repeating the arguments given above, we have . This leads to . By , we have . Hence  is indeed an equivalent relation on .

1emTo see that  is a congruence on , we let . Then, by the definition of , there exists  such that . Hence . By invoking Theorem 1 (4), we have , that is, . This leads to . In other words, we have  and hence  is left compatible. Similarly, we can verify that  is right compatible. Thus  is indeed a congruence on .

We still need to show that  preserve the -classes of . For this purpose, we let  for some . If there are  such that , then there exists  such that  and . Hence, we can find  such that . By  being a semilattice congruence, we can deduce that  and similarly,. This leads to . Clearly,  and consequently,  and , that is, . So we also have . Therefore, by the definition of , there exists  such that  and . On the other hand, since , we have  and hence we deduce that . Similarly,.

Since , we have  and hence  This leads to . Similarly, . Thus, we have . From this relation and its dual, we conclude that  This shows that the relation  on  is preserved in the quotient semigroup , and hence  is a wrpp semigroup.

Finally, we show that the idempotents of  are central. It suffices to show that  for all  and . Since by Theorem 1 (4), , it is clear that . Thus, by , we obtain that . This shows that  is a C-wrpp semigroup.

(3)(1). Suppose that  is a congruence on  such that  is a C-wrpp semigroup. we can easily see that  and hence  is a semilattice. Hence  is a semilattice congruence on , and so  is a right regular band. Now let  be the semilattcie decomposition of  into right zero bands . Clearly,  is isomorphic to . We identify  with . By  is a C-wrpp semigroup, we let  be the semilattice decomposition of the C-wrpp semigroup  into left- cancellative monoids .

Put . Then we define  by . Clearly,  is well defined, and we deduce that

.              (10)

Thus  is a semigroup homomorphism.

Now we prove that  is a semigroup isomorphism. For all , we have  such that  and . It follows that . On the other hand, since , we have . But  is a right regular band, we know that each -class of  contains precisely one element, and thus . Consequently, . This means that  is an epimorphism. To prove  is a monomorphism, now let  and . Then  and . By using the latter formula, we see that there exists  such that , and furthermore, . This shows that  is also a monomorphism. On the other hand,  is a semilattice of direct products  and hence  is a right C-wrpp semigroup.

Now we define a new relation  on a strongly wrpp  as follows:

                            (11)

It is easy to verify that  is a equivalent relation, and

Theorem 3 Let  be a strongly wrpp semigroup. Then  is a right C-wrpp semigroup if and only if  is a semilattice congruence  and  is a right regular band.

Proof Assume that  is a right C-wrpp semigroup. By Lemma 6 (1), we only need to prove that  is a semilattice congruence. For this purpose, we let  is a semilattice of the direct products  for , where  is a left- cancellative monoid and  is a right zero band. We can easily check that  for any , where  is the identity of . Hence it is difficult to verify that identical formula . It follows that  is a semilattice congruence.

Suppose that  is a semilattice congruence on  and  is a right regular band. Since  is a semilattice congruence on ,  is a semilattice of some -classes. But , each -class of  is a strongly wrpp semigroup, therefore it is  -simple. Next we shall show that each  -simple semigroup is also -simple semigroup. For this purpose, we only need to prove . Let . Then . Hence there exists  such that . By , we can see that  is a regular element, and by Lemma 10,  is completely regular. Hence, we can follow that . This means that , so . Conversely, if , then there exist  with  such that . . From the above, we have

  (12)

This shows that . Hence  This shows that . Thus, each  -simple semigroup is also -simple semigroup, it deduces that  is a semilattice of -simple strongly wrpp semigroups.

Also, Since  is a right regular band, by the proof of  in Theorem 2, we know that . Therefore,  is a right C-wrpp semigroup.

Theorem 4 Let  be a strongly wrpp semigroup whose set of idempotents forms a right regular band. Then the following statements are equivalent:

(1).  is a right wrpp semigroup;

(2). ;

(3). .

Proof . By the Corollary 2, clearly.

. Let . Since  is a right regular band, we have (see the proof of Theorem 2). Let  and . Then  and hence . This leads to  by Lemma 4. Thus , that is, . Consequently,  and so .

. Assume that . By Theorem 4, we only need to verify that  is a semilattice congruence on . For this purpose, we prove that  is a semilattice congruence on . Let . Since  is a right congruence, we have . Hence , this means that . Thus, for any , we have  Similarly, we have  and so . Now we let  with . Because  is a right congruence, we have . Similarly, . According to  being a right regular band, we can follow that , thus . Therefore,  is a semilattice congruence, that is,  is a semilattice congruence. Consequently,  is a right C-wrpp.

4. Conclusions

In this paper, we show that a right C-wrpp semigroup can be described as a semilattice of the direct product of left- cancellative monoids and left zero bands, our results further generalizes both results of Shum-Ren on right C-rpp semigroups and Guo on a notes on right dual of left C-rpp semigroups. Last, the characterizations of a C-wrpp semigroup are given, that is, is a right wrpp semigroup if and only if the relatons  or .

Acknowledgment

This research is supported by Foundation of Shandong Province Natural Science (Grant No. ZR2010AL004). The author wish to thank the anonymous referee for the comments to improve the presentation and value suggesting.

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