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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
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
, (Reg
Reg
Reg
), 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.
References