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On right C-rpp semigroups.

Abstract An rpp semigroup S is called a right C - rpp semigroup if [L.sup.*] v R is a congruence on S and [S.sub.e] [subset] eS for all e [member of] E(S). This paper studies some properties on right C - rpp semigroups by using the concept of right [DELTA]-product. And, we obtained that the right C - rpp semigroups whose set of idempotents forms a right normal band are a strong semilattice of direct products of left cancellative monoids and right zero bands.

Keywords Right [DELTA]-product, right normal band, left cancellative monoids.

[subsection]1. Introduction

A semigroup S is called an rpp semigroup if all its principal right ideals a[S.sup.1](a [member of] S), regard as right [S.sup.1]-systems, are projective. This class of semigroups and its subclasses have been extensively studied by J.B.Fountain and other authors (see [1-7]). On a semigroup S, the Green's star relation [L.sup.*] is defined by (a, b) [member of] [L.sup.*] if and only if the elements a, b of S are related by the usual Green's relation L on some oversemigroup of S. It was then shown by J.B.Fountain [2] that a monoid S is rpp if and only if every [L.sup.*]-class contains an idempotent. Thus, a semigroup S is rpp if and only if every [L.sup.*]-class of S contains at least one idempotent. Dually, we can define lpp semigroups and a semigroup which is both rpp and lpp is called abundant[3]. Abundant semigroups and rpp semigroups are generalized regular semigroups.

It is noted that rpp semigroups with central idempotents have similar structure as Clifford semigroups. This kind of rpp semigroups was called the C - rpp semigroups by J.B.Fonutain. He has proved that a C - rpp semigroup can be described as a strong semilattice of left cancellative monoids.

In order to generalize the above result of Fountain, Y.Q.Guo, K.P.Shum and P.Y.Zhu have introduced the concept of strongly rpp semigroups in [4]. They considered an rpp semigroup S with a set of idempotents E(S). For 8a [member of] S, let the enevelope of a be [M.sub.a] = {e [member of] E(S)|[S.sup.1]a [subset] [S.sup.1]e and 8x, y [member of] [S.sup.1], ax = ay ) ex = eyg. Surely, [M.sub.a] consists of the idempotents in the [L.sup.*]class of a. Then the authors in [4] called the semigroup S strongly rpp if there exists a unique e in [M.sub.a] such that ea = a for 8a [member of] S. Now, we call a semigroup S a left C - rpp

semigroup-[4] if S is strongly rpp and [L.sup.*] is a semilattice congruence on S. Y.Q.Guo called an rpp semigroup S a right C - rpp semigroup[5] if [L.sup.*] v R is a congruence on S and [S.sub.e] [subset] eS for 8e [member of] E(S). He has shown that a right C - rpp semigroup S can be expressed as a semilattice Y of direct products [M.sub.a] and Ba, where [M.sub.a] is a left cancellative monoid and Ba is a right zero band for [for all] [alpha] [member of] Y .

In this paper, we will give some properties on right C - rpp semigroups by the result of K.P.Shum and X.M.Ren in [8]. Then, by using the concept of right [DELTA]-product, we study the right C - rpp semigroups whose set of idempotents forms a right normal band. We will see that this kind of semigroups is a strong semilattice of direct products of left cancellative monoids and right zero bands.

Terminologies and notations which are not mentioned in this paper should be referred to [8] and also the text of J.M.Howie [9].

[subsection]2. Preliminaries

In this section, we simply introduce the concept of right [DELTA]-product of semigroups and the structure of right C - rpp semigroups. These are introduced by K.P.Shum and X.M.Ren in [8].

We let Y be a semilattice and M = [Y,[M.sub.[alpha]][[theta].sub.[alpha],[beta]] is a strong semilattice of cancellative monoids Mff with structure homomorphism [[theta].sub.[alpha],[beta]]. Let [LAMBDA] = [[[alpha].sub.2]Y [[LAMBDA].sub.[alpha]] be a semilattice decomposition of right regular band [LAMBDA] into right zero band [[LAMBDA].sub.[alpha]]. For 8ff [member of] Y = we form the Cartesian product [S.sub.[alpha]] = [M.sub.[alpha]] x [[LAMBDA].sub.[alpha]].

Now, for [[LAMBDA].sub.[alpha]], [beta] [member of] Y with [alpha] [greater than or equal to] [beta] and the right transformation semigroup [J.sup.*](*[beta]), we define a mapping

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by u [right arrow] [gamma].sub.[alpha],[beta]] satisfying the following conditions:

(P1): If (a, i) [member of] [S.sub.[alpha]], i' [member of] [[LAMBDA].sub.[alpha]], then i' [[gamma].sup.(a,i).sub.[alpha],[alpha]

(P2): For [for all] (a, i) [member of] [S.sub.[alpha]], (b, j) [member of] [S.sub.[beta], we consider the following situation separately: ([[gamma].sup.(a,i).sub.[alpha],[alpha]) ([[gamma].sup.(a,i).sub.[beta],[beta]) is a constant mapping on [[LAMBDA].sub.[alpha]] and we denote the constant value by

(b) If [alpha],[beta],[delta], [member of] Y with [alpha],[beta][greater than or equal to] [delta] ([[gamma].sup.(a,i).sub.[alpha],[alpha]) ([[gamma].sup.(a,i).sub.[beta],[beta]) is a constant mapping on [[LAMBDA].sub.[alpha]] = k, then, ([[gamma].sup.(a,i).sub.[alpha],[alpha]) ([[gamma].sup.(a,i).sub.[beta],[beta]) = k, then ([[gamma].sup.(ab,k).sub.[alpha],[beta][delta])

(c) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(d) If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [1.sub.[gamma]] is the identity of the monoid [M.sub.[gamma]].

We now form the set union S = [U.sub.[alpha][member of]Y] [S.sub.[alpha]] and define a multiplication "[omicron]" on S by

After straightforward verification, we can verify that the multiplication "[omicron]" satisfies the associative law and hence (S, [omicron]) becomes a semigroup. We call the above constructed semigroup the right [DELTA]-product of semigroup M and [LAMBDA] on Y, under the structure mapping [[psi].sub.[alpha][beta]. We denote this semigroup (S, [omicron]) by S = M[[DELTA].sub.Y],[psi][LAMBDA].

Lemma 2.1. (See [8] Theorem 1.1). Let M = [Y,[M.sub.[alpha]][[theta].sub.[alpha],[beta]] be a strong semilattice of cancellative monoids [M.sub.[alpha] with structure homomorphism [[theta].sub.[alpha],[beta]]. Let * = S[[alpha].sub.2]Y [[LAMBDA].sub.[alpha]] be a semilattice decomposition of right regular band [LAMBDA] into right zero band [[LAMBDA].sub.[alpha]] on the semilattice Y. Then the right [DELTA]-product of M and [LAMBDA] denoted by M[[DELTA].subY,[psi][LAMBDA]., is a right C - rpp semigroup. Conversely, every right C - rpp semigroup can be constructed by using this method.

[subsection]3. Some properties and main result

In this section, we will first give some properties, by using right [DELTA]-product of semigroups, for right C - rpp semigroups which have been stated in the introduction. Then, we will obtain the structure of right C - rpp semigroups whose set of idempotents forms a right normal band.

Theorem 3.1. Let S be a right C - rpp semigroup. Then the following statements hold:

(1) For [[for all].sub.u] [member of] RegS, Su [??] uS,

(2) For 8e [member of] E(S), the mapping [[eta].sub.e]: x [member of] [S.sup.1]) is a semigroup homomorphism from [S.sup.1] onto [S.sup.1]e.

Proof. (1) We first assume that S = M[[DELTA].subY,[psi][LAMBDA] is an arbitrary right C - rpp semigroup. For [[for all].sub.u] = (a, i) [member of] [S.sub.[alpha]] \ RegS, there exists x = (b, j) [member of] [S.sub.[beta]] such that uxu = u and xux = x. we can easily know [alpha] = [beta] by the multiplication of semigroups. Hence for [for all]x [member of] [S.sub.[alpha]], from (*) and (P1) we have (b, j) = x = xux = (bab, j). So bab = b = [1.sub.[beta]] = [1.sub.[alpha]], where [1.sub.[alpha]] is the identity element of [M.sub.[alpha]]. By the left cancelltivity of [M.sub.[alpha]], we immediately obtain ab = [1.sub.[alpha]].

For [gamma] [member of] Y and v = (c, k) [member of] [S.sub.[gamma]], let w = ([[gamma].sup.(a,i).sub.[alpha],[alpha]) ([[gamma].sup.(a,i).sub.[beta],[beta]))(ca), a semigroup homomorphism from [M.sub.[alpha]] onto [M.sub.[gamma]][alpha], and [M.sub.[gamma]][alpha] is a left cancellative monoid. This leads to [1.sub.[alpha]][[theta].sub[alpha][gamma].sub.[alpha]] = 1[[gamma].sub.[alpha]]. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLEIN ASCII.]

= vu.

This shows Su [??] uS.

(2)For [for all]e [member of] E(S) and [for all]x, y [member of] [S.sup.1], if y = 1, then we have immediately

[[eta].sub.e](x * 1) = [[eta].sub.e](x) = xe = xee = [[eta].sub.e](x)[[eta].sub.e] (1).

If x [member of] S, then we know there exists z [member of] S such that ye = ez by using (*) and e [member of] RegS.

Hence,

[[eta].sub.e](xy) = (xy)e = x(ye) = x(ez) = xe(ez) = (xe)(ye) = [[eta].sub.e](x)[[eta].sub.e](y):

This shows that [[eta].sub.e] is a semigroup homomorphism.

In the following section, we proceed to study the structure of right C - rpp semigroups whose set of idempotents forms a right normal band.

Definition 3.2. A band E is called a right normal band if [for all]e, f, g [member of] E such that efg = feg.

Theorem 3.3. Let S = [LAMBDA] be a right C - rpp semigroup. E(S) is the set of idempotents of S. Then the following statements are equivalent:

(1) S is a strong semilattice of [M.sub.[alpha]] - [[LAMBDA].sub.[alpha]],

(2) E(S) is a right normal band.

Where M = [Y,[M.sub.[alpha]][[theta].sub.[alpha],[beta]] be a strong semilattice of left cancellative monoids [M.sub.[alpha]] with structure homomorphism [[theta].sub.[alpha],[beta]], = [[[alpha].sub.2]Y [[LAMBDA].sub.[alpha]] be a semilattice decomposition of right regular band [LAMBDA] into right zero band [[LAMBDA].sub.[alpha]].

Proof. (1)[right arrow] (2). Let S be strong semilattice Y of [S.sub.[alpha]] = [M.sub.[alpha]]-[[LAMBDA].sub.[alpha]] with structure homomorphism [[psi].sub.[alpha],[beta] for [alpha], [beta] [member of] Y and [alpha] > fi. From [8] we knew E(S) = S[[alpha].sub.2]Y f([1.sub.[alpha]], i) [member of] [M.sub.[alpha]] [[LAMBDA].sub.[alpha]]ji [member of] [[LAMBDA].sub.[alpha]]g, where [1.sub.[alpha]] is the identity element of [M.sub.[alpha]]. If ([1.sub.[alpha]], i) [member of] [S.sub.[alpha]], (1fi, j) [member of] [S.sub.[beta]], then ([1.sub.[alpha],[alpha]], i) [[psi].sub.[alpha],[beta] element in [E.sub.[alpha][beta]. Since [E.sub.[alpha][beta] is a right zero band, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Consequently, for any idempotents ([1.sub.[alpha]], i), ([1.sub.[beta]j]) and ([1.sub.[gamma],k]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

([1.sub.[alpha]], i)([1.sub.[beta]j])([1.sub.[gamma],k]) = ([1.sub.[beta]j])([1.sub.[alpha]], i)([1.sub.[gamma],k]).

This shows that E(S) is a right normal band.

(2)[right arrow](1). If E(S) is a right normal band, then we knew that E(S) is a strong semilattice of right zero band, and every right zero band is just a J (= D)-class of E(S). As E(S) itself is a semilattice of right zero bands Eff = f([1.sub.[alpha]], i) | - [member of] [[LAMBDA].sub.[alpha]]g, each Eff is just a J -class of E(S). This means that E(S) is a strong semilattice of Eff. Let the strong semilattice structure homomorphism be [[zeta].sub.[alpha],[beta]], where [alpha] [beta], [beta] [member of] Y and [alpha] [greater than or equal to] [beta]. Then for any idempotents ([1.sub.[alpha],i]), ([1.sub.[beta]j]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let [[theta].sub.[alpha],[beta]] be the strong semilattice structure homomorphism of the C - rpp compotent [M.sub.s] = [U.sub.[alpha][member of]Y] [M.sub.[alpha]] of S. By virtue of the right normality of E(S), for [for all] (a,i) [member of] [S.sub.[alpha]] and [j.sub.1], [j.sub.2] [member of] [[LAMBDA].sub.[beta]] [[LAMBDA].sub.[alpha][beta]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This shows <[[gamma].sup.(a,i).sub.[alpha],[alpha][beta]) ([[gamma].sup.([1.sub.[beta],j1).sub.[beta],[alpha][beta]) = [[gamma].sup.(a,i).sub.[alpha],[alpha][beta]) ([[gamma].sup.([1.sub.[beta],j1).sub.[beta],[alpha][beta])>.

Now, for [for all][alpha], [beta] [member of]Y, [alpha] [greater than or equal to] [beta], - [member of] [[LAMBDA].sub.[alpha], [j.sub.0] [member of] [[LAMBDA].sub.[beta]] we define a mapping [[psi].sub.[alpha][beta]]: [S.sub.[alpha]] [right arrow] [S.sub.3] (a,i} [right arrow] )a[theta].sub.[alpha],[alpha][beta] [[psi].sub.[alpha][beta]] [[gamma].sup.(a,i).sub.[alpha],[alpha][beta]) ([[gamma].sup.([1.sub.[beta],j1).sub.[beta],[alpha][beta]) = [[gamma].sup.(a,i).sub.[alpha],[alpha][beta]) ([[gamma].sup.([1.sub.5]

By the same arguments as the previous one, we can get the above mapping [[psi].sub.[alpha][beta]] is independent of the choices [j.sub.0] [member of] *fi.

Clearly, [[psi].sub.[alpha][beta]] is an identity mapping for [[LAMBDA].sub.[alpha]] [member of] Y .

Furthermore, for [for all] (a,i), (b,j) [member of] [S.sub.[alpha]], [alpha] > [beta] and j0 [member of] [[psi].sub.[alpha][beta]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus, [[psi].sub.[alpha][beta]] is indeed a homomorphism from [S.sub.[alpha]] onto [S.sub.[beta]].

For [[for all].sub.[alpha]], [beta], [gamma] [member of] Y satisfying [alpha] [greater than or equal to] [beta] [greater than or equal to] [gamma], and [j.sub.0] [member of] [[LAMBDA].sub.[gamma], [k.sub.0] [member of] [[LAMBDA].sub.[gamma]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In other words, we have [[psi].sub.[alpha][beta]] [[psi].sub.[alpha][gamma]] = [[psi].sub.[alpha][gamma]]. Summing up all the above discussion, we are now ready to construct a strong semilattice S of [S.sub.[alpha]] with the above structure homomorphism [[psi].sub.[alpha][beta]].

Clearly, [??] = S as sets. The remaining part is to show that [??] = S as semigroup as well.

Denote the multiplication in [??] by *, Then for [for all](a,i) [member of] [S.sub.[alpha]], (b,j) [member of] [S.sub.[beta]] and [j.sub.0] [member of] [[LAMBDA].sub.[alpha]], we

have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus, it can be seen that the multiplication * of S is exactly the same as the usual semigroup multiplication of S. This shows that S is a strong semilattice of [M.sub.[alpha]] - [[LAMBDA].sub.[alpha]]. The proof is completed.

References

[1] J. B. Fountain, Adequate Semigroups, Proc. London Math Soc., 22(1979), 113-125.

[2] J. B. Fountain, Right pp monoids with central idempotents, Semigroup Forum, 13(1977), 229-237.

[3] J. B. Fountain, Abundant Semigroups, Proc. London Math Soc., 44(1982), 103-129.

[4] Y. Q. Guo, K. P. Shum and P. Y. Zhu, The structure of the left C - rpp semigroups, Semigroup Forum, 50(1995), 9-23.

[5] Y. Q. Guo, The right dual of left C - rpp semigroups, Chinese Sci. Bull, 42(1997), No.9, 1599-1603.

[6] K. P. Shum and Y. Q. Guo, Regular semigroups and their generalizations, Lecture Notes in Pure and Applied Math, Marcel Dekker, Inc., New York, 181(1996), 181-226.

[7] K. P. Shum, X. J. Guo and X. M. Ren, (l)-Green's relations and perfect rpp semigroups, Proceedings of the third Asian Mathematical Conference 2000, edited by T.Sunnada, P. W. Sy and Y. Lo, World Scientific Inc., Singapore, 2002, 604-613.

[8] K. P. Shum and X. M. Ren, The structure of right C - rpp semigroups, Semigroup Forum, 68(2004), 280-292.

[9] J. M. Howie, An introduction to semigroup theory, Academic Press, London, 1976.

Huaiyu Zhou

School of Statistics, Xi'an University of Finance and Economics,

Xi'an 710061, China

E-mail: zhouhy9526@126.com
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