F_富足半群_英文_

收稿日期:2005-03-31

基金项目:江西省自然科学基金资助项目(0311038).

作者简介:倪翔飞(1981-),女,江西鹰潭市人,理学硕士研究生,主要从事半群理论的研究.

文章编号:1000-5862(2005)05-0407-04

F -Abundant Semigroup

NI X iang -fei ,　CHEN Wei ,　G UO X iao -jiang

(Institute of M athematics and In formatics ,Jiangxi N ormal University ,Nangchang 330027,China )

Abstract :In this paper ,we introduce the concept of pre -hom om orphism and characterize F -abundant semigroups

in terms of pre -hom om orphisms.

K ey w ords :F -abundant semigroup ;canceled semigroups ;pre -hom om orphism

1　Introduction and Preliminaries

A semigroupis S called abundant if each R 3-class and each L 3-class contains an idem potent.An abundant semigroup is called quasi -adequate if its idem potents form a subsemigroup.Abundant semigroups are generalizations of regular semi -groups while quasi -adequate seigroups those of orthodox semigroups.As a class of semigroups intermedi 2ate between the class of abundant semigroups and the class of regular ones E L -Qallali and F ountain [1]defined and stud 2ied idem potent connected abundant semi -groups.S o -called an idem potent connected (IC )abundant semigroups of

defined as an abundant semigroup in which for each a ∈S and for s ome a +∈R 3a ∩E (S ),a 3∈L 3a ∩E (S )there

is a bijection θ:〈a +〉→〈a 3〉such that xa =a (x θ),for all x ∈〈a +〉,where 〈a +〉is the subsemigroup of S gener 2

ated by eE (S )e ,Indeed ,in this case θis an is om orphism [1].Various kinds of abundant semigroups have been investi 2gating by many authors [1～5].

An F -inverse semigroup is an inverse semigroup whose congruence classes m odulo the least group (congruence contain greatest elements with respect to the natural partial order.McFadden and O ’Carroll [6]determined a structure of such semi -groups.A fter that Edwards [7]studied regular semigroups satis fying the same condition ,called F -regular semigroups.She established the construction of F -regular semi -groups.In [8],X iojiang G uo introduced (strongly )F -abundant semigroups and ,in particular ,determined the structure of strongly F -abundant semigroups.In this paper we shall continue [8]to establish the construction of F -abundant semigroups.

Throughout this paper we shall use the terminology a notations of [2,5].

Lemm a 1[4]　Let S be a semigroup and a ,b ∈S .The following statements are equivalent :

(1)aL 3b ;　(2)for all x ,y ∈S 1,ax =ay Ζbx =by.

As an easy but useful consequence ,we have

Corollary 1[4]　Let a ,e 2=e ∈S.The following statements are equivalent :

(1)aL 3e ;　(2)a =ae and for all x ,y ∈S 1,ax =ay Ζex =ey.

F or a semigroup S ,we use E (S )to denote the set of idem potents of S.F or the sake of sim plicity ,a typical idem 2potent in the L 3-class [resp.R 3-class]of an element a of S will be denoted by a 3[resp.a +].

Let S be an abundant semigroup.F ollowing Laws on [5],define partial orders on S as follows :

a ≤l

b ΖL

3(a )ΑL 3(b )　and a =bf for s ome f ∈E (S );

第29卷第5期

2005年9月　江西师范大学学报(自然科学版)JOURNA L OF J I ANG XI NORM A L UNI VERSITY (NAT URA L SCIE NCE )V ol.29N o.5　Sep.2005