Injective Hulls of Acts over Left Zero Semigroups

In this paper, using the notion of "Cauchy sequences", we study the injectivity of acts over left zero semigroups, and give some equivalent conditions to it. We see that Baer criterion is true if an identity is adjoined to the semigroup. Further, we find injective hulls of acts over left zero semigroups. We hope that the technique used here will be useful to study injectivity of acts over some other more general semigroups.     

Various notions of weak injectivity over Clifford semigroups

We study various notions of weak injectivity of acts over a Clifford semigroup. In addition we characterize weak self-injectivities of several classes of Clifford semigroups constructed via particular kinds of groups and group homomorphisms. Research of X. Zhang supported by the China Scholarship Council No. 2006101056. Research of Y. Wang supported by National Natural Science Foundation of China Research Grant 10571181.     

Characterizing Semigroups by Sequentially Dense Injective Acts

Sequentially dense monomorphisms and injectivity with respect to these monomorphisms were first introduced and studied by Giuli for acts over the monoid (N^∞, min). In this paper we generalize these notions to acts over a general semigroup, and study the behaviour of this notion of injectivity with respect to products, coproducts, and direct sums. As a result we give some characterizations of semigroups.     

PI-强Lρ-富足半群的构造

S为半群,如果S中的每个Lρ-类都含幂等元,称S为Lρ-富足半群．特别地,如果对任意的α∈S,集合Iα∩Lα^ρ都只含唯一的元素,称S为强Lρ-富足半群．在S上通过一个非恒等置换σ,给出了PI-强Lρ-富足半群的结构定理．     

Note on the orderability of idempotent semigroups

In [3] , we gave a condition for the orderability of finite idempotent semigroups. Recently, in [5], T. Saitô gives a condition for the orderability of idempotent semigroups in the general case. As Corollary ( 4–13 of [5] ), he obtains an idempotent semigroup S is orderable if and only if every finite subsemigroup of S is orderable. The purpose of this note is to give a direct proof of this result.     

Levi’s Commutator Theorems for Cancellative Semigroups

For every semigroup of finite exponent whose chains of idempotents are uniformly bounded we construct an identity which holds on this semigroup but does not hold on the variety of all idempotent semigroups. This shows that the variety of all idempotent semigroups E is not contained in any finitely generated variety of semigroups. Since E is locally finite and each proper subvariety of E is finitely generated [1, 3, 4], the variety of all idempotent semigroups is a minimal example of an inherently non-finitely generated variety.     

Intuitionistic fuzzy semiprime ideals in ordered semigroups

We give characterizations of different classes of ordered semigroups by using intuitionistic fuzzy ideals. We prove that an ordered semigroup is regular if and only if every intuitionistic fuzzy left (respectively, right) ideal of S is idempotent. We also prove that an ordered semigroup S is intraregular if and only if every intuitionistic fuzzy two-sided ideal of S is idempotent. We give further characterizations of regular and intra-regular ordered semigroups in terms of intuitionistic fuzzy left (respectively, right) ideals. In conclusion of this paper we prove that an ordered semigroup S is left weakly regular if and only if every intuitionistic fuzzy left ideal of S is idempotent.     

具有理想收缩性质的某些GV-半群(英文)

如果半群S的每一个理想都是它的幂等同态像,称半群S具有理想收缩性质。GV-半群是完全正则半群在π-正则半群范围内的推广。本文刻画了某些具有理想收缩性质的GV-半群。     

Semigroups which contain magnifying elements are factorizable

A semigroup S is factorizable if it contains two proper subsemigroups A and B such that S = AB. An element a of a semigroup 5 is a left ( resp. right) magnifier if there exists a proper subset M of S such that S = aM (resp. S - Ma).

In this paper we prove that every semigroup containing magnifying elements is factorizable. Thus we solve a problem raised up by F. Catino and F. Migliorini in [2], namely to find necessary and sufficient conditions in order that a semigroup with magnifying elements be factorizable. Partial answers to this problem have been obtained by K. Tolo ([14]), F. Catino and F. Migliorini ([2]), for semigroups with left magnifiers and which are regular or have left units or right magnifiers, by V. M. Klimov ([9]), for Baer-Levi and Croisot-Teissier semigroups, and by M. Gutan ([4]), for right cancellative, right simple, idempotent free semigroups.     

保等价部分变换半群的变种半群上的正则元

在现有的保等价部分变换半群的基础上,引入了一个新的运算,得出保等价部分变换半群的变种半群的概念,利用格林关系及幂等元的正则性,讨论了这类半群中元素的正则性,给出了保等价部分变换半群的变种半群中一个元是正则元的充要条件     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

Semilattices of bisimple inverse semigroups

The purpose of this paper is to give a structure for a semigroup which is a semilattice of bisimple inverse semigroups and satisfies certain conditions. For such a semigroup, we characterize the idempotent separating congruences.     

具有弱中间幂等元的正则半群

本文在正则半群上引入弱中间幂等元和拟中间幂等元,着重探讨了这两类幂等元的性质特征.构造了若干具有弱(拟)中间幂等元的正则半群,确定了弱中间幂等元和拟中间幂等元之间的关系,给出了弱中间幂等元和拟中间幂等元各自的等价判定,利用拟中间幂等元刻画了纯正半群.最后,得到了构造具有拟中间幂等元的正则半群的一般途径,并在此基础上进一步给出了判定正则半群是否具有乘逆断面的方法.     

Idempotent bounded C-semigroups

A semigroup with zero isidempotent bounded (IB) if it is the 0-direct union of idempotent generated principal left ideals and the 0-direct union of idempotent generated principal right ideals. Notable examples are completely 0-simple semigroups and the wider class of primitive abundant semigroups. Significant to the structure of these semigroups is that they are all categorical at zero. In this paper we describe IB semigroups that are categorical at zero in terms ofdouble blocked Rees matrix semigroups. This generalises Fountain's characterisation of primitive abundant semigroups via blocked Rees matrix semigroups [1], which in turn yields the Rees theorem for completely 0-simple semigroups.     

Basic properties of e-inversive semigroups

Generalizing a property of regular resp. finite semigroups a semigroup S is called E-(0-) inversive if for every a ∈ S4(a ≠ 0) there exists x ∈ S such that ax (≠ 0) is an idempotent. Several characterizations are given allowing to identify the (completely, resp. eventually) regular semigroups in this class. The case that for every a ∈ S4(≠ 0) there exist x,y ∈ S such that ax = ya(≠ 0) is an idempotent, is dealt with also. Ideal extensions of E- (0-)inversive semigroups are studied discribing in particular retract extensions of completely simple semigroups. The structure of E- (0-)inversive semigroups satisfying different cancellativity conditions is elucidated. 1991 AMS classification number: 20M10.     

The Smallest Regular Semigroups with all Four Skew Pairs of Idempotents

An ordered pair (e, f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. We have shown previously that there are four distinct types of skew pairs of idempotents. Here we consider the smallest regular semigroups that contain precisely one of each of these four types. We show that, to within isomorphism and dualisomorphism, there are six such semigroups and characterise them as quotient semigroups of certain regular Rees matrix semigroups.     

Archimedean ordered semigroups as ideal extensions

The main result of the paper is a structure theorem concerning the ideal extensions of archimedean ordered semigroups. We prove that an archimedean ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if an ordered semigroup S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean. As a consequence, an ordered semigroup is archimedean and contains an idempotent if and only if it is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup.     

On the translational hull of a type B semigroup

In this paper, the translational hull of a type B semigroup is considered. We prove that the translational hull of a type B semigroup is itself a type B semigroup, and give some properties and characterizations of the translational hulls of such semigroups. Moreover, we consider the translational hulls of some special type B semigroups. These results strengthen the results of Fountain and Lawson (Semigroup Forum 32:79–86, 1985) on adequate semigroups. Finally, we give a new proof of a problem posted by Petrich on translational hulls of inverse semigroups in Petrich (Inverse Semigroups, Wiley, New York, 1984).     

Semigroups generated by a group and an idempotent

It is well known that the semigroup of all transformations on a finite set X of order n is generated by its group of units, the symmetric group, and any idempotent of rank n ? 1. Similarly, the symmetric inverse semigroup on X is generated by its group of units and any idempotent of rank n ? 1 while the analogous result is true for the semigroup of all n × n matrices over a field.

In this paper we begin a systematic study of the structure of a semigroup S generated by its group G of units and an idempotent ? . The first section consists of preliminaries while the second contains some general results which provide the setting for those which follow.

In the third section we shall investigate the situation where G is a permutation group on a set X of order n and ? is an idempotent of rank n ? 1. In particular, we shall show that any such semigroup S is regular. Furthermore we shall determine when S is an inverse or orthodox semigroup or completely regular semigroup.

The fourth section deals with a special case, that in which G is cyclic. The fifth, and last, deals with the situation where G is dihedral. In both cases, the resulting semigroup has a particularly delicate structure which is of interest in its own right. Both situations are replete with interesting combinatorial gems.

The author was led to the results of this paper by considering the output of a computer program he was writing for generating and analyzing semigroups.