A CLASS OF NATURALLY ORDERED ABUNDANT SEMIGROUPS WITH A GREATESTIDEMPOTENT

1PreliminariesInthissectionwepresentsomebasicresultsonPalterlyorderedSendgroups,whichwillbeusedfrequentlyinthisPaperThefollowingresultsarequotedfromBlythandMcFaddenl3],justastheysay)whichisalmostpartofthebaulk-lore"ofthetheoryoforderedsetwouPS.Theorem1.1Let(S,S)beanorderedsendgroupwiththesetEofidempotents.Supposethate,jEEwitheSI.Thenel,fe,jef,efeEE.Moreover,ifSisnaturallyorderedthenefe=e.Corollary1.2IfSisnaturallyorderedandhasagreatestidempotentattheneke=eforanyeeE.Inwhatfollows,Sde…     

具有正规中间幂等元的富足半群的构造

本文研究富足半群上中间幂等元的若干性质，并给出了具有正规中间幂等元的富半群的构造。作为推论，我们得到Ｂｌｙｔｈ和ＭｃＦａｄｄｅｎ［１］及Ｅ１－Ｑａｌｌａｌｉ［２］中相应结果。     

一类abundant半群的构造

本文讨论ａｂｕｎｄａｎｔ半群上中间幂等元的性质，研究具有正规中间幂等元的ｑｕａｓｉ－ａｄｅｑｕａｔｅ半群及若干极端情形，并分别给出各类半群的构造．     

Derived Rees Matrix Semigroups as Semigroups of Transformations

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. Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described (as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that T_X contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that T_X contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7.     

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.     

The structure of medial weakly U-abundant semigroups

We consider the class of weakly U-abundant semigroups satisfying the congruence condition (C) containing both the class of regular semigroups and the class of abundant semigroups as its subclasses. The class of weakly U-abundant semigroups with a medial projection satisfying the congruence condition (C) will be particularly studied. This kind of semigroups will be called medial weakly U-abundant semigroups. In this paper, we establish a structure theorem for such semigroups. It is proved that every medial weakly U-abundant semigroup can be expressed by some kind of bands and quasi-Ehresmann semigroups. Our theorem generalizes and enriches the structure theorem given by M. Loganathan in 1987 for regular semigroups with a medial idempotent.     

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.     

含有中间幂等元满足同余条件的◇-富足半群

本文引入了--格林关系和--富足半群,研究了满足同余条件含有中间幂等元的--富足半群.利用具有中间幂等元的由幂等元生成的正则半群和◇-拟恰当半群建立了满足同余条件含有中间幂等元的◇-富足半群的结构.     

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.     

Some subdirectly irreducible idempotent semigroups

Subdirectly irreducible idempotent semigroups were characterized in [3], and in that paper, their connection with the various equational classes of idempotent semigroups was discussed. All these results are in terms of identities, so that examples of subdirectly irreducibles in the equational classes are explicitly known only for small classes. It is easy to show from general considerations (see the last section of the present paper) that every proper equational subclass of the class of idempotent semigroups is generated (as an equational class) by one or two subdirectly irreducibles. In this paper we give an example of a subdirectly irreducible for each join irreducible equational class of idempotent semigroups, which generates the class. This list, together with known results, gives explicit examples of one or two finite subdirectly irreducibles which generate the various equational classes. Research supported by the National Research Council of Canada.     

纯正半群上的同余扩张(一)

刻划半群上的同余及其扩张是半群的代数理论中的一个非常重要的课题．本文讨论了带上的同余的正规性和不变性以及在其Ｈａｌｌ半群上的扩张，从同余扩张的角度刻划了带上的同余的性质，给出了扩张的极大、极小同余的描述．     

Compact totally disconnected k-divisible semigroups

One of the early results [5] regarding divisibility in semigroups states that no finite non-degenerate group is divisible. A sequel to this (which in view of well-known results on compact semigroups is a generalization) is that a compact semigroup is divisible if and only if each component is a divisible subsemigroup [2]. Consequently, a finite semigroup is divisible if and only if it is an idempotent semigroup. However, it is of some interest to know which finite semigroups are k-divisible for a given positive integerk≥2. In this note we present a complete characterization of finitek-divisible semigroups, and use this along with a result of K. Numakura [8], to characterize compact totally disconnected k-divisible semigroups     

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.     

Subgroups of free idempotent generated regular semigroups

In [6] it is shown that the maximal subgroups of the free idempotent generated regular semigroup which is determined by the biordered set of a completely O-simple semigroup are free. In this note we shall extend this result to a wider class of semigroups.     

Topological regular semigroups and topological inductive groupoids

The theory of topological semigroups has emerged as a separate discipline during the fifties (cf. Hofmann and Mostert [5] and Carruth, Koch and Hildebrant [3]), but the idea of topologically consistent regularity in such semigroups does not seem to have gained much attention. In this paper topological regular semigroups are introduced where the regularity is topologically consistent. The topological version of the structure theorem for regular semigroups given by Nambooripad [7] is provided here. Some special cases are also discussed. The necessary terminology on topological groupoids are available in Brown and Hardy [1] and Mackenzie [6].