Regular semigroups of polynomial growth

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 58–64, February, 1990.     

Regular J-classes of subspace semigroups

Subspace semigroup S(M _n (K)) of the matrix algebra over a field K is studied. Idempotents and regular J-classes are characterized and some symmetries of S(M _n (K)) are established.     

P-反演半群上的正则P-同余

令S(P)为P-反演半群，本文借助于P-核正规系来刻画S(P)上的强P-同余，证明了S(P)上的任一正则P-同余可以决定S(P)的一个P-核正规系；反之，S(P)的任一P-核正规系可以决定S(P)上的一个正则P-同余．     

Regular semigroups with a split map

Communicated by R. B. McFadden     

Regular semigroups with weakly simplistic orthodox transversals

Weakly simplistic orthodox transversals are introduced in this paper and some characterizations associated with them are obtained. An example is given to demonstrate that weakly simplistic orthodox transversals are proper generalisations of left simplistic orthodox transversals. The related results of Blyth and Almeida Santos on left simplistic inverse transversals obtained in 1996 and Kong and Luo on left simplistic orthodox transversals obtained in 2011 are generalised and amplified. A structure theorem of regular semigroups with weakly simplistic orthodox transversals is also established.     

Regular semigroups with skew pairs of idempotents

Let S be a regular semigroup with set of idempotents E(S) . Given x,y ∈ S , we say that (x,y) is a skew pair if x y \notin E(S) whereas y x ∈ E(S) . Here we use this concept to characterise certain regular Rees matrix semigroups.     

具有左简纯正断面的正则半群

本文引入并研究了左简纯正断面, 得到了与之相关的若干刻画; 推广并丰富了Blyth 和AlmeidaSantos 于1996 年得到的关于左简逆断面及Kong 于2007 年得到的关于纯正断面的相关结果; 同时,给出了具有左简纯正断面的正则半群的结构定理.     

一类变换半群的正则元

在等价关系E(∈)F的假设下,给出了变换半群TFE(X)的正则元的性质.利用这些性质,简化了正则元的格林关系,得到了更为简单的描述.     

Regular elements in sandwich semigroups of binary relations

This note gives a counterexample to a published characterization of regular elements in sandwich semigroups of binary relations. The method of that paper is used to characterize those elements having a right identity. A characterization of regular elements is obtained following the approach of Schein.     

Regular semigroups: amalgamation and the lattice of existence varieties

A class of regular semigroups is called an existence variety (ore-variety) if it is closed under , and , the operations of taking all morphic images, all regular subsemigroups, and all direct products, respectively; and such classes were previously shown (like varieties) to be determined by sets of identities. We determine whiche-varieties of regular semigroups have the amalgamation property. Preliminary results include (1) a construction of the lattice ofe-varieties of generalized inverse semigroups and (2) the result that thee-variety of combinatorial, strict, regular semigroups is the leaste-variety which is notE-solid.Dedicated to Professor G. B. Preston, on the occasion of his 65th birthdayPresented by Ralph Freese.     

On embedding of finitely generated profinite semigroups into 2-generated profinite semigroups

Koryakov recently asked the following question: when a pseudovariety V does not satisfy any non-trivial identity, does there exist an embedding from any finitely generated V-free profinite semigroup into the 2-generated V-free profinite semigroup? During the conference “Semigroups, Automata and Languages” in Porto (June 1994 [2]), a positive answer to this question was conjectured. We give here a counterexample to this conjecture. Received December 15, 1994; accepted in final form June 5, 1997.     

Regular semigroups with (REP) and (REP) op are not necessarily amalgamation bases

In the previous paper [12], we gave a necessary and sufficient condition for finite CN-bands to have the representation extension property (\REP) and its dual (\REP) ^op . In this paper, we construct a finite CN-band which has the representation extension property (\REP) and its dual (\REP) ^op , but is not a semigroup amalgamation base. This gives a negative solution to an open problem [9, Problem 2], which is originally due to T. E. Hall. October 6, 1999