Finite aperiodic semigroups with commuting idempotents and generalizations

It is proved that any pseudovariety of finite semigroups generated by inverse semigroups, the subgroups of which lie in some proper pseudovariety of groups, does not contain all aperiodic semigroups with commuting idempotents. In contrast we show that every finite semigroup with commuting idempotents divides a semigroup of partial bijections that shares the same subgroups. Finally, we answer in the negative a question of Almeida as to whether a result of Stiffler characterizing the semidirect product of the pseudovarieties ofR-trivial semigroups and groups applies to any proper pseudovariety of groups.     

A note on the power semigroup of a completely simple semigroup

We prove the pseudovariety generated by power semigroups of completely simple semigroups is the semidirect product of the pseudovariety of block groups with the pseudovariety of right zero semigroups, and hence is decidable. This answers a question of Almeida from over 15 years ago. The author was supported in part by NSERC.     

Some pseudovariety joins involvinglocally trivial semigroups

This paper is concerned with the computation of pseudovariety joins involving the pseudovariety L I of locally trivial semigroups. We compute, in particular, the join of L I with any subpseudovariety of CR(m in circle)N, the Mal’cev product of the pseudovariety of completely regular semigroups and the pseudovariety of nilpotent semigroups. Similar studies are conducted for the pseudovarieties K, D and N, where K (resp. D) is the pseudovariety of all semigroups S such that eS=e (resp. Se=e ) for each idempotent e of S .     

The pseudovariety generated by completely 0-simple semigroups

A finite basis of pseudoidentities of the pseudovariety generated by all finite completely 0-simple semigroups is constructed. Thus this pseudovariety is decidable. Partially supported by Israel Ministry of Absorption     

Small inherently nonfinitely based finite semigroups

We show how to construct all ``forbidden divisors' for the pseudovariety of not inherently nonfinitely based finite semigroups. Several other results concerning finite semigroups that generate an inherently nonfinitely based variety that is miminal amongst those generated by finite semigroups are obtained along the way. For example, aside from the variety generated by the well known six element Brandt monoid \tb , a variety of this type is necessarily generated by a semigroup with at least 56 elements (all such semigroups with 56 elements are described by the main result). September 23, 1999     

The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

We use classical results on the lattice of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice Ps (DA) of subpseudovarieties of DA, – where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. We bring forward a lattice congruence on Ps (DA), whose quotient is isomorphic to , and whose classes are intervals with effectively computable least and greatest members. Also we characterize the pro-identities satisfied by the members of an important family of subpseudovarieties of DA. Finally, letting V _k be the pseudovariety generated by the k-generated elements of DA (k≥ 1), we use all our results to compute the position of the congruence class of V _k in . Received April 24, 1996; accepted in final form April 3, 1997.     

Some Pseudovariety Joins InvolvingLocally Trivial Semigroups

This paper is concerned with the computation of pseudovariety joins involving the pseudovariety L I of locally trivial semigroups. We compute, in particular, the join of L I with any subpseudovariety of CR(m in circle)N, the Mal'cev product of the pseudovariety of completely regular semigroups and the pseudovariety of nilpotent semigroups. Similar studies are conducted for the pseudovarieties K, D and N, where K (resp. D) is the pseudovariety of all semigroups S such that eS=e (resp. Se=e ) for each idempotent e of S . May 5, 1999     

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.     

Semigroups of order preserving mappings on a finite chain: A new class of divisors

In this paper we aim to prove that every semigroup of the pseudovariety generated by all semigroups of partial, injective and order preserving transformations on a finite chain belongs to the pseudovariety generated by all semigroups of order preserving mappings on a finite chain. This research was done within the project SAL (JNICT, PBIC/C/CEN/1021/92), and the activities of the “Centro de álgebra da Universidade de Lisboa”.     

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.     

Irreducibility of certain pseudovarieties 1

We prove that the pseudovarieties of all finite semigroups, and of all aperiodic finii e semigroups are irreducible for join, for semidirect product and for Mal’cev product. In particular, these pseudovarieties do not admit maximal proper subpseudovarieties. More generally, analogous results are proved for the pseudovariety of all finite semigroups all of whose subgroups are in a fixed pseudovariety of groups H, provided th.it H is closed under semidirect product.     

Normally Ordered Inverse Semigroups

NO of all normally ordered inverse semigroups. We show that the pseudovariety of inverse semigroups PCS generated by all semigroups of injective and order partial transformations on a finite chain consists of all aperiodic elements of NO . Also, we prove that NO is the join pseudovariety of inverse semigroups. PCS V G , where G is the pseudovariety of all finite groups.     

On semidirectly closed pseudovarieties of aperiodic semigroups

The aim of this work is to study the unknown intervals of the lattice of aperiodic pseudovarieties which are semidirectly closed and answer questions proposed by Almeida in his book “Finite Semigroups and Universal Algebra”. The main results state that the intervals [V^*(B₂),ER∩LR] and [V^*(B₂¹),ER∩A] are not trivial, and that both contain a chain isomorphic to the chain of real numbers. These results are a consequence of the study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B₂.     

Reducibility of pointlike problems

We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all finite semigroups in which the order of every subgroup is a product of elements of a fixed set \(\pi \) of primes; the pseudovariety of all finite semigroups in which every regular \(\mathcal J\)-class is the product of a rectangular band by a group from a fixed pseudovariety of groups that is reducible for the pointlike problem, respectively graph reducible. Allowing only trivial groups, we obtain \(\omega \)-reducibility of the pointlike and idempotent pointlike problems, respectively for the pseudovarieties of all finite aperiodic semigroups (\(\mathsf{A}\)) and of all finite semigroups in which all regular elements are idempotents (\(\mathsf{DA}\)).     

Reducibility of Joins Involving Some Locally Trivial Pseudovarieties

Abstract

In this paper, we show that σ-reducibility is preserved under joins with K, where K is the pseudovariety of semigroups in which idempotents are left zeros. Reducibility of joins with D, the pseudovariety of semigroups in which idempotents are right zeros, is also considered. In this case, we were able to prove that σ- reducibility is preserved for joins with pseudovarieties verifying a certain property of cancellation. As an example involving the semidirect product, we prove that Sl*K is κ-tame, where Sl stands for the pseudovariety of semilattices.