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.     

Direct sums and refinement

In this paper we present a division theorem for the pseudovariety of semigroups OP generated by all semigroups of orientation preserving full transformations on a finite chain, which is achieved by using a natural representation of a certain monoid of transformations of a set X as a monoid of transformations of a subset of X.     

Abelian Kernels of Some Monoids of Injective Partial Transformations and an Application

In this paper we compute the abelian kernels of the monoids POI_n and POPI_n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI_n (n epsilon N) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI_n (n epsilon N) with the pseudovariety Ab of all finite abelian groups.     

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.     

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     

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 (\(\mathsf{DA}\)).     

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.     

On the hyperdecidability of semidirect products of pseudovarieties

The notion of hyperdecidability has been introduced as a tool which is particularly suited for granting decidability of semidirect products. It is shown in this paper that the semidirect product of an hyperdecidable pseudovariety with a pseudovariety whose finitely generated free objects are finite and effectively computable is again hyperdecidable. As instances of this result, one obtains, for example, the hyperdecidability of the pseudovarieties of ail finite completely simple semigroups and of all finite bands of left groups.     

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.     

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.     

E-local pseudovarieties

Generalizing a property of the pseudovariety of all aperiodic semigroups observed by Tilson, we call E -local a pseudovariety V which satisfies the following property: for a finite semigroup, the subsemigroup generated by its idempotents belongs to V if and only if so do the subsemigroups generated by the idempotents in each of its regular $\mathcal{D}$ -classes. In this paper, we present several sufficient or necessary conditions for a pseudovariety to be E-local or for a pseudoidentity to define an E-local pseudovariety. We also determine several examples of the smallest E-local pseudovariety containing a given pseudovariety.     

Power Exponents of Aperiodic Pseudovarieties

n such that the iterated power P ⁿV is the pseudovariety of all finite semigroups.     

Some open problems concerning relatively free and free profinite semigroups

Besides constituting an important and often crucial tool in studying semigroup varieties and pseudovarieties, relatively free and free profinite semigroups have proved to be of interest from many other natural standpoints. We present three groups of new problems concerning certain relatively free and free profinite semigroups and give background and motivation for them. The problems deal with the first-order theories of free profinite semigroups, the structure of free Burnside semigroups with commuting idempotents and of free Burnside inverse semigroups, and the representation of finitely generated free bands by order preserving transformations of a finite chain. April 19, 2001     

Filters in (Quasiordered) Semigroups and Lattices of Filters

A filter in a semigroup is a subsemigroup whose complement is an ideal. (Alternatively, in a quasiordered semigroup, a slightly more general definition can be given.) We prove a number of results related to filters in a semigroup and the lattice of filters of a semigroup. For instance, we prove that every complete algebraic lattice can be the lattice of filters of a semigroup. We prove that every finite semigroup is a homomorphic image of a finite semigroup whose lattice of filters is boolean and which belongs to the pseudovariety generated by the original semigroup. We describe filter lattices of some well-known semigroups such as full transformation semigroups of finite sets (which are three-element chains) and free semigroups (which are boolean).     

The structure of finite monoids satisfying the relation ℛ = ℋ

It is shown that any finite monoid S on which Green’s relations R and H coincide divides the monoid of all upper triangular row-monomial matrices over a finite group. The proof is constructive; given the monoid S, the corresponding group and the order of matrices can be effectively found. The obtained result is used to identify the pseudovariety generated by all finite monoids satisfying R = H with the semidirect product of the pseudovariety of all finite groups and the pseudovariety of all finite R-trivial monoids.     

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 .