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.     

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.     

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.     

Cryptic Semigroup Varieties

We describe all minimal noncryptic periodic semigroup [monoid] varieties. We prove that there are exactly three distinct maximal cryptic semigroup [monoid] varieties contained in the variety determined by xⁿ ≈ x ^n+m, n ≥ 2, m ≥ 2. Analogous results are obtained for pseudovarieties: there are exactly three maximal cryptic pseudovarieties of semigroups [monoids]. It is shown that if a cryptic variety or pseudovariety of monoids contains a nonabelian group, then it consists of bands of groups only. Several characterizations are given of the cryptic overcommutative semigroup [monoid] varieties.     

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 .     

On a class of completely join prime <Emphasis Type="Italic">J</Emphasis>-trivial semigroups with unique involution

A finite algebra is completely join prime if whenever it belongs to the complete join of some collection of pseudovarieties, then it belongs to one of the pseudovarieties. An infinite class of completely join prime J-trivial semigroups with unique involution is introduced to demonstrate the incompatibility between the lattice of pseudovarieties of involution semigroups and the lattice of pseudovarieties of semigroups. Examples are also exhibited to show that a finite involution semigroup and its semigroup reduct need not be simultaneously completely join prime.     

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.     

Tannakian formalism for fiber functors over tensor categories

It is well known that the pseudovariety \(\mathbf {J}\) of all \(\mathscr {J}\)-trivial monoids is not local, which means that the pseudovariety \(g\mathbf {J}\) of categories generated by \(\mathbf {J}\) is a proper subpseudovariety of the pseudovariety \(\ell \mathbf {J}\) of categories all of whose local monoids belong to \(\mathbf {J}\). In this paper, it is proved that the pseudovariety \(\mathbf {J}\) enjoys the following weaker property. For every prime number p, the pseudovariety \(\ell \mathbf {J}\) is a subpseudovariety of the pseudovariety \(g(\mathbf {J}*\mathbf {Ab}_p)\), where \(\mathbf {Ab}_p\) is the pseudovariety of all elementary abelian p-groups and \(\mathbf {J}*\mathbf {Ab}_p\) is the pseudovariety of monoids generated by the class of all semidirect products of monoids from \(\mathbf {J}\) by groups from \(\mathbf {Ab}_p\). As an application, a new proof of the celebrated equality of pseudovarieties \(\mathbf {PG}=\mathbf {BG}\) is obtained, where \(\mathbf {PG}\) is the pseudovariety of monoids generated by the class of all power monoids of groups and \(\mathbf {BG}\) is the pseudovariety of all block groups.     

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 (\(\mathsf{DA}\)).     

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

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.     

On the lattice of sub-pseudovarieties of DA

The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of sub-pseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal’cev products with the pseudovarieties of definite and reverse definite semigroups.     

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     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.