A semilattice of varieties of completely regular semigroups

summary:Completely regular semigroups are unions of their (maximal) subgroups with the unary operation within their maximal subgroups. As such they form a variety whose lattice of subvarieties is denoted by $\mathcal L(\mathcal C\mathcal R)$. \endgraf We construct a 60-element $\cap$-subsemilattice and a 38-element sublattice of $\mathcal L(\mathcal C\mathcal R)$. The bulk of the paper consists in establishing the necessary joins for which it uses Polák's theorem

Characterizing pure, cryptic and Clifford inverse semigroups

summary:An inverse semigroup $S$ is pure if $e=e^2$, $a\in S$, $e<a$ implies $a^2=a$; it is cryptic if Green's relation $\mathcal {H}$ on $S$ is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and Clifford semigroups in a similar way by means of divisors. The paper also contains characterizations of completely semisimple inverse and of combinatorial inverse semigroups in a similar manner. It ends with a description of minimal non-$\mathcal {V}$ varieties, for varieties $\mathcal {V}$ of inverse semigroups considered

Subprojective lattices and projective geometry

AbstractThe class of lattices we are interested in (subprojective lattices), can be gotten by taking the MacNeille completions of the class of complemented, modular, atomic lattices. McLaughlin showed that subprojective lattices can be represented as the lattices of W-closed subspaces of a vector space U in duality with a vector space W. In this paper, we give a characterization of subprojective lattices in terms of atoms and dual atoms, by means of an incidence space satisfying self-dual axioms. In the finite-dimensional case, a subprojective lattice is projective, and hence our self-dual axioms characterize finite-dimensional projective spaces in terms of points and hyperplanes. No numerical parameters appear explicitly in these axioms. For each subprojective lattice L with at least three elements, we define a projective envelope P(L) for it. P(L) is a projective lattice and there is a natural inf-preserving injection of L into P(L). This injection has other important properties which we take as the definition of a geometric map. In the course of studying geometric maps, we obtain a lattice theoretic proof of Mackey's result that the join of a U-closed subspace of V and a finite-dimensional subspace is U-closed, where (U, V) form a dual pair of vector spaces over a division ring. Furthermore, we show that if L is a subprojective lattice, P a projective lattice, and ψ: L → P a geometric map, then P is isomorphic to the projective envelope P(L) of L. The paper presents many other properties of subprojective lattices. It concludes with a characterization of subprojective lattices which are also projective

Unary semigroups with an associate subgroup

A subgroup H of a regular semigroup S is said to be an associate subgroup of S if for every s ∈ S, there is a unique associate of s in H. An idempotent z of S is said to be medial if czc = c, for every c product of idempotents of S. Blyth and Martins established a structure theorem for semigroups with an associate subgroup whose identity is a medial idempotent, in terms of an idempotent generated semigroup, a group and a single homomorphism. Here, we construct a system of axioms which characterize these semigroups in terms of a unary operation satisfying those axioms. As a generalization of this class of semigroups, we characterize regular semigroups S having a subgroup which is a transversal of a congruence on S.Fundação para a Ciência e a Tecnologia (FCT

The normal hull of a completely simple semigroup

AbstractA completely simple subsemigroup K of a completely simple semigroup S is a normal subsemigroup of S, or S is a normal extension of K, if x−1Kx ⊆ K (x ϵ S). A normal extension S of K is an essential extension if each non-trivial congruence on S restricts to a non-trivial congruence on K. For any completely simple semigroup S, the union Φ(S) of the automorphism groups of the maximal subgroups of S is endowed with a semigroup structure such that the mapping θs of each element of S to the associated inner automorphism of the maximal subgroup containing it is a homomorphism of S into φ(S). It is shown that S has a maximal essential extension if and only if the metacentre of S (that is, the union of the centres of the maximal subgroups) is simply the set of idempotents of S. When such a maximal essential extension T exists, θs is one-to-one and there exists a monomorphism of T onto Φ(S) extending θs. A related semigroup ∑(S) whose elements are transformations of S with certain special properties (such as H-class preserving, isomorphisms on ℘-classes) is introduced and studied. A homomorphism of S into the product of ∑(S) and its left-right dual is constructed which induces the same congruence on S as θs