On a sublattice of the lattice of congruences on a simple regular ω-semigroup

Abstract

The set E of idemponents of a semigroup S can be partially ordered by defining e ≦f and only if ef = fe = e (e,f ∈ E). If E = {ei: i = i = 0,1…} and under this ordering e0 > e1 > e2… then we call S an ω-semigroup. Munn [10] has given a complete classification of simple regular ω-semigroups in terms of groups and group homomorphisms. Let ∧0(S) denote the set of congruences on a simple regular w-semigroup S consisting of those congruences which either are idempotent-separating or are group congruences on S. It is evident that ∧0(S) is a sublattice of the lattice of all congruences on S.

Congruences on simple regular ω-semigroups

“…Let 3^ be the usual Green's relation and a the minimal group congruence on S. In this paper we determine all congruences contained in [i s , a\J 3f\. Our work extends the results of [1] where all congruences contained in [i s , 3^~\ U \a, a \] 3T\ are determined.…”

“…It was showniin [1] that^f is in fact a congruence and that S/Jf S B d . A congruence p on a semigroup S is called a group congruence if Sjp is a group.…”

Congruences on simple ω-semigroups

“…1^, e jt m)(p, e k , q) is obtained by substituting 7 for i in (2). Now using hypothesis (1), the definition of p, and (2) .…”

Modularity of the lattice of congruences of a regular ω-semigroup

“…Baird [1] studied simple co-semigroups and gave a necessary and sufficient condition for the sublattice of L(S) consisting of those congruences which are either idempotent separating or group congruences to be a modular lattice. This paper completes the previous results giving the characterization of regular cosemigroups whose congruences lattice is modular.…”