A certain sublattice of the lattice of congruences on a regular semigroup

Abstract

It is well known that, with respect to the natural partial ordering, the set of all congruences on a group forms a modular lattice. In the present paper we develop an extension of this result to the case of a regular semigroup S (α ∈ α S α for all α in S ). Let Σ(ℋ) denote the set of all congruences on S with the property that congruent elements generate the same principal left ideal and the same principal right ideal of S . We show (Theorem 1) that, under the natural partial ordering, Σ(ℋ) is a modular lattice with a greatest and a least element.