Congruences on regular semigroups

Abstract

For any regular semigroup S the relation θ is defined on the lattice, ⋀(S), of congruences on S by: (p, τ) ∈ θ if and only if p and τ induce the same partition of the idempotents of S. Then θ is an equivalence relation on ⋀(S) such that each equivalence class is a complete modular sublattice of ⋀(S). If S is an inverse semig oup then θ is a congruence on ⋀(S), ⋀(S)/θ is complete and the natural homomorphism of A(S) onto A(S)/θ is a complete lattice homomorphism. Any congruence on an inverse semigroup S can be characterized in terms of its kernel, namely, the set of congruence classes containing the idempotents of S. In particular, any congruence on S induces a partition of the set Es of idempotents of S satisfying certain normality conditions. In this note, those partitions of Es which are induced by congruences on S and the largest and smallest congruences on S correspond ing so such a partition of Es are characterized. © 1967 by Pacific Journal of Mathematics.