Kernels of inverse semigroup homomorphisms

Abstract

The aim of this note is to give an analogue, for an inverse semigroup S, of the theorem for a group G which says that if G is the set of normal subgroups of G, then the map N → (N) = {(a, b) ∈ G x G: ab^-1 ∈ N}; for N ∈ G is a 1: 1 order preserving map of G onto ∧(G), the lattice of congruences on G. It will be shown that if E is the semilattice of idempotents of S, P = {E: α ∈ J} is a normal partition of E, and K is a certain collection of self conjugate inverse subsemigroups of S, then the map K →(X) = {(a, b)∈ S x S: a^-la, b^-1b∈ E_α for some α ∈ J and ab^-l ∈ K) for K e Jf is a 1:1 map of K onto the set of congruences on S which induce P.