Kernels of inverse semigroup homomorphisms

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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∈ Eα 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.

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