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A mapping 𝜃 of an inverse
semigroup S into an inverse semigroup T is called a v-prehomomorphism if, for each
a,b ∈ S, (ab)𝜃 ≦ a𝜃b𝜃 and (a−1)𝜃 = (a𝜃)−1. The congruences on an E-unitary inverse
semigroup P(G,𝒳,𝒴) are determined by the normal partition of the idempotents,
which they induce, and by v-prehomorphisms of S into the inverse semigroup of
cosets of G.
Inverse semigroups, with v-prehomomorphisms as morphisms, constitute a
category containing the category of inverse semigroups, and homomorphisms, as a
coreflective subcategory. The coreflective map η : S → V (S) is an isomorphism if the
idempotents of S form a chain and the converse holds if S is E-unitary or a
semilattice of groups. Explicit constructions are given for all v-prehomomorphisms on
S in case S is either a semilattice of groups or is bisimple.
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