Abstract
We introduce a preorder on an inverse semigroup
S
associated to any normal inverse subsemigroup
N
, that lies between the natural partial order and Green’s
J
–relation. The corresponding equivalence relation
≃
N
is not necessarily a congruence on
S
, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on
S
. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.