1.Ministry of Higher Education,University of Dammam,Dammam,Kingdom of Saudi Arabia;2.School of Mathematical and Computer Sciences and the Maxwell Institute for the Mathematical Sciences,Heriot-Watt University,Edinburgh,UK
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 ({mathcal {J}})–relation. The corresponding equivalence relation (simeq _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.