Categorical analysis of faithful representation of translational hull of ample semigroups

Abstract

Representation of a semigroup has to do with obtaining a homomorphism that maps the semigroup into the full transformation semigroup. The representation is said to be faithful when it is an embedding. It is an existing result that ample monoid is embeddable into an inverse semigroup. This result has been extended to translational hulls and, in effect, a faithful representation of translational hull of ample semigroup is also an existing result. This faithful representation will be called categorical if we can establish that it is a class consisting of systems of the same type, referred to as objects and between any pair of objects A and B in the class, there are arrows f:A→B and each arrow is a structure preserving map referred to as morphism. In this paper, therefore, we want to carry out categorical analysis of faithful representation of translational hull of ample semigroup. The commutative diagrams of the faithful representation of translational hull of ample semigroups shall be very useful tools in the analysis.