CATEGORY OF THE TRANSLATIONAL HULL OF TYPE A SEMIGROUP AND ITS CONGRUENCE-PRESERVING *-HOMOMORPHIC REPRESENTATION

CATEGORY OF THE TRANSLATIONAL HULL

Abstract

It is an established result that Type A semigroup is embeddable in an inverse semigroup with the characterization that $S$ is a Type A semigroup if and only if there exist inverse semigroups $S_1$, $S_2$, and embeddings $\phi_1: S \rightarrow S_1$, $\phi_2: S \rightarrow S_2$, such that
                    \[
                    \phi_1 a^* = (\phi_1 a)^* = (\phi_1 a)^{-1} (\phi_1 a), \quad \phi_2 a^\dagger = (\phi_2 a)^\dagger = (\phi_1 a) (\phi_1 a)^{-1}.
                    \]
                    
            With full transformation semigroup, this characterization leads to faithful representation of Type A semigroup and the representation has been extended to the translational hull of Type A semigroup and further to the category of the translational hull of Type A monoid. In this paper, we married up this categorical embedding with some extensions of congruences and homomorphisms, and found out that the categorical representation is structure-preserving and congruence-preserving, among other useful results.