Inner Automorphisms of Clifford Monoids

An automorphism φ of a monoid S is called inner if there exists g in US, the group of units of S, such that φ(s)=gsg⁻¹ for all s in S; we call S nearly complete if all of its automorphisms are inner. In this paper, first we prove several results on inner automorphisms of a general monoid and subsequently apply them to Clifford monoids. For certain subclasses of the class of Clifford monoids, we give necessary and sufficient conditions for a Clifford monoid to be nearly complete. These subclasses arise from conditions on the structure homomorphisms of the Clifford monoids: all being either bijective, surjective, injective, or image trivial.