A Problem of B. Plotkin for S-Acts: Automorphisms of Categories of Free S-Acts

In this paper, we prove that all automorphisms of categories of free S-acts are semi-inner, which solves a variation of a well known B. Plotkin's problem in the case of monoids. We also give a description of automorphisms of categories of finitely generated free algebras of varieties of unary algebras, and show that among varieties of unary algebras only the variety of mono-unary algebras is perfect.