On automorphisms of the endomorphism semigroup of a free universal algebra

Let U be a universal algebra. An automorphism a of the endomorphism semigroup of U defined by α(φ) = sφs^-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.