On Automorphisms of Categories of Universal Algebras

Abstract

Abstract. Given a variety V of universal algebras. A new approach is suggested to characterize algebraically automorphisms of the category of free V-algebras. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category are inner or not. This question is important for universal algebraic geometry [5,9]. Most of results will actually be proved to hold for arbitrary categories with a represented forgetful functor.

Automorphisms of categories of free algebras of some varieties

“…For the sake of completeness we include here some results from [15] in the form adjusted to the case of the categories of free universal algebras.…”

“…This method extends some ideas from [15]. For the reader's convenience we give below a sketch of the method.…”

Multi-Sorted Logic and Logical Geometry: Some Problems

“…Note that if there exists ϕ subject to the conditions of Theorems 3.22 or 3.23, then algebras H 1 and H 2 are called automorphically equivalent (see [27], [39]- [43], [34], [33] for definitions and discussions).…”

Automorphisms of the endomorphism semigroup of a polynomial algebra

“…For every small category C, denote the group of all its automorphisms by Aut C. We distinguish the following classes of automorphisms of C. [8,15,20].) An automorphism ϕ : C → C is equinumerous if ϕ(D) ∼ = D for any object D ∈ Ob C; ϕ is stable if ϕ(D) = D for any object D ∈ Ob C; and ϕ is inner if ϕ and 1 C are naturally isomorphic, i.e., ϕ ∼ = 1 C .…”

“…Consider a constant morphism ν 0 : F (X) → F (X) such that ν 0 (x) = x 0 , x 0 ∈ F (X), for every x ∈ X . [8,13,16,20,23].) Let the free algebra F (X) generate a variety Θ, and ϕ ∈ St Aut Θ 0 .…”