Definable and Autostable Congruences

Abstract

UDC 519.48 We establish a relationship between autostable and point-definable congruences on countable universal algebras. Bibliography :9 titles. The source of this paper is two-fold. First, since the main objects studied in a universal algebra are universal algebras up to an isomorphism, it is natural to require, working with derived objects (subalgebras, endomorphisms, automorphisms, congruences, and so on), that the derived objects (for isomorphic algebras) should be in a one-to-one correspondence independently of a particular isomorphism between the algebras. This leads to consideration of those derived objects on universal algebras that are stable under automorphisms of these algebras (in particular, autostable congruences). Second, it is natural to consider derived objects that are definable on the algebra in some language of logic L (L-definable congruences). In this paper, we study the main simplest properties of autostable and L-definable (for different languages L of logic) congruences and their interaction. A congruence θ on a universal algebra A = � A; σ� is autostable if for any a, b ∈ A and