Action of endomorphism semigroups on definable sets

Abstract

The aim of the paper is to construct, discuss and apply the Galois-type correspondence between subsemigroups of the endomorphism semigroup [Formula: see text] of an algebra [Formula: see text] and sets of logical formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of [Formula: see text] on definable sets over [Formula: see text]. We treat some applications of this Galois correspondence. The first one concerns logic geometry. Namely, it gives a uniform approach to geometries defined by various fragments of the initial language. The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way, one can get an effective syntactical expression by semantic tools. Yet another advantage is a common approach to generalizations of the main model theoretic concepts to the sublanguages of the first-order language and revealing new connections between well-known concepts. The fourth application concerns the generalization of the unification theory, or more generally Term Rewriting Theory, to the logic unification theory.