On the Representation of the Lattices of the Algebraic Sets of the Universal Algebras

Abstract

The concept of an algebraic set is a basic concept of the classical algebraic geometry over fields. This concept, along with the concept of an algebraic lattice of algebraic sets is the basic concept of so-called algebraic geometry of universal algebras. Moreover, there are traditionally two approaches to the definition of algebraic sets: the first is a direct generalization of the classical situation of the concept of the algebraic set over a field and connected with the homomorphisms of free algebras in the considered algebra, the second is formulated within the framework of the traditional model theory. In this paper we propose another approach to the characterisation of algebraic sets based on the concept of the inner homomorphisms of some extensions of considered algebra. Based on this approach we introduce the other representation of the lattices of the algebraic sets of universal algebras. Also we propose the criterion, in terms of internal homomorphisms, of the coincidence of the families of the algebraic sets of universal algebras with identical basic sets.

Keywords

• algebraic set
• lattice
• innere homomorphism