Unityped algebras

Abstract

The paper is essentially a continuation of B.Plotkin, G.Zhitomirski, "Some logical invariants of algebras and logical relations between algebras", St.Peterburg Math. J., {19:5}, (2008) 859 -- 879, whose main notion is that of logic-geometrical equivalence of algebras (LG-equivalence of algebras). This equivalence of algebras is more strict than elementary equivalence. In the paper we introduce the notion of unityped algebras and relate it to LG-equivalence. We show that these notions coincide. The idea of the type is one of the central ideas in M odel Theory. The correspondence introduced in the paper stimulates a bunch of problems which connect universal algebraic geometry and Model Theory. The paper consists of five sections: 1. General view 2. Logical noetherianity 3. Unitypeness and isomorphism 4. Logically perfect algebras 5. Some facts from algebraic logic. We provide a new general view on the subject, arising "on the territory" of universal algebraic geometry, which yield applications of algebraic logic and universal geometry in Model Theory.