Types of points and algebras

Abstract

The connection between classical model theoretical types (MT-types) and logically-geometrical types (LG-types) introduced by B. Plotkin is considered. It is proved that MT-types of two [Formula: see text]-tuples in two universal algebras coincide if and only if their LG-types coincide. Two problems set by B. Plotkin are considered: (1) let two tuples in an algebra have the same type, does it imply that they are connected by an automorphism of this algebra? and (2) let two algebras have the same type, does it imply that they are isomorphic? Some varieties of universal algebras are considered having in view these problems. In particular, it is proved that if a variety is hopfian or co-hopfian, then finitely generated free algebras of such a variety are completely determined by their type.