On countable locally described structures

Abstract

We consider classes of locally X-structures of the form LX = {Gƒ∀a 1, … a n ϵ G ∃A ϵ Ч such that α 1, …, a n ϵ A ⩽ G} where Ч is a countable set of isomorphism types of finitely generated structures. In LЧ we study (1) closed structures also known as basically saturated structures which are algebraic analogues of existentially closed structures and (2) saturated structures which are algebraic analogues of existentially universal structures. We do not suppose that LЧ have the amalgamation property as in Jónsson type model theory nor that LЧ be first-order axiomatizable as in classical model theory. The main theorems of the general theory give characterizations for (1) uniqueness, up to isomorphism, of the countable closed structure and (2) the existence of a countable saturated structure. Our criterions are given in terms of local amalgamation properties of Ч. New among the examples are the results that there exists a unique countable closed structure in the classes of locally finitely presented groups, torsion-free locally nilpotent groups, and torsion-free nilpotent groups of class c ( c≥ 2).