Stability of Universal Equivalence of Groups under Free Constructions

In his important paper in [3] J. Stallings introduced a generalisation of amalgamated products of groups – called a pregroup, which is a particular kind of a partial group. He then defined the universal group U(P ) of a pregroup P to be a universal object (in the sense of category theory) extending the partial operations on P to group operations on U(P ). The universal group turned out to be a versatile and convenient generalisation of classical group constructions: HNN-extensions and amalgamated products. In this respect the following general question arises.Which properties of pregroups, or relations between pregroups, carry over to the respective universal groups? The aim of this paper is to prove that universal equivalence of pregroups extends to universal equivalence of their universal groups. We begin by some preliminary model-theory results. We refer the reader to [2] for a detailed introduction to model theory. The main goal here is to give a criterion of universal equivalence of two models in the form that best suits our needs.