Stability of Universal Equivalence of Groups under Free Constructions
  In 1971 J. Stallings introduced a generalisation of amalgamated products of groups -- called a pregroup, which is a particular kind of a partial group. He 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). This turns 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? In this paper it is proved that universal equivalence of pregroups extends to universal equivalence of universal groups. Applications to free products with amalgamation and HNN-extensions are then described. 
A. J. Duncan, I.V. Kazachkov and V.N. Remeslennikov