Relatively Homogeneous Structures

Publisher Summary The א 0- categoricity is a model theoretic notion. An elementary theory is called א 0 -categorical if its denumerable (i.e., countably infinite) models are all isomorphic. A structure is called א 0 -categorical if its elementary theory is א 0 -categorical. The following question arises immediately: What morphisms are available to establish isomorphisms between the denumerable models of א 0 -categorical theories? To answer this question in full generality apparently one needs the notion of a model theoretic morphism. Because this notion generally does not have a clearly discernible algebraic meaning, algebraists tend to find access to the subject of categoricity difficult. This is very unfortunate because in applications to algebra usually more special algebraic morphisms are utilized, which play an important role in the structure theory of the algebras under consideration. Hence, the techniques used to establish applied categoricity results are often familiar and of considerable interest to the algebraist. Moreover, applied categoricity results can frequently be converted into structure theorems of algebra.