Initial and Quasiinitial Models of Theories

Abstract

We have shown earlier that the theories which uniformly admit initial families of models resp. disjoint initial families of models resp. initial models are exactly those which are closed under equalizers resp. equalizers and pullbacks (= connected limits) resp. equalizers and products (= arbitrary limits). In addition, syntactical characterizations had been given.

To obtain an analogous result for pullbacks we have to weaken the initiality notions. Weakening the uniqueness of the morphism to a uniqueness up to an isomorphism we arrive at the corresponding quasiinitiality notions. As a new result we shall show that the theories which uniformly admit disjoint quasiinitial families of models resp. quasiinitial models are exactly those which are closed under pullbacks resp. pullbacks and products (= arbitrary limits). As a tool we use a localized version of initiality which was suggested by J.Adamek. Using our result Hébert recently obtained a syntactical characterization for the case of disjoint quasiinitial families. In the case of quasiinitial families the characterization by means of certain class of limit constructions is still an open problem.