INJECTIVITY IN CATEGORIES TO REPRESENT ALL FIRST ORDER FORMULAS, I
Hajnal Andréka, Istvàn Németi · Demonstratio Mathematica · 1979
There are a steadily growing number of versions of model theory and universal algebra which differ from classical model theory not only in syntax but first of all in semantics.I.e. the mathematical objects called models are different.Also their homomorphisms are different.One of the examples is partial algebras: The universal algebra of partial algebras is basically different from the classical one not because of syntax but because the models and their homomorphisms lead to a quite new category.Cf.Burmeister [6], Andréka-Nemeti [l], [2], Hoehnke [16] , John [1 7]."Nonclassical model theory", "intensional model theory", the theory of "generalised Kripke-models" etc. are important examples, cfi.Dahn [a], Kutschera [l9], Makowsky-Marcja [23].The kinds of generalised algebras investigated by the ADJ team (cf.Goguen et alj.[13] ) are other examples.Abstract model theory was started by Makowsky, Shelah, Barwise and others to unify at least some of the different model theories, cf.Makowsky [22], Barwise [4].There is a version of abstract model theory in which stress is laid on the case when the models themselves and their homomorphisms are allowed to vary.In this approach, by a language we understand a triple L = , where P and M are arbitrary classes and ^ is a relation between them, i.e.Q M»F.P is called "the class of formulas" of L, M Is called "the olass of modela" of L and t= Is called -717