INJECTIVITY IN CATEGORIES TO REPRESENT ALL FIRST ORDER FORMULAS, I

TL;DRAbstract

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], Andrka-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.

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