Generations of generative classes

S. V. Sudoplatov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 22 (2017), pp. 106-117

Résumé

We study generating sets of diagrams for generative classes. The generative classes appeared solving a series of model-theoretic problems. They are divided into semantic and syntactic ones. The fists ones are witnessed by well-known Fraïssé constructions and Hrushovski constructions. Syntactic generative classes and syntactic generic constructions were introduced by the author. They allow to consider any $\omega$-homogeneous structure as a generic limit of diagrams over finite sets. Therefore any elementary theory is represented by some their generic models. Moreover, an information written by diagrams is realized in these models. We consider generic constructions both in general case and with some natural restrictions, in particular, with the self-sufficiency property. We study the dominating relation and domination-equivalence for generative classes. These relations allow to characterize the finiteness of generic structure reducing the construction of generic structures to maximal diagrams. We also have that a generic structure is finite if and only if given generative class is finitely generated, i.e., all diagrams of this class are reduced to copying of some finite set of diagrams. It is shown that a generative class without maximal diagrams is countably generated, i.e., reduced to some at most countable set of diagrams if and only if there is a countable generic structure. And the uncountable generation is equivalent to the absence of generic structures or to the existence only uncountable generative structures.

Détail

Keywords: generative class, generic structure, generation of generative class.