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