ИСТИНА

In Model Theory, the results called Definability Criteria play a fundamental role. These are theorems that give necessary and sufficient conditions for a class of structures to be definable by a single formula or a set of formulas in the chosen language, or to be representable as the union of (finitely) axiomatizable classes (these four options are exhaustive, in some sense). Here we introduce, in a general setting that is applicable to a wide range of languages and types of structures, the notions of a compact and a saturated class of structures. In these terms, we formulate general Definability Criteria theorems. We also introduce the notions of a compatification operation on structures and a saturation operation on structures; the closure of a class of structures under these operations guarantee the compactness and saturatedness of the class, respectively. We formulate Definability Criteria in terms of closure under these operations.