Abstract
Abstract. In this work, we follow the algebraic approach using definability by formulas presentable in both existential and universal forms. The class of algebraic Cartesian interpretations of theories is studied presenting a foundation of the finitary first-order combinatorics. Common properties of first-order definability in finite models are studied. Some relations are obtained between automorphism groups of finite models and isomorphisms of Cartesian extensions of their theories. A formal definition of the notion of a model-theoretic property is analyzed based on a separate consideration of cases of theories with finite and infinite models. A description of model-theoretic properties defined via finite domains is found. It is established that the class of all finite models with first-order definable elements as well as the corresponding class of theories of such models forms the only model-theoretic property and, therefore, is of little interest as a database with an interface based on the first-order logic language. Key words: first-order logic, Cartesian extension of a theory, Tarski-Lindenbaum algebra, model-theoreticproperty, computable isomorphism.