Infinitely uniform theories and various methods of their construction

The article considers possible generalizations of the concept of logical uniformity, as well as theories whose variables run through some classes of elements of the universal set. Relations that do not coincide with ordering, topological generalizations, and generalizations that work with arbitrary closure systems are considered. A model of logical uniformity in covering theory is proposed. Infinitely uniform theories and various methods of their construction 2026, ORCID Volodymyr. M. Zhuravlov msc: 03B99, 03H99, udc: 510.65 Keywords: axiom scheme, preorder relation, interval, homeomorphism, covering, uniform theories and sets. References [1] Joseph R. Shoenfield. Oxford.Addison-Wesley Publishing Company, Mathematical Logic, (1967). [2] Haskell B. Curry, McGraw-Hill Book Company, INC. Foundations of Manthematical logic, (1984). [3] Helena Rasiowa and Roman Sikorski, Panstwowe Wydawnlctwo Naukowe Warszawa, The Mathemayics of Metamathematics, (1963). [4] Alonzo Church, Princeton University Press, Introduction to Mathematical Logic, (1956). [5] Efimov N.V. Moscow. Science, Higher geometry, (1971) [6] P.M. Kon, D. Reidel Publishing Company, Universal Algebra, (1961). [7] R. Goldblatt, North-Holland Publishing Company Amsterdam New York Oxford Topoi. The categorial analysis of logic, (1979) [8] D. Hilbert und P. Bernays. Springer-Verlag Berlin — Heidelberg — New York, Grundlagen der Mathematik. I, (1968).