Approximations of Theories of Unars - Markhabatov, N.D. (Karagandy University of the name of academician E.A. Buketov, 2025)
Lo´s’s theorem states that a first-order formula holds in an ultraproduct of structures if and only if it holds in “almost all” factors, where “almost all” is understood in terms of a given ultrafilter. This fundamental result plays a key role in understanding the behavior of first-order properties under ultraproduct constructions. Pseudofinite structures – those that are elementarily equivalent to ultraproducts of finite models–serve as an important bridge between the finite and the infinite, allowing the transfer of finite combinatorial intuition to the study of infinite models. In the context of unary algebras (unars), a classification of unar theories provides a foundation for analyzing pseudofiniteness within this framework. Based on this classification, a characterization of pseudofinite unar theories is obtained, along with several necessary and sufficient conditions for a unar theory to be pseudofinite. Furthermore, various forms of approximation to unar theories are investigated. These include approximations not only for arbitrary unar theories but also for the strongly minimal unar theory. Different types of approximating sequences of finite structures are examined, shedding light on the model-theoretic and algebraic properties of unars and enhancing our understanding of their finite counterparts.
Keywords: pseudofinite theory; pseudofinite structure; strongly minimal unar; smoothly approximated structure; unar; Collatz Hypotesis; connected unar; bounded unar; !-categorical unar