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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2026. Vol 57

Some Model-Theoretic Properties of the Class of 𝑇-Pseudofinite 𝑆-Acts

Author(s)

Alena A. Stepanova, Evgeniy L. Efremov, Sergey G. Chekanov

Far Eastern Federal University, Vladivostok, Russian Federation

Abstract
The structure M in a language 𝐿 is called pseudofinite if every sentence in a language 𝐿 true in M has a finite model. A model M of a theory 𝑇 in a language 𝐿 is called 𝑇-pseudofinite if every sentence in a language 𝐿 true in M has a finite model, which is a model of the theory 𝑇. In this work we prove that for any theory 𝑇 the class of all pseudofinite (𝑇-pseudofinite) models of this theory is axiomatizable. A (left) 𝑆–act over monoid 𝑆 is a set 𝐴 upon which 𝑆 acts unitarily on the left. If 𝑇 is the theory of all 𝑆-acts, then completeness, model completeness, and totally categoricalness of the class of all 𝑇-pseudofinite 𝑆-acts are equivalent to the following condition: every 𝑇-pseudofinite 𝑆-act is a coproduct of one-element 𝑆-acts. If 𝑆 is a finite monoid or a commutative Noetherian and Artinian monoid then the theory of every 𝑆-act (𝑇-pseudofinite 𝑆-act) is stable iff 𝑆 is a linearly ordered monoid, where 𝑇 is the theory of all 𝑆-acts.
About the Authors

Alena A. Stepanova, Dr. Sci. (Phys.–Math.), Prof., Far Eastern Federal University, Vladivostok, 690922, Russian Federation, stepltd@mail.ru 

Evgeniy L. Efremov, Cand. Sci. (Phys.-Math.), Assoc. Prof., Far Eastern Federal University, Vladivostok, 690922, Russian Federation, efremov-el@mail.ru 

Sergey G. Chekanov, Cand. Sci. (Phys.-Math.), Assoc. Prof., Far Eastern Federal University, Vladivostok, 690922, Russian Federation, chekanov.sg@dvfu.ru 

For citation
Stepanova A. A., Efremov E. L., Chekanov S. G. Some Model-Theoretic Properties of the Class of 𝑇-Pseudofinite 𝑆-Acts. The Bulletin of Irkutsk State University. Series Mathematics, 2026, vol. 57, pp. 143–152. https://doi.org/10.26516/1997-7670.2026.57.143
Keywords
𝑇-pseudofinite structure, act over monoid, axiomatizable theory, complete theory, stable theory
UDC
510.67:512.57
MSC
03C60
DOI
https://doi.org/10.26516/1997-7670.2026.57.143
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