MATHEMATICS

Theories of Boolean algebras with a distinguished subalgebra that do not have a prime model

D. E. Pal'chunov^a, A. V. Trofimov<sup>b

a</sup> Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^b Novosibirsk State University

Abstract: Boolean algebras in a signature enriched with a distinguished subalgebra are investigated. The question of the existence of complete theories of Boolean algebras with a distinguished subalgebra that do not have a prime model is studied. A continuum of Boolean algebras with a distinguished subalgebra is constructed, the elementary theories of which are different and do not have prime models, and for a fairly narrow class of very simply structured algebras. Namely, these are superatomic Boolean algebras with a distinguished subalgebra that is isomorphic to the Boolean algebra itself and almost coincides with it.

Keywords: Boolean algebra, superatomic Boolean algebra, Boolean algebra with a distinguished subalgebra, elementary theory, prime model.

UDC: 512.563

Presented: S. S. Goncharov
Received: 01.09.2025
Revised: 14.10.2025
Accepted: 14.10.2025

DOI: 10.7868/S3034504925060045

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