ISSN 0037–4466, Siberian Mathematical Journal, 2020, Vol. 61, No. 3, pp. 516–527. c  Pleiades Publishing, Ltd., 2020. Russian Text c  The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 3, pp. 654–668. THE THEORIES OF SUPERATOMIC BOOLEAN ALGEBRAS WITH DISTINGUISHED SUBALGEBRA WHICH LACK COUNTABLY SATURATED MODELS D. E. Palchunov and A. V. Trofimov UDC 512.563 Abstract: Studying Boolean algebras with distinguished subalgebras, we establish the existence of continuum many simple superatomic Boolean algebras with distinguished subalgebra whose elementary theories are distinct and each of them lacks countably saturated models. The subalgebras coincide with the Boolean algebras modulo their Fr´ echet ideals. DOI: 10.1134/S0037446620030131 Keywords: Boolean algebra, Boolean algebra with distinguished subalgebra, local algebra, elementary theory, finitely axiomatizable theory, solvable theory, elementary equivalence Introduction This article deals with the model-theoretic properties of Boolean algebras with distinguished dense subalgebra of finite width. The key question is the presence of simple and countably saturated models of elementary theories. While a simple model is a countable model in which only the main types are realized [1], a countably saturated model is a countable model with the opposite property: All types consistent with this theory are realized. In other words, a simple model is the “smallest” model of the theory, but a countably saturated model is the “largest” among the countable models of the theory. It is known that each theory with a countably saturated model also has a simple model. The elementary theory of each Boolean algebra has both simple and countably saturated models [2]. If we add to the signature of Boolean algebras some unary predicate that distinguishes an ideal then the situation with the existence of saturated and simple models changes drastically. Boolean algebras with distinguished ideal were studied in [3–9]. The following is shown in particular: There exist contin- uum many theories of Boolean algebras with one distinguished ideal possessing a simple model but not a countably saturated model; there exist continuum many theories lacking simple models, and there exist continuum many elementary theories with countably saturated models. However, each elementary theory of a superatomic Boolean algebra with one distinguished ideal has both simple and countably saturated models [3, 4]. This article shows that there exists a superatomic Boolean algebra with a distinguished dense sub- algebra of finite width whose theory lacks countably saturated models. In this algebra, the subalgebra is isomorphic to the Boolean algebra itself; moreover, the subalgebra coincides with the Boolean algebra modulo its Fr´ echet ideal. 1. Main Definitions and Preliminary Results Consider Boolean algebras of signature σ = 〈∪, ∩, C, 0, 1〉. Put σ ∗ = 〈σ, P 〉, where P is the sym- bol of an unary predicate distinguishing a subalgebra. Refer to a Boolean algebra with distinguished subalgebra just as an algebra. Denote the underlying set of the distinguished subalgebra of A by P A = {a ∈ A | A | = P (a)}. Refer to a superatomic Boolean algebra with a distinguished subalgebra as a superatomic algebra. Given a countable superatomic algebra A, denote by o(A) the smallest ordinal α such that the identity element of the algebra belongs to the (α + 1)st Fr´ echet ideal of the Boolean algebra: 1 ∈ F α+1 (A). Original article submitted March 10, 2020; revised April 6, 2020; accepted April 8, 2020. 516