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Elementary theory of \({\mathfrak D}\)-degrees. (English. Russian original) Zbl 0602.03006
Algebra Logic 23, 358-363 (1984); translation from Algebra Logika 23, No. 5, 530-537 (1984).
The author first adapts his characterization of \(\kappa\)-saturated Boolean algebras [Sib. Mat. Zh. 15, 1414-1415 (1974; Zbl 0311.02059)], in terms of \(\kappa\)-separation and \(\kappa\)-compactness, to Boolean lattices. (A Boolean lattice is to a Boolean algebra what a ring of sets is to a field of sets.) He then turns to [\({\mathfrak D}]{\mathfrak M}\) (the ”\({\mathfrak D}\)-degree of \({\mathfrak M}\)” of the translator), where \({\mathfrak D}\) is a Boolean lattice and \({\mathfrak M}\) is an algebraic structure (with a one-element subalgebra). This notion was introduced by Yu. L. Ershov [Algebra Logika 18, 680-722 (1979; Zbl 0451.06013)]; [\({\mathfrak D}]{\mathfrak M}\) is a direct limit of certain direct powers of \({\mathfrak M}\). The author shows: (1) The elementary theory of \([{\mathfrak D}]{\mathfrak M}\) only depends on the elementary theory of \({\mathfrak D}\) and the elementary theory of \({\mathfrak M}\). (2) If \({\mathfrak M}\) is finite and \({\mathfrak D}\) is \(\kappa\)-saturated, then [\({\mathfrak D}]{\mathfrak M}\) is \(\kappa\)-saturated. Finally, the author extends his algebra of elementary types of Boolean algebras [Algebra Logika 12, 74-82 (1973; Zbl 0286.02055)] to the elementary types of Boolean lattices. (Let \(\epsilon({\mathfrak D})\) be the elementary type of the Boolean lattice \({\mathfrak D}\). Then: \(\epsilon ({\mathfrak D}')+\epsilon ({\mathfrak D}'')=\epsilon ({\mathfrak D}'\times {\mathfrak D}'')\) and \(\epsilon ({\mathfrak D}')\times \epsilon ({\mathfrak D}'')=\epsilon ([{\mathfrak D}']{\mathfrak D}'').)\)
03C20 Ultraproducts and related constructions
03G10 Logical aspects of lattices and related structures
06D99 Distributive lattices
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