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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 $$\epsilon ({\mathfrak D}')\times \epsilon ({\mathfrak D}'')=\epsilon ([{\mathfrak D}']{\mathfrak D}'').)$$
##### MSC:
 03C20 Ultraproducts and related constructions 03G10 Logical aspects of lattices and related structures 06D99 Distributive lattices
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##### References:
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