The Lindenbaum-Tarski algebra for Boolean algebras with distinguished ideals

Palchunov, D. E.

doi:10.1007/bf00750556

Cited by 3 publications

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“…It relies on a proof, due to MONK, of the atomicity of the Lindenbaum-Tarski algebra of TX . The atomicity of the Lindenbaum-Tarski algebra also follows from the fact that it is isomorphic to intalg (ww x (1 + q ) ) which is believed to have been proven by PAL'CHUNOV [12].…”

Section: The Lindenbaum-tarski Algebra and The Algebra Of Formulas Of Txmentioning

confidence: 86%

Some Boolean Algebras with Finitely Many Distinguished Ideals I

Aragón

1995

Mathematical Logic Qtrly

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We consider the theory Th,,i, of Boolean algebras with a principal ideal, the theory Th,,, of Boolean algebras with a maximal ideal, the theory Thac of atomic Boolean algebras with an ideal where the supremum of the ideal exists, and the theory Th,, of atomless Boolean algebras with an ideal where the supremum of the ideal exists. First, we find elementary invariants for Th,,;, and Th,,. If T is a theory in a first order language and a is a linear order with least element, then we let Sentalg(T) be the Lindenbaum-Tarski algebra with respect to T, and we let intalg(a) be the interval algebra of a. Using rank diagrams, we show that Sentalg (Th,,i,) intalg(w'), Sentalg(Th,,,) E intalg(w3) Z Sentalg(Thac), and Sentalg(Th,,) 2 intalg(w2 + w z ) . For Th,,, and Th,, we use Ershov's elementary invariants of these theories. We also show that the algebra of formulas of the theory Tx of Boolean algebras with finitely many ideals is atomic. Mathematics Subject Classification: 03C35, 03C65, 06399.

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Section: The Lindenbaum-tarski Algebra and The Algebra Of Formulas Of Txmentioning

confidence: 86%

Some Boolean Algebras with Finitely Many Distinguished Ideals I

Aragón

1995

Mathematical Logic Qtrly

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Some Boolean algebras with finitely many distinguished ideals II

Aragón

2003

Mathematical Logic Qtrly

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We describe the countably saturated models and prime models (up to isomorphism) of the theory Ì ÔÖ Ò of Boolean algebras with a principal ideal, the theory Ì Ñ Ü of Boolean algebras with a maximal ideal, the theory Ì of atomic Boolean algebras with an ideal such that the supremum of the ideal exists, and the theory Ì × of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove that there are infinitely many completions of the theory of Boolean algebras with a distinguished ideal that do not have a countably saturated model. Also, we give a sufficient condition for a model of the theory Ì of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of Ì . IntroductionThis paper is concerned with some questions about the theory Ì of Boolean algebras (BA's) with distinguished ideals. We prove that there are infinitely many completions of the theory of BA's with a distinguished ideal that do not have a countably saturated model. We give a sufficient condition in terms of the Heyting algebra of definable ideals for a model of the theory Ì of BA's with distinguished ideals to be elementarily equivalent to a countably saturated model of Ì . These results are related to results in Pal'chunov [9], where sufficient conditions are given for a model of Ì to be a prime model and for the elementary theory of a model of Ì to have no countably saturated models.In a more concrete direction, we give a description of the countably saturated models and prime models (up to isomorphism) of the theory Ì ÔÖ Ò of BA's with a principal ideal, the theory Ì Ñ Ü of BA's with a maximal ideal, the theory Ì of atomic BA's with an ideal such that the supremum of the ideal exists, and the theory Ì × of atomless BA's with an ideal such that the supremum of the ideal exists. NotationOur notation for BA's follows Koppelberg [6], while for model theory we follow, with some modifications, Chang and Keisler [2]. We also assume some notation and results from Aragón [1], but we recall here the basic new notation, referring to Aragón [1] for some notions with lengthy definitions.If is an element of a BA, we let ½ and ¼ . For a sequence ¼ Ò ½ of elements and ¾ Ò ¾If is a subset of a BA , then , or simply , denotes the subalgebra of generated by . We denote by È´ µ the principal ideal generated by an element . Interval algebras play a large role in this paper. The interval algebra of a linear order Ä is denoted by ÒØ Ð ´Äµ. We use reverse-lexicographic ordering on the products of linear orders. Let denote the order type of the rational numbers. If

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Palchunov

2011

Lecture Notes in Computer Science

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The Lindenbaum-Tarski algebra for Boolean algebras with distinguished ideals

Cited by 3 publications

References 4 publications

Some Boolean Algebras with Finitely Many Distinguished Ideals I

Some Boolean Algebras with Finitely Many Distinguished Ideals I

Some Boolean algebras with finitely many distinguished ideals II

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