Effectiveness of the representation theory for Boolean algebras

J. Łoś; Czesław Ryll-Nardzewski

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The independence of the axiom of choice from the Boolean prime ideal theorem

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B. R. Salinas, F. Bombal (1973)

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From accessible to inaccessible cardinals (Results holding for all accessible cardinal numbers and the problem of their extension to inaccessible ones)

H. Keisler, Alfred Tarski (1964)

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Remarks on Henkin's paper: Boolean representation trough, propositional calculus

J. Łoś (1957)

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On an unsolved problem from the theory of Boolean algebras

Roman Sikorski (1949)

Colloquium Mathematicum

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Kelley's specialization of Tychonoff's Theorem is equivalent to the Boolean Prime Ideal Theorem

Eric Schechter (2006)

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The principle that "any product of cofinite topologies is compact" is equivalent (without appealing to the Axiom of Choice) to the Boolean Prime Ideal Theorem.

A proof of the independence of the Axiom of Choice from the Boolean Prime Ideal Theorem

Miroslav Repický (2015)

Commentationes Mathematicae Universitatis Carolinae

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We present a proof of the Boolean Prime Ideal Theorem in a transitive model of ZF in which the Axiom of Choice does not hold. We omit the argument based on the full Halpern-Läuchli partition theorem and instead we reduce the proof to its elementary case.

A set-theoretical equivalent of the prime ideal theorem for Boolean algebras

J. van Benthem (1975)

Fundamenta Mathematicae

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Partially additive states on orthomodular posets

Josef Tkadlec (1991)

Colloquium Mathematicae

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We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s:P → [0,1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P. We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P. Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result...