### A remark on Sikorski's extension theorem for homomorphisms in the theory of Boolean algebras

W. Luxemburg (1964)

Fundamenta Mathematicae

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W. Luxemburg (1964)

Fundamenta Mathematicae

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Janusz Czelakowski (1981)

Colloquium Mathematicae

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Brian Wynne (2008)

Fundamenta Mathematicae

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Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.

Bernasconi, Anna (2001)

Discrete Mathematics and Theoretical Computer Science. DMTCS [electronic only]

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Paul R. Halmos (1954-1956)

Compositio Mathematica

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Bernhard Banaschewski (1993)

Commentationes Mathematicae Universitatis Carolinae

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The functor taking global elements of Boolean algebras in the topos $\mathrm{\mathbf{S}\mathbf{h}}\U0001d505$ of sheaves on a complete Boolean algebra $\U0001d505$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\U0001d505$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.

Roman Sikorski (1961)

Colloquium Mathematicum

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Robert Lagrange (1974)

Colloquium Mathematicae

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Martin Gavalec (1981)

Colloquium Mathematicae

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Žarko Mijajlović (1977)

Publications de l'Institut Mathématique

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Sergiu Rudeanu (1998)

Mathware and Soft Computing

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An abstract form of modus ponens in a Boolean algebra was suggested in [1]. In this paper we use the general theory of Boolean equations (see e.g. [2]) to obtain a further generalization. For a similar research on Boolean deduction theorems see [3].