Algebraic logic, I. Monadic boolean algebras
Paul R. Halmos (1954-1956)
Compositio Mathematica
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Paul R. Halmos (1954-1956)
Compositio Mathematica
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William Hanf (1976)
Fundamenta Mathematicae
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Roman Sikorski (1961)
Colloquium Mathematicum
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L. Szczerba (1973)
Fundamenta Mathematicae
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W. Luxemburg (1964)
Fundamenta Mathematicae
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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 of sheaves on a complete Boolean algebra 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 -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.
Leon Henkin (1955)
Fundamenta Mathematicae
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Enric Trillas, Susana Cubillo (1996)
Mathware and Soft Computing
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In a Boolean Algebra B, an inequality f(x,x --> y)) ≤ y satisfying the condition f(1,1)=1, is considered for defining operations a --> b among the elements of B. These operations are called Conditionals'' for f. In this paper, we obtain all the boolean Conditionals and Internal Conditionals, and some of their properties as, for example, monotonicity are briefly discussed.