Topic
Two-element Boolean algebra
About: Two-element Boolean algebra is a research topic. Over the lifetime, 2585 publications have been published within this topic receiving 43758 citations.
Papers published on a yearly basis
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Papers
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TL;DR: The empirical set theory of as mentioned in this paper is a subcategory of the category of Boolean localic toposes and geometric morphisms, and it is shown that observables can be identified with real numbers.
Abstract: We share with Foulis and Randall the evangel that it is not orthomodular posets or the like, but manuals of operations that are of primary importance in the foundations of the empirical sciences. In sharp contrast to them, we regard an operation not as a set of possible outcomes, but as a complete Boolean algebra of observable events, which we adopt, following the lines of Davis and of Takeuti, as a building block of our empirical set theory. Just as a smooth manifold is covered by open subsets of a Euclidean space interconnected by smooth mappings, our empirical set theory is covered by the Scott-Solovay universesV(B) over complete Boolean algebrasB interconnected by geometric morphisms. Using the nomenclature of topos theory, our empirical set theory is a subcategory of the categoryBIop of Boolean localic toposes and geometric morphisms. It is shown that in this set theory observables can be identified with real numbers. This is the first step of formal development of Davis' ambitious program.
19 citations
TL;DR: In this paper, it was shown that for every operator A E 6 there is a sequence of projections belonging to the above mentioned Boolean algebra of projections which increases to the identity and A multiplied by any element of this sequence is a spectral operator.
Abstract: The algebra of operators commuting with a Boolean algebra of projections of finite uniform multiplicity (in the sense of Bade [3]) has been studied by Foguel [10] and the author [12] The aim of the present paper is to show that most properties described in these two papers can be extended (sometimes, under additional conditions) to the algebra 6 of all the operators which commute with a complete countably decomposable Boolean algebra of projections containing no projections of infinite uniform multiplicity It should be mentioned that one cannot get interesting general results in the case in which there are projections of infinite uniform multiplicity in the Boolean algebra of projections, since every operator on a Banach space commutes with the Boolean algebra of projections composed from the identities 0 and I We shall start by proving that for every operator A E 6 there is a sequence of projections belonging to the above mentioned Boolean algebra of projections which increases to the identity and A multiplied by any element of this sequence is a spectral operator Relying on this result we study the spectrum of operators of E and give a necessary and sufficient condition for such an operator to be spectral In the following section we generalize Theorem 8 of [12] showing that in 6 a strong limit of spectral operators on a Hilbert space is spectral provided that they are of the same finite type and their resolutions of the identity are uniformly bounded Adequate examples elucidate why we require the boundedness of the type of the spectral operators in most theorems
19 citations
5 Feb 1986
TL;DR: The relational algebra operator division is explained and HAS, a generalization of division which is useful for answering questions about an M-M relationship between entities is presented.
Abstract: The relational algebra operator division is misnamed, hard to understand and insufficient. This paper explains division and presents HAS, a generalization of division which is useful for answering questions about an M-M relationship between entities.
19 citations
TL;DR: This work investigates classes of Boolean algebras related to the notion of forcing that adds Cohen reals, and introduces and study generalizations of Cohen algeBRas: semi-Cohen alge Bras, pseudo-Cohens alge bras and potentially Cohen al gebras.
19 citations
24 Feb 2005
TL;DR: This work defines and studies the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise, and shows that every Boolean function can be computed by O(n) quantum queries even in the model with noise.
Abstract: We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We show that, in contrast to the classical model of Feige et al., every Boolean function can be computed by O(n) quantum queries even in the model with noise. This implies, for instance, the somewhat surprising result that every Boolean function has robust degree bounded by O(n).
19 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 5 |
| 2024 | 16 |
| 2023 | 21 |
| 2022 | 36 |
| 2021 | 2 |
| 2020 | 1 |