Kevin A. Clarke · Political Analysis · 2020
Researchers employing qualitative comparative analysis (QCA) and its variants use two-element Boolean algebra to compare cases and identify putative causal conditions. I show that the two-element Boolean algebra constrains research in three important ways: it restricts what we can say about sets and the interactions between sets, it embodies a logical language that is too weak to capture modern social science theories, and it restricts our analysis of causation to necessity and sufficiency accounts and does not allow for counterfactuals. Modern quantitative analysis suffers none of these restrictions and provides a much richer way to understand the social world.
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Léon LeBlanc · Proceedings of the American Mathematical Society · 1962
Introduction.Boolean equalities occur naturally in Logic and especially in the theory of polyadic algebras.For instance, they have been used in [2] under the name of Champ Logique.The purpose of this paper is to develop a duality theory for Boolean equalities and by doing so establish a representation theory for them.0. Notation.The basic notation for Boolean algebras is the standard one (for instance, see [l]).Throughout this paper, the letter O will denote the two-element Boolean algebra; whenever a topology is assumed to be defined on 0, then this topology is the discrete one.If 73 is a Boolean algebra, then a valuation of B is a homomorphism from 73 onto 0. We make precise the terminology we shall be using in connection with the Stone duality theory for Boolean algebras.A Boolean space is a compact Hausdorff space whose topology is generated by the clopen sets.If X is a Boolean space, then the dual algebra of X is the Boolean algebra of clopen sets of X.If 73 is a Boolean algebra, then the dual space of 73 is the Boolean space X of all valuations of 73 (the topology on X is the topology generated by all sets of the form {v: vp = 1} where p belongs to B).
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Leon Leblanc · Proceedings of the American Mathematical Society · 1962
Introduction.Boolean equalities occur naturally in Logic and especially in the theory of polyadic algebras.For instance, they have been used in [2] under the name of Champ Logique.The purpose of this paper is to develop a duality theory for Boolean equalities and by doing so establish a representation theory for them.0. Notation.The basic notation for Boolean algebras is the standard one (for instance, see [l]).Throughout this paper, the letter O will denote the two-element Boolean algebra; whenever a topology is assumed to be defined on 0, then this topology is the discrete one.If 73 is a Boolean algebra, then a valuation of B is a homomorphism from 73 onto 0. We make precise the terminology we shall be using in connection with the Stone duality theory for Boolean algebras.A Boolean space is a compact Hausdorff space whose topology is generated by the clopen sets.If X is a Boolean space, then the dual algebra of X is the Boolean algebra of clopen sets of X.If 73 is a Boolean algebra, then the dual space of 73 is the Boolean space X of all valuations of 73 (the topology on X is the topology generated by all sets of the form {v: vp = 1} where p belongs to B).
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Bohuslav Balcar, Thomas Jech · Bulletin of Symbolic Logic · 2006
This article investigates the weak distributivity of Boolean σ-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a σ-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure. Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann's Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article. § 1. Complete Boolean algebras and weak distributivity . A Boolean algebra is a set B with Boolean operations a ˅ b (join), a ˄ b (meet) and − a (complement), partial ordering a ≤ b defined by a ˄ b = a and the smallest and greatest element, 0 and 1 . By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty set S , under operations a ∪ b , a ∩ b , S − a , ordered by inclusion, with 0 = ∅ and 1 = S . Complete Boolean algebras and weak distributivity. A Boolean algebra is a set B with Boolean operations a ˅ b (join), a ˄ b (meet) and - a (complement), partial ordering a ≤ b defined by a ˄ b = a and the smallest and greatest element. 0 and 1 . By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty set S , under operations a ∪ b , a ∩ b , S - a , ordered by inclusion, with 0 = ϕ and 1 = S .
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Mori Toru · Yokohama National University Repository (Yokohama National University) · 1964
Introduction. This paper is concerned with the properties of a minimal extention of a Boolean algebra.T. Traczyk [2] had $1$) $roved$ that a minimal extension of a weakly a-distributive Boolean algebra satisfying the a-chain condition is also weakly a-distributive.The tools with which he had proved this theorem was the family of a-partitions of a Boolean algebra $B$ which has a covering of $B$ which refines every member of the family.In this way his proof was so algebraic but so elegant.Motiviated by this theorem, I thought that whether this theorem can be proved by another method $i.c$ .topological one or not.However, an answer for this problem is affirmative by showing the theorem which is as follows: A minimal extension of an $\alpha\cdot representative$ Boolean algebra satisfying the a-chain condition is also a-representative.In the first place I prove this theorem by using topological properties of the Stone space of a Boolean algebra and in the second place givc another proof of the previous Traczyk's theorem.All sorts of theorems, concepts and terminologies which are used in this paper due to the work of Prof. Sikorski [1]. Terminology and notation.Throughout this $P^{a}I$ ) $er,$ $\alpha$ denotes an infinite cardinal number.The symbol $\cup$ will be used both for the Boolean $|oin$ and for set-theoretical union.The symbol $\cap$ will be used both for the Boolean meet and for the set-theoretical intersection.The zero element of a Boolean algebra will be denoted by $0$ and the unit by 1.The empty set will be denoted by $\phi$ and the complement of a set $A$ will be denoted by $A^{\prime}$ .The difference of two sets $A$ and $C$ is written by $A-C$.The symbol $|\mathcal{T}$ stands for the cardinal number of the set $T$ .If $A$ is a subalgebra of a Boolean algebra $B$ and $a_{t}cA$ for every $t\epsilon T$ , then the set $\{a_{t} : \ell\epsilon T\}$ may have two joins, one taken in $A$ and the other in $B$ ; we denote these joins whenever they exist, by $\bigcup_{\iota_{\epsilon T}}^{A}a_{t}$ and $\bigcup_{\ell\epsilon T}^{l}a_{t}$ respectively.$A$ subalgebra $A$ of a Boolean algebra $B$ is said to be a-rcgular subalgebra of $B$ provided, for $\{a_{t} : a_{t}\epsilon A, t\epsilon T, |T|\leqq\alpha\}$ , if the join $\bigcup_{*r}^{A}a_{t}$ exists, it is also the join of all $a_{t}$ in $B,$ $i.e$ .the equality $\bigcup_{eI'}^{B}a_{t}=\bigcup_{eT}^{A}a_{t}$ holds.A set $D$ of clements of a Boolean algcbra $B$ is said to be dense in $B$ provided, for every element $a\epsilon B,$ $a\neq 0.$ ' then there exists an element bcD such
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Zdenka Rie Čanová · 2011
doi:10.3842/SIGMA.2010.001 Abstract. We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra E is separable and modular then there exists a faithful state on E. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra Ê and the compatiblity center of E is not a Boolean algebra then there exists an (o)-continuous subadditive state on E. Key words: effect algebra; state; sharp element; center; compatibility center 2010 Mathematics Subject Classification: 06C15; 03G12; 81P10 1 Introduction, basic definitions and some known facts The classical (Kolmogorovian) probability theory is assuming that every two events are simultaniously measurable (as every two elements of a Boolean algebra are mutually compatible). Thus this theory cannot explain events occuring, e.g., in quantum physics, as well as in economy and many other areas.
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Ján Jakubík · Mathematica Slovaca · 2010
Abstract The notion of idempotent modification of an algebra was introduced by Ježek; he proved that the idempotent modification of a group is always subdirectly irreducible. In the present note we show that the idempotent modification of a generalized MV -algebra having more than two elements is directly irreducible if and only if there exists an element in A which fails to be boolean. Some further results on idempotent modifications are also proved.
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Radim Bělohlávek, Markéta Krmelová · 2013
Boolean matrix factorization (BMF), or decomposition, received a considerable attention in data mining research. In this paper, we argue that research should extend beyond the Boolean case toward more general type of data such as ordinal data. Technically, such extension amounts to replacement of the two-element Boolean algebra utilized in BMF by more general structures, which brings non-trivial challenges. We first present the problem formulation, survey the existing literature, and provide an illustrative example. Second, we present new theorems regarding decompositions of matrices with ordinal data. Third, we propose a new algorithm based on these results along with an experimental evaluation.
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Andrey Mokhov · IET Computers & Digital Techniques · 2015
A switch, mechanical or electrical, is a fundamental building element of digital systems. The theory of switching networks, or simply circuits, dates back to Shannon's thesis (1937), where he employed Boolean algebra for reasoning about the functionality of switching networks, and graph theory for describing and manipulating their structure. Following this classic approach, one can deduce functionality from a given structure via analysis, and create a structure implementing a specified functionality via synthesis. The use of two mathematical languages leads to a 'language barrier' – whenever a circuit description is changed in one language, it is necessary to translate the change into the other one to keep both descriptions synchronised. This work presents a unified algebra of switching networks. Its elements are circuits rather than just Boolean functions (as in Boolean algebra) or vertices/edges (as in graph theory). This approach allows one to express both the functionality and structure of switching networks in the same mathematical language and brings new methods of circuit composition for greater reuse of components and interfaces. In this paper we demonstrate how to use the algebra to formally transform circuits, reason about their properties, and even solve equations whose 'unknowns' are circuits.
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Ballew, Walter David, · SHAREOK (University of Oklahoma) · 2013
The upward compatibility of binary Boolean algebra with Post algebra was examined. There exists in a Post algebra, P(m), a single two-element Boolean algebra, B(2) {4}. If the complement operation in P(m) is the pseudo-complement or strong negation, then more than one B(2) to P(m) mapping is possible. For m = 2('N), N an integer greater than one, there are 2('N-1) homomorphic mappings of B(2) into P(m). Standard B(2):P(4) building blocks were designed and constructed with integrated injection logic to demonstrate the practical aspects of the dual radix concept. An algorithm for finding the maximum compatible mapping from B(2) to P(m) for completely and incompletely specified functions was developed. Finally, memory elements, bus design, and basic architecture to support a B(2):P(4) processor were considered and comments concerning a B(2) machine operating in a P(m) host were made.
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Mykola Kovalov · System research and information technologies · 2023
The method of Boolean function representation in terms of positional logic algebra in compact operator form is offered. Compared with the known method, it uses position operators with a complexity of no more than two and only one type of equivalent transformations. The method is less labor intensive. It allows parallelizing logic calculations. The corresponding way of Boolean function implementation is developed. It competes with some known ways in terms of hardware complexity, resource intensity, and speed when implemented on an FPGA basis. Possibilities open up for creating effective automating means of representing Boolean functions from a large number of variables, synthesizing the corresponding LCs, and improving modern element bases.
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Antoni Torrens · arXiv (Cornell University) · 2023
We present a family of local involutive integral bounded residuated lattice-ordered commutative monoids (involutive residuated lattices, for short) having Boolean term, radical term (see \cite{CT12} and \cite{T23}), and satisfying GAP (Generalized Appel property) (see \cite{T23}).The construction of this family is based in the examples given in \cite[Subsection 5.1]{CT12}, by taking $\bfm{\LLn}$ in place of $\bfm{L_{n+1}}$ and $\bfm{L_{p+1}}$ in place of the two element Boolean algebra. In some sense this paper is a companion of the paper \cite{T23}, in which the General Apple Property (GAP) and Boolean terms are studied.
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Микола Ковальов · The Scientific Issues of Ternopil Volodymyr Hnatiuk National Pedagogical University Series pedagogy · 2023
The method of Boolean function representation in terms of positional logic algebra in compact operator form is offered. Compared with the known method, it uses position operators with a complexity of no more than two and only one type of equivalent transformations. The method is less labor intensive. It allows parallelizing logic calculations. The corresponding way of Boolean function implementation is developed. It competes with some known ways in terms of hardware complexity, resource intensity, and speed when implemented on an FPGA basis. Possibilities open up for creating effective automating means of representing Boolean functions from a large number of variables, synthesizing the corresponding LCs, and improving modern element bases.
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Lior Tzafriri · Transactions of the American Mathematical Society · 1967
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 S 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 7.We shall start by proving that for every operator AeS.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 5 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 H 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.1. Preliminaries.For convenience we shall summarize here some definitions and results concerning the Bade theory of multiplicity for Boolean algebras of projections on Banach spaces (see [2] and [3] and the general theory of spectral operators (see [4], [6], [7], [8], and [13])).According to Bade [3], a Boolean algebra (B.A.) of projections S will be called complete if for every family {Fa}f the projections V Fa and A. Ea exist in S and, moreover (1.1) (V Ea)X = cl m{EaX) ; (A Ea)X = f| EaX.
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L. Tzafriri · Transactions of the American Mathematical Society · 1967
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 S 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 7.We shall start by proving that for every operator AeS.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 5 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 H 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.1. Preliminaries.For convenience we shall summarize here some definitions and results concerning the Bade theory of multiplicity for Boolean algebras of projections on Banach spaces (see [2] and [3] and the general theory of spectral operators (see [4], [6], [7], [8], and [13])).According to Bade [3], a Boolean algebra (B.A.) of projections S will be called complete if for every family {Fa}£«f the projections V Fa and A. Ea exist in S and, moreover (1.1) (V Ea)X = cl m{EaX) ; (A Ea)X = f| EaX.
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Juei-Ling Ho · Taiwanese Journal of Mathematics · 2010
Robert has established a global convergence theorem in $\{0,1\}^{n}$: If a map $\hat{F}$ from $\{0,1\}^{n}$ to itself is contracting relative to the boolean vector distance $d$, then there exists a positive integer $p \leq n$ such that $\hat{F}^{p}$ is constant. In other words, $\hat{F}$ has a unique fixed point $\xi$ such that for any $x$ in $\{0,1\}^{n}$, we have $\hat{F}^{p}(x) = \xi$. The structure ($\{0,1\},+,\cdot,-,0,1$) may be regarded as the two-element boolean algebra. In this paper, this result is extended to any map $F$ from the product $X$ of $n$ finite boolean algebras to itself.
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Salibra A., Ledda A., Paoli F. · ARCA (Università Ca' Foscari Venezia) · 2018
We introduce Boolean-like algebras of dimension n (nBA) having n constants e 1 , . . ., en, and an (n + 1)-ary operation q (a "generalised if-then-else") that induces a decomposition of the algebra into n factors through the so-called n-central elements.Varieties of nBAs share many remarkable properties with the variety of Boolean algebras and with primal varieties.Exploiting the concept of central element, we extend the notion of Boolean power to that of semiring power and we prove two representation theorems: (i) Any pure nBA is isomorphic to the algebra of n-central elements of a Boolean vector space; (ii) Any member of a variety of nBAs with one generator is isomorphic to a Boolean power of this generator.This gives a new proof of Foster's theorem on primal varieties.The nBAs provide the algebraic framework for generalising the classical propositional calculus to the case of n -perfectly symmetric -truth-values.Every finite-valued tabular logic can be embedded into such an n-valued propositional logic, nCL, and this embedding preserves validity.We define a confluent and terminating first-order rewriting system for deciding validity in nCL, and, via the embeddings, in all the finite tabular logics.
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Volodymyr Mazorchuk, Bridget Eileen Tenner · arXiv (Cornell University) · 2021
We prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's a-function. Our arguments are combinatorial, and include a description of the intersection of two principal order ideals when at least one permutation is boolean. An important object in our work is a reduced word written as minimally many runs of consecutive integers, and one step of our argument shows that this minimal quantity is equal to the length of the second row in the permutation's shape under the Robinson-Schensted correspondence. We also prove that a simple module over the above-mentioned incidence algebra is perfect if and only if its index is the longest element of a parabolic subgroup.
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Tracy Wall · 2010
This research studies three-element unary algebras to determine which of them are standard. A quasi-variety generated by an algebra M is standard if it consists exactly of those algebraic structures being the same type as M which carry a compatible Boolean topology and are models of the quasi-equational theory of M. This work shows that two previously unclassified structures, M, are standard. --P.ii.
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Ming–Huat Lim, Sin-Chee Tan · Linear and Multilinear Algebra · 2013
AbstractLet denote the two element Boolean algebra and be the semimodule of all n-square symmetric Boolean matrices with zero diagonal. We characterize (i) subsemimodules of whose nonzero members all have factor rank 2, (ii) linear mappings from to that send distinct elements of term rank 2 to distinct elements of factor rank 2 and (iii) linear mappings from to that preserve elements of factor rank 2 and also elements of factor rank k for some Keywords: semimoduleBoolean symmetric matrixfactor rankterm ranklinear mappingAMS Subject Classifications: 15A8615A0315A04 AcknowledgementThe research of the first author was supported by the University of Malaya Research Grant RG157-11AFR.
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