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About 536 papers for “Complete Boolean algebra”

The stationarity preserving part of a complete Boolean algebra

Joji Takahashi · Kobe University Repository Kernel (Kobe University) · 1993

An algebraic formula expressing the A-value of the statement that all stationary sets in V remain stationary in Vᴬ is obtained, where A is an arbitrary complete Boolean algebra. It is also shown how this formula leads to some of the known characterizations of A being proper. Forcing that preserves the stationarity of stationary sets in the ground model, named proper forcing by S. Shelah, has been a subject of considerable interest in set theory. Here, stationarity means that of subsets of P ≤ ω(X) = {M ⊆ X | Card(M) ≤ ω} with the base set X allowed to be arbitrary. In this paper, I will look at the Boolean-value || All stationary sets in V remain stationary || A for an arbitrary complete Boolean algebra A. Working with a complete Boolean algebra rather than with a poset, I will obtain an algebraic formula that expresses this value in terms of the infinite meet and join operations of A (Theorem 1.9). I will also show how this formula leads naturally to some of the known characterizations of A being proper (Theorems 2.1 and 2.3). I am grateful to the referee for suggesting a number of improvements on my original manuscript.

Cited by 0Open access

A Countably Distributive Complete Boolean Algebra Not Uncountably Representable

John Gregory · Proceedings of the American Mathematical Society · 1974

It is proved from the Continuum Hypothesis that there exists an co-distributive complete Boolean algebra which is not cu^representable.Karp [1, Corollary to main theorem] shows that there is an todistributive non-^l-representable ^-complete

Cited by 3Open access

Every complete atomic Boolean algebra is the ideal lattice of a cBCK-algebra

C. Matthew Evans · arXiv (Cornell University) · 2021

Given a complete atomic Boolean algebra, we show there is a commutative BCK-algebra whose ideal lattice is that Boolean algebra. This result is shown to exist within a larger framework involving BCK-algebras of functions, whose ideals and prime ideals are analyzed by way of a specific Galois connection. As a corollary of the main theorem, we show that every discrete topological space is the prime spectrum of a cBCK-algebra.

Cited by 0Open access

Quantifier-Free Boolean Algebra with Presburger Arithmetic is NP-Complete

2008

1 Introduction This paper considers the satisfiability problem for a logic that allows reasoningabout sets and their cardinalities. We call this logic quantifier-free Boolean Algebra with Presburger Arithmetic and denote it QFBAPA. Figure 1 shows thesyntax of QFBAPA. The logic contains 1) arbitrary boolean algebra (BA) ex-pressions denoting sets, 2) arbitrary quantifier-free Presburger arithmetic (

Cited by 0

A countable Boolean algebra that is Reichenbach's common cause complete

Dominika Burešová · ArXiv.org · 2025

The common cause completeness (CCC) is a philosophical principle that asserts that if we consider two positively correlated events then it evokes a common cause. The principle is due to H. Reichenbach and has been largely studied in Boolean algebras and elsewhere.The results published so far bring about a question whether there is a small (countable) Boolean algebra with CCC. In this note we construct such an example.

Cited by 0Open access

Complete Boolean Algebra for Memristive and Spintronic Asymmetric Basis Logic Functions

V. Vyas, Joseph S. Friedman · ACM Journal on Emerging Technologies in Computing Systems · 2025

The increasing advancement of emerging device technologies that provide alternative basis logic sets necessitates the exploration of innovative logic design automation methodologies. Specifically, emerging computing architectures based on the memristor and the bilayer avalanche spin-diode offer non-commutative or “asymmetric” operations, namely the inverted-input AND (IAND) and implication as basis logic gates. Existing logic design techniques inadequately leverage the unique characteristics of asymmetric logic functions, resulting in insufficiently optimized logic circuits. This article presents a complete Boolean algebraic framework specifically tailored to asymmetric logic functions, introducing fundamental identities, theorems, and canonical normal forms that lay the groundwork for efficient synthesis and minimization of such logic circuits without relying on conventional Boolean algebra. Further, this article establishes a logical relationship between implication and IAND operations. A previously proposed modified Karnaugh map method based on a subset of the presented algebraic principles demonstrated a 28% reduction in computational steps for an algorithmically designed memristive full adder; the presently proposed algebraic framework lays the foundation for much greater future improvements.

Cited by 0

A SIMPLE PROOF THAT (s)/(s⁰) IS A COMPLETE BOOLEAN ALGEBRA

Baldwin, Brown · Real Analysis Exchange · 1998

Let $X$ be a complete separable metric space, let $(s)$ be the set of all Marczewski \cite{sm} measurable subsets of $X$, and let $(s^0)$ be the the set of all Marczewski null subsets of $X$. It is already known that $(s)/(s^0)$ is a complete Boolean algebra, but the known proofs of this involve complicated preliminaries. We present a simple proof that $(s)/(s^0)$ is a complete Boolean algebra.

Cited by 1

Complete Boolean Algebra for Memristive and Spintronic Asymmetric Basis Logic Functions

V. Vyas, Joseph S. Friedman · arXiv (Cornell University) · 2024

The increasing advancement of emerging device technologies that provide alternative basis logic sets necessitates the exploration of innovative logic design automation methodologies. Specifically, emerging computing architectures based on the memristor and the bilayer avalanche spin-diode offer non-commutative or `asymmetric' operations, namely the inverted-input AND (IAND) and implication as basis logic gates. Existing logic design techniques inadequately leverage the unique characteristics of asymmetric logic functions resulting in insufficiently optimized logic circuits. This paper presents a complete Boolean algebraic framework specifically tailored to asymmetric logic functions, introducing fundamental identities, theorems and canonical normal forms that lay the groundwork for efficient synthesis and minimization of such logic circuits without relying on conventional Boolean algebra. Further, this paper establishes a logical relationship between implication and IAND operations. A previously proposed modified Karnaugh map method based on a subset of the presented algebraic principles demonstrated a 28% reduction in computational steps for an algorithmically designed memristive full adder; the presently-proposed algebraic framework lays the foundation for much greater future improvements.

Cited by 0Open access

Embedding theorems for boolean algebras and consistency results on ordinal definable sets

Petr Štěpánek, Bohuslav Balcar · Journal of Symbolic Logic · 1977

The existence of complete rigid Boolean algebras was first proved by McAloon [8] who also showed that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra. McAloon was interested in consistency results on ordinal definable sets. His approach was based on forcing. Recently, Shelah [10] proved that for every uncountable cardinal κ there exists a Boolean algebra of power κ with rigid completion. Extending his method, we get the following theorems. Theorem 1. Any Boolean algebra B can be completely embedded in a complete Boolean algebra C with no nontrivial σ-complete one-one endomor-phism. If B satisfies the κ-chain condition for an uncountable cardinal κ, the same holds true for C . Since every automorphism is a complete endomorphism, it follows from Theorem 1 that C is rigid. The other extreme case of Boolean algebras are homogeneous algebras. It was proved by Kripke [7] that every Boolean algebra can be completely embedded in a homogeneous complete Boolean algebra. In his proof, the homogeneous algebra contains antichains of cardinality equal to the power of the embedded Boolean algebra. The following result shows that this is essential: the analogue of Theorem 1 is not provable in set theory even for Boolean algebras with a very weak homogeneity property. We use a Suslin tree with particular properties constructed by Jensen [6] in conjunction with a forcing argument.

Cited by 11

A note on the representation of 𝛼-complete Boolean algebras

Carol Karp · Proceedings of the American Mathematical Society · 1963

It is a fundamental theorem of representation theory for Boolean algebras that every No-complete Boolean algebra is an No-homomorphic image of an No-field of sets. It is also well known that there is a 2s-complete Boolean algebra which is not a 2No-homomorphic image of a 2No-field of sets. The usual proof goes by constructing a complete Boolean algebra that is not (No, 2)-distributive;

Cited by 1Open access