9. Boolean lattices, Boolean algebras and Stone’s theorem

2018 ◽  
pp. 245-278
Keyword(s):  
Boolean Algebras ◽  
Boolean Lattices ◽  
Stone’S Theorem

scholarly journals LOGICS AND ALGEBRAS FOR MULTIPLE PLAYERS

2010 ◽  
Vol 3 (3) ◽  
pp. 485-519 ◽  
Author(s):  
LOES OLDE LOOHUIS ◽  
YDE VENEMA
Keyword(s):  
Computational Complexity ◽  
Winning Strategy ◽  
Boolean Algebras ◽  
Game Theoretic ◽  
Np Complete ◽  
Game Theoretic Semantics ◽  
Stone’S Theorem

We study a generalization of the standard syntax and game-theoretic semantics of logic, which is based on a duality between two players, to a multiplayer setting. We define propositional and modal languages of multiplayer formulas, and provide them with a semantics involving a multiplayer game. Our focus is on the notion of equivalence between two formulas, which is defined by saying that two formulas are equivalent if under each valuation, the set of players with a winning strategy is the same in the two respective associated games. We provide a derivation system which enumerates the pairs of equivalent formulas, both in the propositional case and in the modal case. Our approach is algebraic: We introduce multiplayer algebras as the analogue of Boolean algebras, and show, as the corresponding analog to Stone’s theorem, that these abstract multiplayer algebras can be represented as concrete ones which capture the game-theoretic semantics. For the modal case we prove a similar result. We also address the computational complexity of the problem whether two given multiplayer formulas are equivalent. In the propositional case, we show that this problem is co-NP-complete, whereas in the modal case, it is PSPACE-hard.

On Boolean Lattices of Module Classes

2018 ◽  
Vol 25 (02) ◽  
pp. 285-294 ◽  
Author(s):  
Alejandro Alvarado-García ◽  
César Cejudo-Castilla ◽  
Hugo Alberto Rincón-Mejía ◽  
Ivan Fernando Vilchis-Montalvo ◽  
Manuel Gerardo Zorrilla-Noriega
Keyword(s):  
Torsion Theory ◽  
Boolean Lattices

Some properties of and relations between several (big) lattices of module classes are used in this paper to obtain information about the ring over which modules are taken. The authors reach characterizations of trivial rings, semisimple rings and certain rings over which every torsion theory is hereditary.

Information storage and retrieval systems with incomplete information I

1979 ◽  
Vol 2 (1) ◽  
pp. 17-41
Author(s):  
Michał Jaegermann
Keyword(s):  
Incomplete Information ◽  
Information Storage And Retrieval ◽  
Boolean Algebras ◽  
Information Storage ◽  
Storage And Retrieval ◽  
Retrieval Systems ◽  
Theory Of Information

In the paper is developed a theory of information storage and retrieval systems which arise in situations when a whole possessed information amounts to a fact that a given document has some feature from properly chosen set. Such systems are described as suitable maps from descriptor algebras into sets of subsets of sets of documents. Since descriptor algebras turn out to be pseudo-Boolean algebras, hence an “inner logic” of our systems is intuitionistic. In the paper is given a construction of systems and are considered theirs properties. We will show also (in Part II) a formalized theory of such systems.

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

scholarly journals Handbook of boolean algebras

1990 ◽  
Vol 84 (1) ◽  
pp. 136
Author(s):  
Gian-Carlo Rota
Keyword(s):  
Boolean Algebras

On Poset Boolean Algebras

Order ◽  
2003 ◽  
Vol 20 (3) ◽  
pp. 265-290 ◽  
Author(s):  
Uri Abraham ◽  
Robert Bonnet ◽  
Wiesław Kubiś ◽  
Matatyahu Rubin
Keyword(s):  
Boolean Algebras

Common extension of Boolean informational systems

1979 ◽  
Vol 2 (1) ◽  
pp. 63-70
Author(s):  
Tadeusz Traczyk
Keyword(s):  
Information Systems ◽  
Boolean Algebras ◽  
Numerical Characterization ◽  
Common Extension

The notion of numerical characterization of Boolean algebras and coproducts are used to define information systems and to develop the theory of such systems.

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

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