On measures on complete Boolean algebras

1971 ◽  
Vol 36 (3) ◽  
pp. 395-406 ◽  
Author(s):  
Karel Prikry
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Measurable Cardinal ◽  
Boolean Algebras

In this paper we prove some theorems concerning measures on complete Boolean algebras. Among other things, in §I of this paper, we construct a counterexample to the following conjecture of W. Luxemburg: Every measure on a nonatomic hyperstonian Boolean algebra is normal. (See [3, p. 57].) This result is expressed by Theorem 1, §I. In order to construct this example we have to suppose that a real-valued measurable cardinal exists. This hypothesis is independent of the usual axioms of set theory. Luxemburg proved that our assumption is necessary. Our second result is stated in Theorem 2 near the end of the paper.

Boolean algebras in a localic topos

1986 ◽  
Vol 100 (1) ◽  
pp. 43-55 ◽  
Author(s):  
B. Banaschewski ◽  
K. R. Bhutani
Keyword(s):  
Boolean Algebra ◽  
Universal Algebra ◽  
Set Theory ◽  
Abelian Groups ◽  
Boolean Algebras

When a familiar notion is modelled in a certain topos E, the natural problem arises to what extent theorems concerning its models in usual set theory remain valid for its models in E, or how suitable properties of E affect the validity of certain of these theorems. Problems of this type have in particular been studied by Banaschewski[2], Bhutani[5], and Ebrahimi[6, 7], dealing with abelian groups in a localic topos and universal algebra in an arbitrary Grothendieck topos. This paper is concerned with Boolean algebras, specifically with injectivity and related topics for the category of Boolean algebras in the topos of sheaves on a locale and with properties of the initial Boolean algebra in .

Continuum cardinals generalized to Boolean algebras

2001 ◽  
Vol 66 (4) ◽  
pp. 1928-1958 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Boolean Algebras ◽  
Infinite Cardinal ◽  
Cardinal Numbers ◽  
Almost Disjoint ◽  
Infinite Sets ◽  
Maximal Almost Disjoint ◽  
Almost Disjoint Families

A number of specific cardinal numbers have been defined in terms of /fin or ωω. Some have been generalized to higher cardinals, and some even to arbitrary Boolean algebras. Here we study eight of these cardinals, defining their generalizations to higher cardinals, and then defining them for Boolean algebras. We then attempt to completely describe their relationships within each of several important classes of Boolean algebras.The generalizations to higher cardinals might involve several cardinals instead of just one as in the case of ω, For example, the number a associated with maximal almost disjoint families of infinite sets of integers can be generalized to talk about maximal subsets of [κ]μ subject to the pairwise intersections having size less than ν. (For this multiple generalization of . see Monk [2001].) For brevity we do not consider such generalizations, restricting ourselves to just one cardinal. The set-theoretic generalizations then associate with each infinite cardinal κ some other cardinal λ, defined as the minimum of cardinals with a certain property.The generalizations to Boolean algebras assign to each Boolean algebra some cardinal λ, also defined as the minimum of cardinals with a certain property.For the theory of the original “continuum” cardinal numbers, see Douwen [1984]. Balcar and Simon [1989]. and Vaughan [1990].I am grateful to Mati Rubin for some conversations concerning these functions for superatomic algebras, and to Bohuslav Balcar for information concerning the function h.The notation for set theory is standard. For Boolean algebras we follow Koppelberg [1989], but recall at the appropriate place any somewhat unusual notation.

Zorn's lemma and complete Boolean algebras in intuitionistic type theories

1997 ◽  
Vol 62 (4) ◽  
pp. 1265-1279 ◽  
Author(s):  
J. L. Bell
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Natural Kind ◽  
Extension Theorem ◽  
Axiom Of Choice ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Law Of Excluded Middle ◽  
The Axiom Of Choice ◽  
Excluded Middle

AbstractWe analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some familiar results in the theory of (complete) Boolean algebras—notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent.

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.

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.

On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers

1980 ◽  
Vol 45 (2) ◽  
pp. 265-283 ◽  
Author(s):  
Matatyahu Rubin ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Automorphism Groups ◽  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Countably Compact

AbstractTheorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

On the Representation of Boolean Algebras

1962 ◽  
Vol 5 (1) ◽  
pp. 37-41 ◽  
Author(s):  
Günter Bruns
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Two Systems ◽  
Image Position

Let B be a Boolean algebra and let ℳ and n be two systems of subsets of B, both containing all finite subsets of B. Let us assume further that the join ∨M of every set M∊ℳ and the meet ∧N of every set N∊n exist. Several authors have treated the question under which conditions there exists an isomorphism φ between B and a field δ of sets, satisfying the conditions:

scholarly journals The Application of Boolean Algebra in Modelling of Leakage Condition of a Car Hydraulic Braking System

2013 ◽  
Vol 18 (2) ◽  
pp. 353-363
Author(s):  
A. Idzikowski ◽  
S. Salamon
Keyword(s):  
Boolean Algebra ◽  
Mathematical Models ◽  
Set Theory ◽  
Boolean Functions ◽  
General Model ◽  
Graphical Model ◽  
Braking System

A general characteristics of a car hydraulic braking system (CHBS) is presented in this publication. A graphical model of properties-component objects is developed for the above-mentioned system. Moreover, four mathematical models in terms of logic, the set theory and the Boolean algebra of Boolean functions are developed. The examination is ended with a general model of the CHBS for n - Boolean variables and the construction and mathematical-technical interpretation of this model is presented.

scholarly journals Wreath Product Action on Generalized Boolean Algebras

10.37236/4831 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ashish Mishra ◽  
Murali K. Srinivasan
Keyword(s):  
Finite Group ◽  
Boolean Algebra ◽  
Wreath Product ◽  
Boolean Algebras ◽  
Product Action ◽  
Finite Set ◽  
A Finite Group ◽  
Multiplicity Free ◽  
Generalized Boolean Algebra ◽  
Generalized Boolean Algebras

Let $G$ be a finite group acting on the finite set $X$ such that the corresponding (complex) permutation representation is multiplicity free. There is a natural rank and order preserving action of the wreath product $G\sim S_n$ on the generalized Boolean algebra $B_X(n)$. We explicitly block diagonalize the commutant of this action.

Boolean Algebra Retracts

1971 ◽  
Vol 23 (2) ◽  
pp. 339-344
Author(s):  
Timothy Cramer
Keyword(s):  
Boolean Algebra ◽  
Dual Problem ◽  
Sufficient Conditions ◽  
Boolean Algebras ◽  
Necessary And Sufficient Conditions ◽  
Infinite Cardinal ◽  
Necessary And Sufficient

A Boolean algebra B is a retract of an algebra A if there exist homomorphisms ƒ: B → A and g: A → B such that gƒ is the identity map B. Some important properties of retracts of Boolean algebras are stated in [3, §§ 30, 31, 32]. If A and B are a-complete, and A is α-generated by B, Dwinger [1, p. 145, Theorem 2.4] proved necessary and sufficient conditions for the existence of an α-homomorphism g: A → B such that g is the identity map on B. Note that if a is not an infinite cardinal, B must be equal to A. The dual problem was treated by Wright [6]; he assumed that A and B are σ-algebras, and that g: A → B is a σ-homomorphism, and gave conditions for the existence of a homomorphism ƒ:B → A such that gƒ is the identity map.

Sign in / Sign up

Export Citation Format

Share Document