The Boolean algebras of abelian groups and well-orders

1974 ◽  
Vol 39 (3) ◽  
pp. 452-458 ◽  
Author(s):  
Dale Myers
Keyword(s):  
Boolean Algebra ◽  
Finite Number ◽  
Linear Order ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Order Type ◽  
Equivalence Classes ◽  
Boolean Algebras ◽  
Partial Ordering ◽  
Elementary Classes

For any linear order type τ with first element let be the Boolean set algebra generated by the left-closed right-open (including [x, ∞)) intervals of some linear order of type τ. Let η and ω be the order types of the rationals and natural numbers respectively (when not used as an order type, ω will, as usual, be the set of nonnegative integers). We show that the Boolean algebra of the elementary theory of well-orders, i.e., the Boolean algebra of elementary classes of well-orders or, equivalently, the algebra of equivalence classes of sentences of the elementary theory of well-orders, is isomorphic to and that the Boolean algebra of the elementary theory of abelian groups is isomorphic to . Both results are obtained by applying Hanf's structure diagram technique to the work of Mostowski, Tarski, and Szmielew. One may formalize discussion of algebras of proper classes by assuming the classes are included in a universe which is a set in a larger universe. Given a Boolean algebra, let 0 be its zero and 1 its unit, let ≤ be its associated partial ordering, and, for any elements a, b, and c of the algebra, let “a + b = c ” be the assertion that c is the disjoint sum of a and b. A subset of the algebra disjointly generates the algebra iff each of the algebra's elements is a disjoint sum of a finite number of the subset's elements.

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 .

Weak Distributivity, A Problem of Von Neumann and the Mystery of Measurability

2006 ◽  
Vol 12 (2) ◽  
pp. 241-266 ◽  
Author(s):  
Bohuslav Balcar ◽  
Thomas Jech
Keyword(s):  
Boolean Algebra ◽  
Representation Theorem ◽  
Chain Condition ◽  
Boolean Algebras ◽  
Partial Ordering ◽  
Additive Measure ◽  
Boolean Operations ◽  
Von Neumann ◽  
Countable Chain Condition ◽  
Countable Chain

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. ABoolean algebrais a setBwith Boolean operationsa˅b(join),a˄b(meet) and −a(complement), partial orderinga≤bdefined bya˄b=aand the smallest and greatest element,0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S−a, ordered by inclusion, with0= ∅ and1=S.Complete Boolean algebras and weak distributivity.A Boolean algebrais a setBwith Boolean operationsa˅b(join),a˄b(meet) and -a(complement), partial orderinga≤bdefined bya˄b=aand the smallest and greatest element.0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S-a, ordered by inclusion, with0= ϕ and1=S.

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.

A Fast 3-D Visualization Methodology Using Characteristic Views of Objects

1998 ◽  
Vol 08 (01) ◽  
pp. 97-114 ◽  
Author(s):  
Soochan Hwang ◽  
Sang-Young Cho ◽  
Taehyung Wang ◽  
Phillip C.-Y. Sheu
Keyword(s):  
Finite Number ◽  
Real Time ◽  
Time Complexity ◽  
Object Oriented ◽  
Equivalence Classes ◽  
Visualization Method ◽  
Tree Algorithm ◽  
Projected Image ◽  
Characteristic Views ◽  
Run Time

This paper describes a 3-D visualization method based on the concept of characteristic views (CVs). The idea of characteristic views was derived based on the observation that the infinite possible views of a 3-D object can be grouped into a finite number of equivalence classes so that within each class all the views are isomorphic in the sense that they have the same line-junction graphs. To visualize the changes of scenes in real time, the BSP tree algorithm is known to be efficient in a static environment in which the viewpoint can be changed easily. However, if a scene consists of many objects and each object consists of many polygons, the time complexity involved in traversing a BSP tree increases rapidly so that the original BSP tree algorithm may not be efficient. The method proposed in this paper is object-oriented in the sense that, for all viewpoints, at the preprocessing stage the ordering for displaying the objects is determined. At run time, the objects are displayed based on a pre-calculated ordering according to the viewpoint. In addition, a CV is used as a basic 2-D projected image of a 3-D object.

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 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.

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