scholarly journals The pointfree version of grills

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2667-2681
Author(s):  
Ali Estaji ◽  
Toktam Haghdadi
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Boolean Algebras ◽  
New Concepts

In this paper, the pointfree version of grills is introduced. We consider a Boolean algebra B and a subframe L instead of a topological space (X,?), and present the concept of approximation ? over B. Moreover, some properties of them are given. Also, we introduce and study the new concepts grill and ?-grill on Boolean algebras.

Games played on Boolean algebras

1983 ◽  
Vol 48 (3) ◽  
pp. 714-723 ◽  
Author(s):  
Matthew Foreman
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Closed Subset ◽  
Winning Strategy ◽  
Boolean Algebras ◽  
Algebra Structure ◽  
Partial Converse ◽  
Image Position ◽  
Special Case

In this paper we consider the special case of the Banach-Mazur game played on a topological space when the space also has an underlying Boolean Algebra structure. This case was first studied by Jech [2]. The version of the Banach-Mazur game we will play is the following game played on the Boolean algebra:Players I and II alternate moves playing a descending sequence of elements of a Boolean algebra ℬ.Player II wins the game iff Πi∈ωbi ≠ 0. Jech first considered these games and showed:Theorem (Jech [2]). ℬ is (ω1, ∞)-distributive iff player I does not have a winning strategy in the game played on ℬ.If ℬ has a dense ω-closed subset then it is easy to see that player II has a winning strategy in this game. This paper establishes a partial converse to this, namely it gives cardinality conditions on ℬ under which II having a winning strategy implies ω-closure.In the course of proving the converse, we consider games of length > ω and generalize Jech's theorem to these games. Finally we present an example due to C. Gray that stands in counterpoint to the theorems in this paper.In this section we give a few basis definitions and explain our notation. These definitions are all standard.

scholarly journals Topologies on Boolean algebras defined by ideals and dual ideals

1973 ◽  
Vol 14 (1) ◽  
pp. 13-20
Author(s):  
R. Beazer
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Binary Operation ◽  
Boolean Algebras ◽  
Arbitrary Lattice

In the paper [5], Rema used the well-known fact that in a Boolean algebra the binary operation d: B × B → B defined by is a “metric“ operation to show that, if D is any dual ideal of ^, then the sets Up = {(x, y): d(x, y) <p}, where p ∈ D, form a base for a uniformity of }, the resulting topological space <B; T[D]> being called an auto-topologized Boolean algebra. Recently, Kent and Atherton [1, 4] exhibited a family of topologies on an arbitrary lattice ℒ defined in terms of ideals and dual ideals. More specifically, if I and D are respectively an ideal and a dual ideal of ℒ, then the T[I:D] topology on ℒ is the topology defined by taking the sets of the form a*⋂b+, where , as sub-base for the open sets. It is these topologies that are studied in this paper.

scholarly journals Some Modifications of Pairwise Soft Sets and Some of Their Related Concepts

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1781
Author(s):  
Samer Al Ghour
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Soft Sets ◽  
New Concepts ◽  
Bitopological Spaces ◽  
The Relationship ◽  
Soft Topological Space

In this paper, we first define soft u-open sets and soft s-open as two new classes of soft sets on soft bitopological spaces. We show that the class of soft p-open sets lies strictly between these classes, and we give several sufficient conditions for the equivalence between soft p-open sets and each of the soft u-open sets and soft s-open sets, respectively. In addition to these, we introduce the soft u-ω-open, soft p-ω-open, and soft s-ω-open sets as three new classes of soft sets in soft bitopological spaces, which contain soft u-open sets, soft p-open sets, and soft s-open sets, respectively. Via soft u-open sets, we define two notions of Lindelöfeness in SBTSs. We discuss the relationship between these two notions, and we characterize them via other types of soft sets. We define several types of soft local countability in soft bitopological spaces. We discuss relationships between them, and via some of them, we give two results related to the discrete soft topological space. According to our new concepts, the study deals with the correspondence between soft bitopological spaces and their generated bitopological spaces.

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.

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.

scholarly journals Balleans, hyperballeans and ideals

2019 ◽  
Vol 20 (2) ◽  
pp. 431 ◽  
Author(s):  
Dikran Dikranjan ◽  
Igor Protasov ◽  
Ksenia Protasova ◽  
Nicolò Zava
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Connected Components ◽  
Coarse Structure

<p>A ballean B (or a coarse structure) on a set X is a family of subsets of X called balls (or entourages of the diagonal in X × X) dened in such a way that B can be considered as the asymptotic counterpart of a uniform topological space. The aim of this paper is to study two concrete balleans dened by the ideals in the Boolean algebra of all subsets of X and their hyperballeans, with particular emphasis on their connectedness structure, more specically the number of their connected components.</p>

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.

scholarly journals Densities of Ultraproducts of Boolean Algebras

1995 ◽  
Vol 47 (1) ◽  
pp. 132-145
Author(s):  
Sabine Koppelberg ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Cardinal Invariants ◽  
The Third ◽  
Topological Density

AbstractWe answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density πA resp. the topological density cL4 of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.

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