A Generalization of Certain Rings of A. L. Foster

1963 ◽  
Vol 6 (1) ◽  
pp. 55-60 ◽  
Author(s):  
Adil Yaqub
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Boolean Ring ◽  
Boolean Rings ◽  
Image Position

The concept of a Boolean ring, as a ring A in which every element is idempotent (i. e., a2 = a for all a in A), was first introduced by Stone [4]. Boolean algebras and Boolean rings, though historically and conceptually different, were shown by Stone to be equationally interdefinable. Indeed, let (A, +, x) be a Boolean ring with unit 1, and let (A, ∪, ∩, ') be a Boolean algebra, where ∩, ∪, ', denote "union", " intersection", and "complement". The equations which convert the Boolean ring into a Boolean algebra are:IConversely, the equations which convert the Boolean algebra into a Boolean ring are:II

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:

Yap Hian Poh. Postulational study of an axiom system of Boolean algebra. Majallah Tahunan 'Ilmu Pasti—Shu Hsüeh Nien K'an—Bulletin of Mathematical Society of Nanyang University (1960), pp. 94–110. - R. M. Dicker. A set of independent axioms for Boolean algebra. Proceedings of the London Mathematical Society, ser. 3 vol. 13 (1963), pp. 20–30. - P. J. van Albada. A self-dual system of axioms for Boolean algebra. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, series A vol. 67 (1964), pp. 377–381; also Indagationes mathematicae, vol. 26 (1964), pp. 377–381. - Antonio Diego and Alberto Suárez. Two sets of axioms for Boolean algebras. Portugaliae mathematica, vol. 23 nos. 3–4 (for 1964, pub. 1965), pp. 139–145. (Reprinted from Notas de lógica matemática no. 16, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca 1964, 13 pp.) - P. J. van Albada. Axiomatique des algèbres de Boole. Bulletin de la Société Mathématique de Belgique, vol. 18 (1966), pp. 260–272. - Lawrence J. Dickson. A short axiomatic system for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 6 (1967), pp. 253–257. - Leroy J. Dickey. A shorter axiomatic system for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 8 (1968), p. 336. - Chinthayamma . Independent postulate sets for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 9 (1968), pp. 378–379. - Kiyoshi Iséki. A simple characterization of Boolean rings. Proceedings of the Japan Academy, vol. 44 (1968), pp. 923–924. - Sakiko Ôhashi. On definitions of Boolean rings and distributive lattices. Proceedings of the Japan Academy, vol. 44 (1968), pp. 1015–1017.

1973 ◽  
Vol 38 (4) ◽  
pp. 658-660
Author(s):  
Donald H. Potts
Keyword(s):  
Boolean Algebra ◽  
London Mathematical Society ◽  
Axiom System ◽  
Dual System ◽  
Boolean Algebras ◽  
Axiomatic System ◽  
Mathematical Society ◽  
Distributive Lattices ◽  
Boolean Rings

Cardinal functions on ultra products of Boolean algebras

1997 ◽  
Vol 62 (1) ◽  
pp. 43-59 ◽  
Author(s):  
Douglas Peterson
Keyword(s):  
Boolean Algebra ◽  
Cardinal Number ◽  
Boolean Algebras ◽  
Open Problems ◽  
Cardinal Functions ◽  
Image Position

This article is concerned with functions k assigning a cardinal number to each infinite Boolean algebra (BA), and the behaviour of such functions under ultraproducts. For some common functions k we havefor others we have ≤ instead, under suitable assumptions. For the function π character we go into more detail. More specifically, ≥ holds when F is regular, for cellularity, length, irredundance, spread, and incomparability. ≤ holds for π. ≥ holds under GCH for F regular, for depth, π, πχ, χ, h-cof, tightness, hL, and hd. These results show that ≥ can consistently hold in ZFC since if V = L holds then all uniform ultrafilters are regular. For π-character we prove two more results: (1) If F is regular and ess , then(2) It is relatively consistent to have , where A is the denumerable atomless BA.A thorough analysis of what happens without the assumption that F is regular can be found in Rosłanowski, Shelah [8] and Magidor, Shelah [5]. Those papers also mention open problems concerning the above two possible inequalities.

A Characterization of Implicative Boolean Rings

1953 ◽  
Vol 5 ◽  
pp. 465-469 ◽  
Author(s):  
A. H. Copeland ◽  
Frank Harary
Keyword(s):  
Boolean Algebra ◽  
Cross Product ◽  
Boolean Ring ◽  
Conditional Probabilities ◽  
Boolean Operations ◽  
Boolean Rings ◽  
The Cross

In the theory of probability, the conditional can be treated by an operation analogous to division. Many properties of the conditional can best be studied by means of the corresponding multiplication (called the cross-product). An implicative Boolean ring is defined [2] in terms of a cross-product and the usual Boolean operations. The cross-product is the only device yet known in which the events corresponding to conditional probabilities are themselves elements of the Boolean ring. The fact that such advice was not introduced by Boole is probably the reason why Boolean algebra has been very little used in the theory of probability, although probability was one of the principal applications which Boole had in mind.

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.

Locally Flat Vector Lattices

1980 ◽  
Vol 32 (4) ◽  
pp. 924-936 ◽  
Author(s):  
Marlow Anderson
Keyword(s):  
Boolean Algebra ◽  
Essential Extension ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Trivial Intersection ◽  
Image Position

Let G be a lattice-ordered group (l-group). If X ⊆ G, then letThen X’ is a convex l-subgroup of G called a polar. The set P(G) of all polars of G is a complete Boolean algebra with ‘ as complementation and set-theoretic intersection as meet. An l-subgroup H of G is large in G (G is an essential extension of H) if each non-zero convex l-subgroup of G has non-trivial intersection with H. If these l-groups are archimedean, it is enough to require that each non-zero polar of G meets H. This implies that the Boolean algebras of polars of G and H are isomorphic. If K is a cardinal summand of G, then K is a polar, and we write G = K⊞K'.

Equational bases of boolean algebras

1964 ◽  
Vol 29 (3) ◽  
pp. 115-124 ◽  
Author(s):  
F. M. Sioson
Keyword(s):  
Boolean Algebra ◽  
Algebraic System ◽  
Boolean Algebras ◽  
Definition Of ◽  
Image Position

It is well-known that a Boolean algebra (B, +, ., ‐) may be defined as an algebraic system with at least two elements such that (for all x, y, z ε B): These axioms or equations are not independent, in the sense that some of them are logical consequences of the others. B. A. Bernstein [1] thought that the first three and their duals form an independent dual-symmetric definition of a Boolean algebra, but R. Montague and J. Tarski [3] proved later that B1 (or B̅1) follows from B2, B3, B̅1, B̅2, B̅3 (from B1, B2, B3, B̅2, B̅3).

scholarly journals On ring-like structures of lattice-Ordered numerical events

2021 ◽  
pp. 2150186
Author(s):  
Dietmar Dorninger ◽  
Helmut Länger
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Boolean Algebra ◽  
Special Interest ◽  
Physical System ◽  
Boolean Algebras ◽  
Boolean Rings ◽  
Term Functions ◽  
Ordered Algebras

Let [Formula: see text] be a set of states of a physical system. The probabilities [Formula: see text] of the occurrence of an event when the system is in different states [Formula: see text] define a function from [Formula: see text] to [Formula: see text] called a numerical event or, more accurately, an [Formula: see text]-probability. Sets of [Formula: see text]-probabilities ordered by the partial order of functions give rise to so-called algebras of [Formula: see text]-probabilities, in particular, to the ones that are lattice-ordered. Among these, there are the [Formula: see text]-algebras known from probability theory and the Hilbert-space logics which are important in quantum-mechanics. Any algebra of [Formula: see text]-probabilities can serve as a quantum-logic, and it is of special interest when this logic turns out to be a Boolean algebra because then the observed physical system will be classical. Boolean algebras are in one-to-one correspondence to Boolean rings, and the question arises to find an analogue correspondence for lattice-ordered algebras of [Formula: see text]-probabilities generalizing the correspondence between Boolean algebras and Boolean rings. We answer this question by defining ring-like structures of events (RLSEs). First, the structure of RLSEs is revealed and Boolean rings among RLSEs are characterized. Then we establish how RLSEs correspond to lattice-ordered algebras of numerical events. Further, functions for associating lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs are studied. It is shown that there are only two ways to assign lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs if one restricts the corresponding mappings to term functions over the underlying orthomodular lattice. These term functions are the very functions by which also the Boolean algebras can be assigned to Boolean rings.

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.

Sign in / Sign up

Export Citation Format

Share Document