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

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.

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

Some remarks on openly generated Boolean algebras

1994 ◽  
Vol 59 (1) ◽  
pp. 302-310 ◽  
Author(s):  
Sakaé Fuchino
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras

AbstractA Boolean algebra B is said to be openly generated if {A : A ≤rcB, ∣A∣ = ℵ0} includes a club subset of . We show:(V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1).(MA+(σ-closed)). Every -free Boolean algebra is openly generated (Theorem 4.2).The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using this characterization we also prove the independence of problem 7 in Ščepin [15] (Proposition 4.3 and Theorem 4.4).

Dense elements characterization of quasi-complemented Almost Distributive Lattices

2014 ◽  
Vol 07 (04) ◽  
pp. 1450062
Author(s):  
G. C. Rao ◽  
G. Nanaji Rao ◽  
A. Lakshmana
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The properties of the set D of dense elements of an ADL are studied. The filter congruence θD generated by D in quasi-complemented ADLs is characterized. Quasi-complemented ADLs is characterized in terms of dense elements. A necessary and sufficient condition for a quasi-complemented ADLs to become a Boolean algebra is established.

Which distributive lattices are lattices of closed sets?

1959 ◽  
Vol 55 (2) ◽  
pp. 172-176 ◽  
Author(s):  
S. Papert
Keyword(s):  
Boolean Algebra ◽  
Complete Lattice ◽  
Boolean Algebras ◽  
Natural Order ◽  
Distributive Lattices ◽  
Real Numbers ◽  
Closed Sets ◽  
Order Form ◽  
Completely Distributive ◽  
Obvious Generalization

1. An elegant theorem due to Tarski states that a completely distributive complete Boolean algebra is isomorphic with a lattice of sets, and in fact the lattice of all the subsets of some aggregate. The obvious generalization of the question underlying this theorem is to ask whether one can pick out by means of a distributivity condition those lattices (not necessarily Boolean algebras) which are isomorphs of lattices of sets. The answer is no. The real numbers with their natural order form a complete lattice which satisfies the strongest possible distributivity conditions and yet is not iso-morphic with any lattice of sets.

The Duality of Distributive Continuous Lattices

1980 ◽  
Vol 32 (2) ◽  
pp. 385-394 ◽  
Author(s):  
B. Banaschewski
Keyword(s):  
Dual Space ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
Continuous Lattice ◽  
Prime Spectrum ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Lattices ◽  
Prime Elements

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.

scholarly journals Existence of nonatomic charges

1978 ◽  
Vol 25 (1) ◽  
pp. 1-6 ◽  
Author(s):  
K. P. S. Bhaskara Rao ◽  
M. Bhaskara Rao
Keyword(s):  
Boolean Algebra ◽  
Complete Characterization ◽  
Boolean Algebras ◽  
Finitely Additive Measures ◽  
Additive Measures

AbstractA complete characterization of Boolean algebras which admit nonatomic charges (i.e. finitely additive measures) is obtained. This also gives rise to a characterization of superatomic Boolean algebras. We also consider the problem of denseness of the set of all nonatomic charges in the space of all charges on a given Boolean algebra, equipped with a suitable topology.

Characterization of prime ideals, filters in ABAs

2019 ◽  
Vol 13 (06) ◽  
pp. 2050106
Author(s):  
U. M. Swamy ◽  
R. Chudamani ◽  
K. Krishna Rao
Keyword(s):  
Boolean Algebras ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Equivalent Conditions

In this paper, we discuss several properties of ideals, filters, annihilators and maximisors in a general Almost Distributive Lattices (ADLs) and in particular, Almost Boolean Algebras (ABAs). Also, we characterize extreme ideals and filters. Further, several equivalent conditions are obtained in terms of ideals and filters for an ADL to become an ABA.

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.

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