scholarly journals The Boolean space of higher level orderings

2007 ◽  
Vol 196 (2) ◽  
pp. 101-117 ◽  
Author(s):  
Katarzyna Osiak
Keyword(s):  
Boolean Space

The Lattice of Subalgebras of a Boolean Algebra

1962 ◽  
Vol 14 ◽  
pp. 451-460 ◽  
Author(s):  
David Sachs
Keyword(s):  
Boolean Algebra ◽  
Lattice Theory ◽  
Closure Operator ◽  
Vector Spaces ◽  
Finite Partition ◽  
Partition Lattice ◽  
Topological Vector Spaces ◽  
Boolean Space ◽  
Lattice Of Subalgebras

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

On Sheaf Representation of a Biregular Near-Ring

1977 ◽  
Vol 20 (4) ◽  
pp. 495-500 ◽  
Author(s):  
George Szeto
Keyword(s):  
Boolean Space ◽  
Sheaf Representation

AbstractIt is shown that R is a biregular near-ring if and only if it is isomorphic with the near-ring of sections of a sheaf of reduced near-rings over a Boolean space. Also, some ideal properties of a biregular near-ring are proved. These are considered as generalizations of some works of R. Pierce on biregular rings.

scholarly journals A note on commutative Baer rings III

1973 ◽  
Vol 15 (1) ◽  
pp. 15-21 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Semiprime Ring ◽  
Boolean Space ◽  
Baer Ring ◽  
Baer Rings

If R is a commutative semiprime ring with identity Kist [4], [5] has shown that R can be embedded into a commutative Baer ring B(R), and has given some properties of this embedding. More recently Mewborn [7] has given a construction which embeds R into a commutative Baer ring with the stronger property that every annihilator is generated by an idempotent. Both of these constructions involve a representation of R as a ring of global sections of a sheaf over a Boolean space.

scholarly journals Systems of Inclusions with Unknowns in Multioperations

2021 ◽  
Vol 38 ◽  
pp. 112-123
Author(s):  
N. A. Peryazev ◽  
Keyword(s):  
Numerical Methods ◽  
Finite Rank ◽  
Logical Inference ◽  
Boolean Space ◽  
Boolean Equation ◽  
Inference Systems

We consider systems of inclusions with unknowns and coefficients in multioperations of finite rank. An algorithm for solving such systems by the method of reduction to Boolean equations using superposition representation of multioperations by Boolean space matrices is given. Two methods for solving Boolean equations with many unknowns are described for completeness. The presentation is demonstrated by examples: the representation of the superposition of multioperations by Boolean space matrices; solving a Boolean equation by analytical and numerical methods; and finding solutions to an inclusion with one unknown. The resulting algorithm can be applied to the development of logical inference systems for multioperator logics.

Commutative regular rings and Boolean-valued fields

1984 ◽  
Vol 49 (1) ◽  
pp. 281-297 ◽  
Author(s):  
Kay Smith
Keyword(s):  
Set Theory ◽  
Regular Ring ◽  
Galois Theory ◽  
Algebraic Closure ◽  
Original Proof ◽  
Valued Fields ◽  
Boolean Space ◽  
Equality Relation ◽  
The One ◽  
Regular Rings

In this paper we present an equivalence between the category of commutative regular rings and the category of Boolean-valued fields, i.e., Boolean-valued sets for which the field axioms are true. The author used this equivalence in [12] to develop a Galois theory for commutative regular rings. Here we apply the equivalence to give an alternative construction of an algebraic closure for any commutative regular ring (the original proof is due to Carson [2]).Boolean-valued sets were developed in 1965 by Scott and Solovay [10] to simplify independence proofs in set theory. They later were applied by Takeuti [13] to obtain results on Hilbert and Banach spaces. Ellentuck [3] and Weispfenning [14] considered Boolean-valued rings which consisted of rings and associated Boolean-valued relations. (Lemma 4.2 shows that their equality relation is the same as the one used in this paper.) To the author's knowledge, the present work is the first to employ the Boolean-valued sets of Scott and Solovay to obtain results in algebra.The idea that commutative regular rings can be studied by examining the properties of related fields is not new. For several years algebraists and logicians have investigated commutative regular rings by representing a commutative regular ring as a subdirect product of fields or as the ring of global sections of a sheaf of fields over a Boolean space (see, for example, [9] and [8]). These representations depend, as does the work presented here, on the fact that the set of central idempotents of any ring with identity forms a Boolean algebra. The advantage of the Boolean-valued set approach is that the axioms of classical logic and set theory are true in the Boolean universe. Therefore, if the axioms for a field are true for a Boolean-valued set, then other properties of the set can be deduced immediately from field theory.

The 𝒞(X,D), a characterization of a Stone almost distributive lattice

2015 ◽  
Vol 08 (03) ◽  
pp. 1550055
Author(s):  
Ch. Santhi Sundar Raj ◽  
K. Rama Prasad ◽  
M. Santhi ◽  
R. Vasu Babu
Keyword(s):  
Distributive Lattice ◽  
Homomorphic Image ◽  
Maximal Element ◽  
Continuous Functions ◽  
Boolean Space

We prove that for any Boolean space [Formula: see text] and a dense Almost Distributive Lattice (ADL) [Formula: see text] with a maximal element, the set [Formula: see text] of all continuous functions of [Formula: see text] into the discrete [Formula: see text] is a Stone ADL. Conversely, it is proved that any Stone ADL is a homomorphic image of [Formula: see text] for a suitable Boolean space [Formula: see text] and a dense ADL [Formula: see text] with a maximal element.

scholarly journals A heuristic method for bi-decomposition of partial Boolean functions

Informatics ◽  
2020 ◽  
Vol 17 (3) ◽  
pp. 44-53
Author(s):  
Yu. V. Pottosin
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Heuristic Method ◽  
Disjunctive Normal Form ◽  
The Other ◽  
Output Function ◽  
Minimal Rank ◽  
Boolean Space ◽  
Definitional Domain ◽  
The Given

The problem of decomposition of a Boolean function is to represent a given Boolean function in the form of a superposition of some Boolean functions whose number of arguments are less than the number of given function. The bi-decomposition represents a given function as a logic algebra operation, which is also given, over two Boolean functions. The task is reduced to specification of those two functions. A method for bi-decomposition of incompletely specified (partial) Boolean function is suggested. The given Boolean function is specified by two sets, one of which is the part of the Boolean space of the arguments of the function where its value is 1, and the other set is the part of the space where the function has the value 0. The complete graph of orthogonality of Boolean vectors that constitute the definitional domain of the given function is considered. In the graph, the edges are picked out, any of which has its ends corresponding the elements of Boolean space where the given function has different values. The problem of bi-decomposition is reduced to the problem of a weighted two-block covering the set of picked out edges of considered graph by its complete bipartite subgraphs (bicliques). Every biclique is assigned with a disjunctive normal form (DNF) in definite way. The weight of a biclique is a pair of certain parameters of   assigned DNF. According to each biclique of obtained cover, a Boolean function is constructed whose arguments are the variables from the term of minimal rank on the DNF. A technique for constructing the mentioned cover for two kinds of output function is described.

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