Generalization Of Boolean Algebras Research Articles

The spectra of quasi-Boolean algebras

Abstract In the present paper, we introduce the notions of quasi-Boolean algebras as the generalization of Boolean algebras. First we discuss the related properties of quasi-Boolean algebras. Second we define filters of quasi-Boolean algebras and investigate some properties of filters in quasi-Boolean algebras. We also show that there is a one-to-one correspondence between the set of filters and the set of filter congruences on a quasi-Boolean algebra. Then we investigate the prime filters and maximal filters of quasi-Boolean algebras, showing that the prime filters of a quasi-Boolean algebra are precisely the maximal filters and the prime spectrum of a quasi-Boolean algebra is a compact Hausdorff topological space. Finally, we define and study the reticulation of a quasi-Boolean algebra.

Relative Pseudo-Complementations on ADL's

The notion of an Almost Distributive Lattice (abbreviated as ADL) is a common abstraction of several lattice theoretic generalization of Boolean algebras and Boolean rings. In this paper we introduce the notion of Relative pseudo-complementation on ADL's and discuss several properties of this.

Associate elements in ADLs

The concept of an Almost Distributive Lattice (ADL) is an abstraction of several lattice theoretic and ring theoretic generalizations of Boolean algebras and Boolean rings. In this paper, we introduce notion of associate elements in an ADL A and mainly prove that the associativity relation is the smallest congruence relation on A with respect to which the quotient of A is a distributive lattice.

Top Element Problem and Macneille Completions of Generalized Effect Algebras

Effect algebras (EAs), introduced by D. J. Foulis and M. K. Bennett, as common generalizations of Boolean algebras, orthomodular lattices and MV-algebras, are nondistributive algebraic structures including unsharp elements. Their unbounded versions, called generalized effect algebras, are posets which may have or may have not an EA-MacNeille completion, or cannot be embedded into any complete effect algebra. We give a necessary and sufficient condition for a generalized effect algebra to have an EA-MacNeille completion. Some examples are provided.

Coreflections in Algebraic Quantum Logic

Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.

A characterization of free generating sets in MV-algebras

MV-algebras are a generalization of Boolean algebras. As is well known, a free generating set \(\{b_i \mid i \in X\}\) for a Boolean algebra is characterized by the following simple algebraic condition: whenever A and B are finite disjoint subsets of X then \(\bigwedge_{b_i\in A}\,b_i\,\wedge\,\,\bigwedge_{b_{j}\in B}\neg b_j\neq 0\). Our aim in this note is to give a similar characterization of free generating sets in MV-algebras.

Some lattice models of bilinear logic

The aim of this article is to present to universal algebraists a generalization of Boolean algebras which do not obey Gentzen's three structural rules. These so-calledGrishin algebras are models ofclassical bilinear propositional logic, a non-commutative version of linear logic which allows two negations. Examples in which these negations coincide are easy to come by, but examples in which they are distinct are more elusive. To this purpose, it was found necessary to generalize the notion of a group to that of alattice ordered monoid with adjoints. While the left inverse and the right inverse of a group element necessarily coincide this is not so for the left adjoint and the right adjoint of an element in a lattice ordered monoid.

Principal congruences in de Morgan algebras

A congruence relation θ on an algebra L is principal if there exist a, b)∈L such that θ is the smallest congruence relation for which (a, b)∈θ. The property that, for every algebra in a variety, the intersection of two principal congruences is again a principal congruence is one that is known to be shared by many varieties (see, for example, K. A. Baker [1]). One such example is the variety of Boolean algebras. De Morgan algebras are a generalization of Boolean algebras and it is the intersection of principal congruences in the variety of de Morgan algebras that is to be considered in this note.

Continuously valued logic

Motivated by the recognized inadequacy of conventional logic for the representationand manipulation of variables in areas related to artificial intelligence, this paper addresses itself to the investigation of the formal systems obtained by extending well-known connectives to continuous arguments. The studied systems, called “soft algebras,” are generalizations of Boolean algebras in that they satisfy all the axioms of the latter ones except the laws of complementarity, i.e., x+ =1 and x =0. It is shown that every soft algebra is a bounded, distributive and symmetric lattice. A specific soft algebra, the family of all functions of n variables in the closed interval [0, 1], is analyzed in great detail. This particularlalgebra is a formal unification of many recent results concerning “fuzzy” logic. It is shown that every “soft” function can be canonically represented by a pair of normal expressions, i.e., each soft function is representable by a double array of tables (a generalization of the truth-table representation of Boolean functions). Also, a synthesis and two-level minimization procedure, which is a generalization of the Quine-McCluskey method, is given.

A generalization of Stone’s representation theorem for Boolean algebras

The well-known theorem of M. H. Stone [10] asserts that every Boolean algebra is isomorphic to a field of sets, that is, to a subalgebra of a complete and atomic Boolean algebra. A generalization of Boolean algebras are the Stone lattices. It is my aim to prove a representation theorem for Stone lattices: Every Stone is isomorphic to a sublattice of the of all ideals of a complete and atomic Boolean algebra. . This theorem was conjectured by O. Frink [3] as a solution to Problem 70 of Garrett Birkhoff's book Lattice Theory (revised edition, New York 1948), which asks for a characterization of Stone lattices. The name ilStone lattice was proposed in [5], which contains the first solution to Problem 70. The author's paper [4J contains a further solution, though less deep than the first one. A Stone is a which is distributive and pseudo-complemented, and in which the formula a* V a** = 1 holds identically. Stone lattices and the problem of characterizing them were first discussed in Stone's paper [l1J on Brouwerian logic. My proof involves applications of mathematical logic to algebra like those of Jonsson and Tarski, L. Henkin, and A. Robinson [6J, [8], [9]. First I omit the assumption that the Boolean algebra is complete, and prove an apparently weaker imbedding theorem; then I show that this weaker theorem is actually equivalent to the original one. This idea is due to Jonsson and Tarski. Next I show that a particular class of Stone lattices forms an arithmetic class in the sense of Tarski. Then by applying a consequence of Godel's theorem concerning the completeness of the first order functional calculus (see Henkin [6] and Hintikka [7]), we conclude that it is sufficient to consider finitely generated Stone lattices, which are finite. A direct decomposition theorem further simplifies the problem; the resulting class of Stone lattices is so simple that a representation is obtained without difficulty.