Boolean Rings Research Articles

Weak pseudo-complementations on ADL’s

The notion of an Almost Distributive Lattice (abbreviated as ADL) was introduced by U. M. Swamy and G. C. Rao [6] as a common abstraction of several lattice theoretic and ring theoretic generalization of Boolean algebras and Boolean rings. In this paper, we introduce the concept of weak pseudo-complementation on ADL’s and discuss several properties of this.

Smarandache Rings and Smarandache Elements

In this paper, we study some Smarandache (S) notions in some types of rings. Conditions are given under which is a Smarandache ring. We study Smarandache ideals, Smarandache subrings and Smarandache weakly Boolean rings. We discuss some types of Smarandache elements in rings. Moreover, we get some other results.

M-Power Commuting MAPS on Semiprime Rings

Let R be a semiprime ring with center Z(R), extended centroid C, U the maximal right ring of quotients of R, and m a positive integer. Let f: R → U be an additive m-power commuting map. Suppose that f is Z(R)-linear. It is proved that there exists an idempotent e ∈ C such that ef(x) = λx + μ(x) for all x ∈ R, where λ ∈C and μ: R → C. Moreover, (1 − e)U ≅ M2(E), where E is a complete Boolean ring. As consequences of the theorem, it is proved that every additive, 2-power commuting map or centralizing map from R to U is commuting.

On duality between order and algebraic structures in Boolean systems

nski Abstract: We present an extension of the known one-to-one corre- spondence between Boolean algebras and Boolean rings with unit being two types of Boolean systems endowed with order and algebraic struc- tures, respectively. Two equivalent generalizations of Boolean algebras are discussed. We show that there is a one-to-one correspondence between any of the two mentioned generalized Boolean algebras and Boolean rings without unit.

Generalized Boolean and Boolean-like rings

A generalized Boolean ring is a ring R such that, for all x, y in R\(N∪C), x n y – xy n ∈ (N∩C), where n is a fixed even integer, and N , C are the set of nilpotents and center of R, respectively. A Boolean-like ring is a ring R such that, for all x ,y ∈ R\(N∪C),

Orthocomplemented difference lattices in association with generalized rings

Abstract Orthocomplemented difference lattices (ODLs) are orthocomplemented lattices endowed with an additional operation of “abstract symmetric difference”. In studying ODLs as universal algebras or instances of quantum logics, several results have been obtained (see the references at the end of this paper where the explicite link with orthomodularity is discussed, too). Since the ODLs are “nearly Boolean”, a natural question arises whether there are “nearly Boolean rings” associated with ODLs. In this paper we find such an association — we introduce some difference ring-like algebras (the DRAs) that allow for a natural one-to-one correspondence with the ODLs. The DRAs are defined by only a few rather plausible axioms. The axioms guarantee, among others, that a DRA is a group and that the association with ODLs agrees, for the subrings of DRAs, with the famous Stone (Boolean ring) correspondence.

Characterizations of Three Classes of Zero-Divisor Graphs

AbstractThe zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy = 0. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

REMARKS ON A GOLDBACH PROPERTY

In this paper, we study Noetherian Boolean rings. We show that if R is a Noetherian Boolean ring, then R is finite and <TEX>$R{\simeq}(\mathbb{Z}_2)^n$</TEX> for some integer <TEX>$n{\geq}1$</TEX>. If R is a Noetherian ring, then R/J is a Noetherian Boolean ring, where J is the intersection of all ideals I of R with |R/I| = 2. Thus R/J is finite, and hence the set of ideals I of R with |R/I| = 2 is finite. We also give a short proof of Hayes's result : For every polynomial <TEX>$f(x){\in}\mathbb{Z}[x]$</TEX> of degree <TEX>$n{\geq}1$</TEX>, there are irreducible polynomials <TEX>$g(x)$</TEX> and <TEX>$h(x)$</TEX>, each of degree <TEX>$n$</TEX>, such that <TEX>$g(x)+h(x)=f(x)$</TEX>.

Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph

Let R be a commutative ring with 1≠0. The zero-divisor graph Γ(R) of R is the (undirected) graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy=0. The relation on R given by r∼s if and only if annR(r)=annR(s) is an equivalence relation. The compressed zero-divisor graph ΓE(R) is the (undirected) graph whose vertices are the equivalence classes induced by ∼ other than [0] and [1], such that distinct vertices [r] and [s] are adjacent in ΓE(R) if and only if rs=0. We investigate ΓE(R) when R is reduced and are interested in when ΓE(R)≅Γ(S) for a reduced ring S. Among other results, it is shown that ΓE(R)≅Γ(B) for some Boolean ring B if and only if Γ(R) (and hence ΓE(R)) is a complemented graph, and this is equivalent to the total quotient ring of R being a von Neumann regular ring.

Boolean rings and reciprocal eigenvalue properties

Let A be the adjacency matrix of the zero-divisor graph Γ(R) of a finite commutative ring R containing nonzero zero-divisors. In this paper, it is shown that Γ(R) is the zero-divisor graph of a Boolean ring if and only if det(A)=-1. Also, A is similar to plus or minus its inverse whenever R is a Boolean ring. As a consequence, it is proved that Γ(R) is the zero-divisor graph of a Boolean ring if and only if the set of eigenvalues (including multiplicities) of Γ(R) can be partitioned into 2-element subsets of the form {λ,±1/λ}. Furthermore, any finite Boolean ring R is characterized by the degree and coefficients of the characteristic polynomial of A.

On zero-divisor graphs of Boolean rings

The zero-divisor graph of a ring R is the graph whose vertices consist of the nonzero zero-divisors of R in which two distinct vertices a and b are adjacent if and only if either ab = 0 or ba = 0. In this paper, we investigate some properties of zero-divisor graphs of Boolean rings. Among other results, we prove that for any two rings R and S with Γ(R) ≃ Γ(S), if R is Boolean and |R| > 4, then R ≃ S.

Boolean Gröbner bases

In recent years, Boolean Gröbner bases have attracted the attention of many researchers, mainly in connection with cryptography. Several sophisticated methods have been developed for the computation of Boolean Gröbner bases. However, most of them only deal with Boolean polynomial rings over the simplest coefficient Boolean ring G F 2 . Boolean Gröbner bases for arbitrary coefficient Boolean rings were first introduced by two of the authors almost two decades ago. While the work is not well-known among computer algebra researchers, recent active work on Boolean Gröbner bases inspired us to return to their development. In this paper, we introduce our work on Boolean Gröbner bases with arbitrary coefficient Boolean rings.

On algebras of multidimensional probabilities

Abstract The probability p(s) of the occurrence of an event pertaining to a physical system which is observed in different states s determines a function p from the set S of states of the system to [0, 1]. The function p is called a multidimensional probability or numerical event. Sets of numerical events which are structured either by partially ordering the functions p and considering orthocomplementation or by introducing operations + and · in order to generalize the notion of Boolean rings representing classical event fields are studied with the goal to relate the algebraic operations + and · to the sum and product of real functions and thus to distinguish between classical and quantum mechanical behaviour of the physical system. Necessary and sufficient conditions for this are derived, as well for the case that the functions p can assume any value between 0 and 1 as for the special cases that the values of p are restricted to two or three different outcomes.

On computation of Boolean involutive bases

Grobner bases in Boolean rings can be calculated by an involutive algorithm based on the Janet or Pommaret division. The Pommaret division allows calculations immediately in the Boolean ring, whereas the Janet division implies use of a polynomial ring over field $$ \mathbb{F}_2 $$ . In this paper, both divisions are considered, and distributive and recursive representations of Boolean polynomials are compared from the point of view of calculation effectiveness. Results of computer experiments with both representations for an algorithm based on the Pommaret division and for lexicographical monomial order are presented.

Royce, Boolean Rings, and the T-Relation

This paper examines in detail the four-place Boolean relation that in his article “An Extension of the Algebra of Logic” Royce calls “the T-Relation,” and the paper proves correct Royce’s major mathematical claims about the T-Relation. The paper also argues that in his article Royce develops the concept of a Boolean Ring as well as an algebra that is in essence identical to the Zhegalkin Algebra developed by the father of the Russian school of logic Ivan Ivanovich Zhegalkin. The paper finds it ironic that Royce seems to have underestimated the importance of his own work. Finally, the paper argues that there is nothing special about the T-Relation’s having four places, and it completely generalizes the T-Relation into a Boolean relation having an arbitrary number of places.

On Brooks–Jewett, Vitali–Hahn–Saks and Nikodým convergence theorems for quasi-triangular functions

Connections are established between Brooks–Jewett, Vitali–Hahn–Saks and Nikodym type convergence theorems for quasi-triangular functions on Boolean rings satisfying the Subsequential Completeness Property and valued into Hausdorff topological spaces.

Role of involutive criteria in computing Boolean Gröbner bases

In this paper, effectiveness of using four criteria in an involutive algorithm based on the Pommaret division for construction of Boolean Grobner bases is studied. One of the results of this study is the observation that the role of the criteria in computations in Boolean rings is much less than that in computations in an ordinary ring of polynomials over the field of integers. Another conclusion of this study is that the efficiency of the second and/or third criteria is higher than that of the two others.

PolyBoRi: A framework for Gröbner-basis computations with Boolean polynomials

This work presents a new framework for Gröbner-basis computations with Boolean polynomials. Boolean polynomials can be modelled in a rather simple way, with both coefficients and degree per variable lying in { 0 , 1 } . The ring of Boolean polynomials is, however, not a polynomial ring, but rather the quotient ring of the polynomial ring over the field with two elements modulo the field equations x 2 = x for each variable x . Therefore, the usual polynomial data structures seem not to be appropriate for fast Gröbner-basis computations. We introduce a specialised data structure for Boolean polynomials based on zero-suppressed binary decision diagrams (ZDDs), which are capable of handling these polynomials more efficiently with respect to memory consumption and also computational speed. Furthermore, we concentrate on high-level algorithmic aspects, taking into account the new data structures as well as structural properties of Boolean polynomials. For example, a new useless-pair criterion for Gröbner-basis computations in Boolean rings is introduced. One of the motivations for our work is the growing importance of formal hardware and software verification based on Boolean expressions, which suffer–besides from the complexity of the problems –from the lack of an adequate treatment of arithmetic components. We are convinced that algebraic methods are more suited and we believe that our preliminary implementation shows that Gröbner-bases on specific data structures can be capable of handling problems of industrial size.

Semivariations of an additive function on a Boolean ring

With an additive function $\varphi $ from a Boolean ring $A$ into a normed space two positive functions on $A$, called semivariations of $\varphi $, are associated. We characterize those functions as submeasures with some additional properties in the gene

On Realizing Zero-Divisor Graphs

An algorithm is presented for constructing the zero-divisor graph of a direct product of integral domains. Moreover, graphs which are realizable as zero-divisor graphs of direct products of integral domains are classified, as well as those of Boolean rings. In particular, graphs which are realizable as zero-divisor graphs of finite reduced commutative rings are classified.