Boolean Semirings Research Articles

Top-heavy phenomena for transformations

Let S be a transformation semigroup acting on a set Ω. The action of S on Ω can be naturally extended to be an action on all subsets of Ω. We say that S is ℓ-homogeneous provided it can send A to B for any two (not necessarily distinct) ℓ-subsets A and B of Ω. On the condition that k ≤ ℓ < k + ℓ ≤ |Ω|, we show that every ℓ-homogeneous transformation semigroup acting on Ω must be k-homogeneous. We report other variants of this result for Boolean semirings and affine/projective geometries. In general, any semigroup action on a poset gives rise to an automaton and we associate some sequences of integers with the phase space of this automaton. When this poset is a geometric lattice, we propose to investigate various possible regularity properties of these sequences, especially the so-called top-heavy property. In the course of this study, we are led to a conjecture about the injectivity of the incidence operator of a geometric lattice, generalizing a conjecture of Kung.

Linear isomorphisms preserving Green's relations for matrices over anti-negative semifields

In this paper we characterize those linear bijective maps on the monoid of all n×n square matrices over an anti-negative semifield (that is, a semifield which is not a field) which preserve each of Green's equivalence relations L, R, H, D, J and the corresponding four pre-orderings ≤L, ≤R, ≤H, ≤J. These results apply in particular to the tropical and boolean semirings.

On locally nilpotent derivations of Boolean semirings

AbstractIn this paper, we consider the composition of derivations in Boolean semirings and investigate the conditions that the composition of two derivations is a derivation. We also show that the nth derivation of a derivation d, denoted by dn, on a Boolean semiring satisfies Leibniz rule. Finally, we show that any locally nipotent derivations on a zero-symmetric Boolean semiring must be zero.

Subdirectly irreducible commutative multiplicatively idempotent semirings

Commutative multiplicatively idempotent semirings were studied by the authors and F. Svrcek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzman. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2n such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.

Commutativity of $\Gamma$-Generalized Boolean Semirings with Derivations

In this paper the notion of derivations on $\Gamma$-generalized Boolean semiring are established, namely $\Gamma$-$(f, g)$ derivation and $\Gamma$-$(f, g)$ generalized derivation. We also investigate the commutativity of prime $\Gamma$-generalized Boolean semiring admitting $\Gamma$-$(f, g)$ derivation and $\Gamma$-$(f, g)$ generalized derivation satisfying some conditions.

How many Boolean polynomials are irreducible?

We present a number of new results on the multiplicative structure of univariate polynomials over the Boolean and tropical semirings. We answer the question asked by Kim and Roush in 2005 by proving that almost all Boolean polynomials with nonzero constant term are irreducible. We also give a lower bound for the number of reducible polynomials, and we discuss the related issues for polynomials over tropical semiring.

EXTREME PRESERVERS OF TERM RANK INEQUALITIES OVER NONBINARY BOOLEAN SEMIRING

The term rank of a matrix A over a semiring S is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we characterize linear operators that preserve the sets of matrix ordered pairs which satisfy extremal properties with respect to term rank inequalities of matrices over nonbinary Boolean semirings.

The dual geometry of Boolean semirings

It is well known that the variety of Boolean semirings, which is generated by the three element semiring S, is dual to the category of partially Stone spaces. We place this duality in the context of natural dualities. We begin by introducing a topological structure \uS and obtain an optimal natural duality between the quasi-variety ISP(S) and the category IS_cP+(\uS). Then we construct an optimal and very small structure \uS_os that yields a strong duality. The geometry of some of the partially Stone spaces that take part in these dualities is presented, and we call them "hairy cubes", as they are n-dimensional cubes with unique incomparable covers for each element of the cube. We also obtain a polynomial representation for the elements of the hairy cube.

A Characterization of Semirings Which Are Subdirect Products of a Distributive Lattice and a Ring

-inversive semiring and a Clifford semiring and show that a semiring S is a subdirect product of a distributive lattice and a ring if and only if S is an E-inversive strong distributive lattice of halfrings. Further a Clifford semiring which is, in fact, an inversive subdirect product of a distributive lattice and a ring, is characterized as a strong distributive lattice of rings. Finally, as a consequence of these results we extend a result of Galbiati and Veronesi [2] in the case of Boolean semirings.

The variety of Boolean semirings

The variety of Boolean semirings BIR is the variety generated by the two 2-element semirings. We find a complete set of laws for this variety, and show that it is equivalent to the category of partially Stone spaces. We get a detailed description of the finitely generated free Boolean semirings and a normal form for their elements. From this, a formula is obtained for the free spectrum. This formula relates the free spectra of BIR and DL , the variety of bounded distributive lattices. Finally, we show that the relational degree of BIR is 3.

On a class of near-rings

In [3], Ligh proved that every distributively generated Boolean near-ring is a ring, and he gave an example to which the above fact can not be extended. That is, let G be an additive group and let the multiplication on G be defined by xy = y for all x, y in G. Ligh called this Boolean near-ring G a general Boolean near-ring. near-ring. Then in [4], Ligh called Ra β-near-ring if for each x in R, x2 = x and xyz = yxz for all x, y, z in R, and he proved that the structure of a β-near-ring is “very close” to that of a usual Boolean ring. We note that general Boolean near-rings and Boolean semirings as defined in [5] are β-near-rings. The purpose of this paper is to generalize the structure theorem on β-near-rings given by Ligh in [4] to a broader class of near-rings.