One of the open problems in the max-plus-algebraic system theory for discrete event systems is the minimal realization problem. We consider a simpliied version of the general minimal realization problem: the boolean minimal realization problem, i.e., we consider models in which the entries of the system matrices are either equal to the max-plus-algebraic zero element or to the max-plus-algebrai...

A boolean term order is a total order on subsets of [n] = {1,. .. , n} such that ∅ ≺ α for all α ⊆ [n], α = ∅, and α ≺ β ⇒ α ∪ γ ≺ β ∪ γ for all γ with γ ∩ (α ∪ β) = ∅. Boolean term orders arise in several different areas of mathematics, including Gröbner basis theory for the exterior algebra, and comparative probability. The main result of this paper is that boolean term orders correspond to o...

One of the open problems in the max-plus-algebraic system theory for discrete event systems is the minimal realization problem. We consider a simplified version of the general minimal realization problem: the boolean minimal realization problem, i.e., we consider models in which the entries of the system matrices are either equal to the max-plus-algebraic zero element or to the maxplus-algebrai...

A Boolean function is bipolar iff it is monotone or antimonotone in each of its arguments. We investigate the number b(n) of n-ary bipolar Boolean functions. We present an (almost) closed-form expression for b(n) that uses the number a(n) of antichain covers of an n-element set. This is closely related to Dedekind’s problem, which can be rephrased as determining the number d(n) of Boolean funct...

Let C be a clone of operations on a two element set. In [2] we have proved that the first order theory of finite Boolean algebras with distinguish subset closed under C is decidable if and only if C is a clone containing the ternary discriminator function. Decidability of the theories referred to in the abstract is a consequence of the fact, that finite discriminator subsets of products of alge...

In this paper the concept of an $Omega$- Almost Boolean ring is introduced and illistrated how a sheaf of algebras can be constructed from an $Omega$- Almost Boolean ring over a locally Boolean space.

one of the models that can be used to study the relationship between boolean random sets and explanatory variables is growth regression model which is defined by generalization of boolean model and permitting its grains distribution to be dependent on the values of explanatory variables. this model can be used in the study of behavior of boolean random sets when their coverage regions variation...

0. Notation. The basic notation for Boolean algebras is the standard one (for instance, see [l]). Throughout this paper, the letter O will denote the two-element Boolean algebra; whenever a topology is assumed to be defined on 0, then this topology is the discrete one. If 73 is a Boolean algebra, then a valuation of B is a homomorphism from 73 onto 0. We make precise the terminology we shall be...

Developments in solving equations over Boolean algebras and their applications are as old as George Boole’s monograph [1], while Shannon pioneered applications of Boolean logic to switching circuits. Boolean equations and their solutions are of central importance to many problems across Sciences such as Chemistry, Biology or Medicine while traditionally Boolean problems arise in domains such as...