Systems of Inclusions with Unknowns in Multioperations

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.

A heuristic method for bi-decomposition of partial Boolean functions

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.

Boolean Curve Fitting with the Aid of Variable-Entered Karnaugh Maps

The Variable-Entered Karnaugh Map is utilized to grant a simpler view and a visual perspective to Boolean curve fitting (Boolean interpolation); a topic whose inherent complexity hinders its potential applications. We derive the function(s) through m points in the Boolean space B^(n+1) together with consistency and uniqueness conditions, where B is a general ‘big’ Boolean algebra of l≥1 generators, L atoms (2^(l-1)<L≤2^l) and 2^L elements. We highlight prominent cases in which the consistency condition reduces to the identity (0=0) with a unique solution or with multiple solutions. We conjecture that consistent (albeit not necessarily unique) curve fitting is possible if, and only if, m=2^n. This conjecture is a generalization of the fact that a Boolean function of n variables is fully and uniquely determined by its values in the {0,1}^n subdomain of its B^n domain. A few illustrative examples are used to clarify the pertinent concepts and techniques.

Spectra of finitely presented lattice-ordered Abelian groups and MV-algebras, part 1

This is the first part of a series of two abstract, the second one being by Daniel McNeill.If X is any topological space, its collection of opens sets O(X) is a complete distributive lattice and also a Heyting algebra. When X is equipped with a distinguished basis D(X) for its topology, closed under finite meets and joins, one can investigate situations where D(X) is also a Heyting subalgebra of O(X).Recall that X is a spectral space if it is compact and T0, its collection D(X) of compact open subsets forms a basis which is closed under finite intersections and unions, and X is sober. By Stone duality, spectral spaces are precisely the spaces arising as sets of prime ideals of some distributive lattice, topologised with the Stone or hull-kernel topology. Specifically, given such a spectral space X, its collection of compact open sets D(X) is (naturally isomorphic to) the distributive lattice dual to X under Stone duality.We are going to exhibit a significant class of such spaces for which D(X) is a Heyting subalgebra of O(X).We work with lattice-ordered Abelian groups and vector spaces. Using Mundici’s Gamma-functor the results can be rephrased in terms of MV-algebras, the algebraic semantics of Lukasiewicz infinite-valued propositional logic.Let (G,u) be a finitely presented vector lattice (or Q-vector lattice, or l-group) G equipped with a distinguished strong order unit u. It turns out that Spec(G,u), i.e. the the space of prime ideals of (G,u) topologised with the hull-kernel topology, is a compact spectral space. Our first main result states that the collection D(Spec(G,u)) of compact open subsets of Spec(G,u) is a Heyting subalgebra of the Heyting algebra of open subsets O(Spec(G,u)).As a consequence, we also prove that the subspace Min(G,u) of minimal prime ideals of G is a Boolean space, i.e. a compact Hausdorff space whose clopen sets form a basis for the topology.Further, for any fixed maximal ideal m of G, the set l(m) of prime ideals of G contained in m, equipped with the subspace topology, is a spectral space, and the subspace Min(l(m)) of l(m) is a Boolean space.

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

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.