On Addition of Sets in Boolean Space

Vladimir Leontiev,Zhirayr Margaryan,Garib Movsisyan

doi:10.4236/jis.2016.74019

Abstract

In many problems of combinatory analysis, operations of addition of sets are used (sum, direct sum, direct product etc.). In the present paper, as well as in the preceding one [1], some properties of addition operation of sets (namely, Minkowski addition) in Boolean space Bn are presented. Also, sums and multisums of various “classical figures” as: sphere, layer, interval etc. are considered. The obtained results make possible to describe multisums by such characteristics of summands as: the sphere radius, weight of layer, dimension of interval etc. using the methods presented in [2], as well as possible solutions of the equation X+Y=A, where , are considered. In spite of simplicity of the statement of the problem, complexity of its solutions is obvious at once, when the connection of solutions with constructions of equidistant codes or existence the Hadamard matrices is apparent. The present paper submits certain results (statements) which are to be the ground for next investigations dealing with Minkowski summation operations of sets in Boolean space.

Highlights

The sum of subsets X + Y is consisted of sums of points belonging to X and Y, respectively
A facet, or sub-cube, or interval in Bn is the set of points satisfying the following condition [5] [6]: J = {u ≤ x ≤ v}, where (≤) is a coordinate-wise partial order relation in Bn : x ≤ y xi ≤ yi, i = 1, n, where x = ( x1x2 xn ), y = ( y1 y2 yn )
=J {α1α2 αn ≤ x ≤ β1β2 βn}, the code λ ( J ) of the interval J is built in the following way

Summary

Sum of Spheres in Bn

The k-dimensional interval we denote by J k. J k + Stn (0) = ∪ Stn ( x), x∈J k i.e. J k + Stn (0) is the union of all spheres of the radii t with centres at the points in the interval Jk , or:. ∪ =z (α1α2 αk , β1β2 ) βn−k ∈ Stn ( x), x∈J k if β1 βn−k ≤ t. In this casezzbelongs to the sphere of the radius t with the centre at (α1α2 αk , 0 0). If z = (α1α2 αk , β1 βn−k ) and β1 βn−k ≥ t +1, ρ ( z, y) ≥ t +1 for any point y in the interval J k , that is:

The Sum of Facets in Bn

Sum of Layers in Bn

Sum of Subsets in Bn

Sum of a Sphere and a Layer in Bn

Multisets