Compositions and decompositions of binary relations

TL;DR

This paper explores the algebraic structure of binary relations through incidence matrices, establishing a multiplication that mirrors relation composition, and provides algorithms for solving related matrix equations and constructing product relations.

Contribution

It introduces a matrix multiplication framework for binary relations and develops algorithms for solving relation equations and constructing relation products.

Findings

Matrix multiplication corresponds to relation composition.

Algorithms for solving R o X = S using incidence matrices.

Method to obtain incidence matrix of a product of relations.

Abstract

It is well-known that to every binary relation on a non-void set I there can be assigned its incidence matrix, also in the case when I is infinite. We show that a certain kind of "multiplication" of such incidence matrices corresponds to the composition of the corresponding relations. Using this fact we investigate the solvability of the equation R o X = S for given binary relations R and S on I and derive an algorithm for solving this equation by using the connections between the corresponding incidence matrices. Moreover, we describe how one can obtain the incidence matrix of a product of binary relations from the incidence matrices of its factors.