Binary Relations in the Social and Mathematical Sciences

Abstract

This is an exposition of some applications of the calculus of binary relations to anthropology, linguistics, computer science and mathematics. Binary relations appear in the rewrite systems used by anthropologists to study kinship terminologies of primitive societies. They also serve to model the syntactic calculus, a form of categorial grammar once proposed by the author. Moreover, if we introduce an operation perp as the complement of the converse, we may take the calculus of binary relations as a model for cyclic linear logic, studied by Yetter in a first attempt to remove commutativity from linear logic, recently of interest in theoretical computer science. Partial recursive functions may best be viewed as recursively enumerable binary relations which happen to be single-valued; they are of the form f composed with g converse, where f and g are primitive recursive functions satisfying the condition: g converse composed with g is contained in f converse composed with f. In mathematics, binary relations were pushed into the background when it was decided that functions be single-valued and universally defined. However, they made a comeback in homological algebra, where they provide the easiest construction for the so-called connecting homomorphism. While such constructions were originally confined to abelian categories, it is now clear that they can be extended to arbitrary categories, provided one chooses appropriate generalizations of ‘exactness’ and other relevant concepts.