Regular relations and bicartesian squares (Q1329746)

A relation \(R\) is regular iff \(R=R*\widehat R*R\), where \(*\) indicates the relational composition and \(\widehat{R}\) is the inverse of \(R\). Regular relations arise in many areas of computer science and are characterized in a number of ways. The purpose of the paper is to recast the properties of regular relations in categorical terms. The result is that regular relations correspond to unique bicartesian squares in Set, where a bicartesian square in a category is a commutative square that is both a pushout and a pullback.