An abstract theory of invertible relations
The purpose of this paper is to present certain results arising from a study of quasi-orderings (pre-orderings). We show that to each relation \(R\subseteq X\times Y\) there are associated unique largest quasi- orderings \(\pi_{\ell}(R)\) on X and \(\pi_ r(R)\) on Y such that \(\pi_{\ell}(R)\circ R\circ \pi_ r(R)=R\); and we present formulas for these quasi-orderings. For a fixed pair of quasi-orders \(\pi_ 1\) and \(\pi_ 2\) we characterize the invertible relations (with respect to the units \(\pi_ 1\) and \(\pi_ 2)\) in terms of isomorphisms between \({}^*\pi_ 1\) and \({}^*\pi_ 2\), where \({}^*\pi_ i\) is the partial ordering naturally induced by \(\pi_ i\). In particular we show that the set of invertible relations with \(\pi_ 1=\pi_ 2=\pi\) is a group isomorphic to the group \(Aut(^*\pi)\) of automorphisms of \({}^*\pi\). We present these results in sections 1-3 in the framework of a general relation algebra. In section 4 we describe an anti-isomorphism between the lattice of quasi-orderings on a set X and a certain lattice of topologies on X. Using this anti-isomorphism, we obtain a characterization of the set of relations \(R\subseteq X\times Y\) with fixed left and right units \(\pi_ 1\) and \(\pi_ 2\).
scientific article; zbMATH DE number 4095451