A survey of congruences and quotients of partially ordered sets

Abstract

A quotient of a poset $P$ is a partial order obtained on the equivalence classes of an equivalence relation $\theta$ on $P$; $\theta$ is then called a congruence if it satisfies certain conditions, which vary according to different theories. The literature on congruences and quotients of partially ordered sets contains a large and proliferating array of approaches, but little in the way of systematic exposition and examination of the subject. We seek to rectify this by surveying the different theories in the literature and providing philosophical discussion on requirements for notions of congruences of posets. We advocate a pluralist approach which recognises that different types of congruence arise naturally in different mathematical situations. There are some notions of congruence which are very general, whilst others capture specific structure which often appears in examples. Indeed, we finish by giving several examples where quotients of posets appear naturally in mathematics.