Lattices of orders

Abstract

Ordering order relations is a well-established topic of order theory. Traditionally, the concept of order extension plays an important role and leads to fundamental results like Dushnik and Miller’s Theorem stating that every order is the intersection of linear extensions [1]. We introduce an alternative but still quite elementary way to order relations. The resulting lattices of orders can be viewed as a generalisation of the lattices of permutations from [2] and accordingly maintain a very high degree of symmetry. Furthermore, the resulting lattices of orders form complete sublattices of it’s quasiorder counterpart, the lattices of quasiorders, which are also introduced. We examine the basic properties of these two classes of lattices and present their contextual representations.