Sorting in Lattices

Abstract

In a totally ordered set the notion of sorting a finite sequence is defined through a suitable permutation of the sequence's indices. In this paper we prove a simple formula that explicitly describes how the elements of a sequence are related to those of its sorted counterpart. As this formula relies only on the minimum and maximum functions we use it to define the notion of sorting for lattices. A major difference of sorting in lattices is that it does not guarantee that sequence elements are only rearranged. However, we can show that other fundamental properties that are associated with sorting are preserved.