Sorting in Lattices

TL;DR

This paper extends the concept of sorting from totally ordered sets to lattices, providing a formula based on min and max functions that preserves key sorting properties despite not merely rearranging sequence elements.

Contribution

It introduces a novel definition of sorting in lattices using a simple min-max formula, generalizing the classical notion beyond total orders.

Findings

Provides an explicit formula for sorting in lattices

Shows that key properties of sorting are preserved in lattices

Highlights differences from sorting in totally ordered sets

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.