Surprising occurrences of order structures in mathematics

TL;DR

This paper uncovers hidden order structures in various mathematical contexts such as topologies, algebras, groups, rings, and graphs, revealing deeper organizational principles.

Contribution

It provides five novel examples where order structures are not initially obvious but are revealed through detailed analysis.

Findings

Finite topologies are governed by underlying order structures.

Associative algebras exhibit hidden order principles.

Subgroups of matrix groups and ideals in polynomial rings have underlying order relations.

Abstract

Order and symmetry are main structural principles in mathematics. We give five examples where on the face of it order is not apparent, but deeper investigations reveal that they are governed by order structures. These examples are finite topologies, associative algebras, subgroups of matrix groups, ideals in polynomial rings, and classes of bipartite graphs.