Upper and Lower Semimodularity of the Supercharacter Theory Lattices of Cyclic Groups

TL;DR

This paper characterizes when the lattice of supercharacter theories of cyclic groups is upper or lower semimodular, providing necessary and sufficient conditions based on the group's order.

Contribution

It offers a complete characterization of semimodularity conditions for supercharacter theory lattices of cyclic groups, advancing understanding of their algebraic structure.

Findings

Identifies necessary and sufficient conditions for upper semimodularity.

Identifies necessary and sufficient conditions for lower semimodularity.

Provides a detailed analysis of the lattice structure for cyclic groups.

Abstract

We consider the lattice of supercharacter theories, in the sense of Diaconis and Isaacs, of the cyclic group of order n. We find necessary and sufficient conditions on n for that lattice to be upper or lower semimodular.