The Rhodes semilattice of a biased graph

TL;DR

This paper reinterprets the Rhodes semilattice of a group using gain graphs, generalizes it to biased graphs, and explores related lattice structures, broadening the algebraic framework for gain graph analysis.

Contribution

It introduces a new interpretation of Rhodes semilattices via gain graphs and extends the concept to biased graphs, connecting algebraic and combinatorial structures.

Findings

Reinterpretation of Rhodes semilattices in terms of gain graphs

Generalization to all gain graphs and biased graphs

Identification of four natural lattices containing these structures

Abstract

We reinterpret the Rhodes semilattices $R_n(\mathfrak{G})$ of a group $\mathfrak{G}$ in terms of gain graphs and generalize them to all gain graphs, both as sets of partition-potential pairs and as sets of subgraphs, and for the latter, further to biased graphs. Based on this we propose four different natural lattices in which the Rhodes semilattices and its generalizations are order ideals.