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The Rhodes semilattice of a biased graph

Michael J. Gottstein, Thomas Zaslavsky

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.

The Rhodes semilattice of a biased graph

Abstract

We reinterpret the Rhodes semilattices of a group 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.
Paper Structure (7 sections, 7 theorems, 10 equations)

This paper contains 7 sections, 7 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. A new perspective on the Rhodes semilattice
  3. Generalization to gain graphs
  4. Generalization to biased graphs
  5. Interlude on groupoids
  6. Four lattices
  7. Acknowledgement

Key Result

Lemma 2.1

In a balanced gain graph, the gain of a path depends only on its initial and final vertices.

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 15 more