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The Rhodes semilattice of a biased graph

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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.

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Notes

  1. Called “balance-closed” in [6, 7].

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Acknowledgements

We are eternally grateful to Stuart Margolis for his generous assistance and advice in understanding the context and literature around the Rhodes semilattice.

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  1. Department of Mathematics and Statistics, Binghamton University (SUNY), Binghamton, NY, 13902-6000, USA

    Michael J. Gottstein & Thomas Zaslavsky

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  1. Michael J. Gottstein
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  2. Thomas Zaslavsky
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Correspondence to Michael J. Gottstein or Thomas Zaslavsky.

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Gottstein, M.J., Zaslavsky, T. The Rhodes semilattice of a biased graph. Aequat. Math. (2024). https://doi.org/10.1007/s00010-024-01039-3

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  • DOI: https://doi.org/10.1007/s00010-024-01039-3

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