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Commutative Monoid Duality

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Abstract

We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For interacting particle systems whose local state space has two elements, this approach yields a unified treatment of the well-known additive and cancellative dualities. For local state spaces with three or more elements, we discover several new dualities.

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Notes

  1. Note that the functions \(\psi (\,\cdot \,,y)\) with y ranging through R are trivially all different from each other, since \(R\ni y\mapsto f_y\in {{\mathcal {H}}}(S,T)\) is an isomorphism.

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Acknowledgements

Work was supported by Grant 20-08468S of the Czech Science Foundation (GAČR)

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  1. The Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou věží 4, 18200, Praha 8, Czech Republic

    Jan Niklas Latz & Jan M. Swart

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  1. Jan Niklas Latz
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Correspondence to Jan M. Swart.

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A Appendix

A Appendix

1.1 A.1 Addition Tables of Commutative Monoids of Order Four

figure d

1.2 A.2 Duality Functions for Commutative Monoids of Order Four

figure e

1.3 A.3 Multiplications in Semirings of Cardinality Four

figure f

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Latz, J.N., Swart, J.M. Commutative Monoid Duality. J Theor Probab 36, 1088–1115 (2023). https://doi.org/10.1007/s10959-022-01197-7

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