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arXiv · 2508.20786

Enumerating submonoids of finite commutative monoids

Caoilainn Kirkpatrick·Amelie el Mahmoud·Kyle Ormsby·Angélica M. Osorno·Dale Schandelmeier-Lynch·Riley Shahar·Lixing Yi·Avery Young·Saron Zhu
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Abstract

Given a finite commutative monoid $M$, we show that submonoids of $M\times [n]$ - where $[n] = \{0,1,\ldots,n\}$ is equipped with the max operation $\vee$ - may be enumerated via the transfer matrix method. When $M$ is also idempotent, we show that there are finitely many integers $\lambda$ and rational numbers $b_\lambda$ (only depending on $M$) such that the number of submonoids of $M\times [n]$ is $\sum_\lambda b_\lambda\lambda^n$. This answers a question of Knuth regarding ternary (and higher order) max-closed relations, and has applications to the enumeration of saturated transfer systems in equivariant infinite loop space theory.

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Caoilainn Kirkpatrick, Amelie el Mahmoud, Kyle Ormsby, Angélica M. Osorno, Dale Schandelmeier-Lynch, Riley Shahar, Lixing Yi, Avery Young, Saron Zhu. 2025-08-28. Enumerating submonoids of finite commutative monoids. https://arxiv.org/abs/2508.20786

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