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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.20786 |
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| _version_ | 1866912557855408128 |
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| author | Kirkpatrick, Caoilainn Mahmoud, Amelie el Ormsby, Kyle Osorno, Angélica M. Schandelmeier-Lynch, Dale Shahar, Riley Yi, Lixing Young, Avery Zhu, Saron |
| author_facet | Kirkpatrick, Caoilainn Mahmoud, Amelie el Ormsby, Kyle Osorno, Angélica M. Schandelmeier-Lynch, Dale Shahar, Riley Yi, Lixing Young, Avery Zhu, Saron |
| contents | 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 $λ$ and rational numbers $b_λ$ (only depending on $M$) such that the number of submonoids of $M\times [n]$ is $\sum_λb_λλ^n$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20786 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumerating submonoids of finite commutative monoids Kirkpatrick, Caoilainn Mahmoud, Amelie el Ormsby, Kyle Osorno, Angélica M. Schandelmeier-Lynch, Dale Shahar, Riley Yi, Lixing Young, Avery Zhu, Saron Combinatorics Algebraic Topology 05A15, 20M10, 55P91 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 $λ$ and rational numbers $b_λ$ (only depending on $M$) such that the number of submonoids of $M\times [n]$ is $\sum_λb_λλ^n$. 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. |
| title | Enumerating submonoids of finite commutative monoids |
| topic | Combinatorics Algebraic Topology 05A15, 20M10, 55P91 |
| url | https://arxiv.org/abs/2508.20786 |