# Enumerating submonoids of finite commutative monoids

## TL;DR

This paper develops a method to enumerate submonoids of finite commutative monoids extended by a max-structured set, revealing a finite sum formula for their counts when the monoid is idempotent, with applications in algebra and topology.

## Contribution

It introduces a transfer matrix approach for enumeration and establishes a finite sum formula for submonoid counts in idempotent cases, answering Knuth's question.

## Key findings

- Enumeration via transfer matrix method.
- Finite sum formula for submonoid counts in idempotent monoids.
- Applications to saturated transfer systems in topology.

## 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.