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September 24, 2025Open Access

Enumerating submonoids of finite commutative monoids

Key Points

The number of submonoids of M×[n] can be expressed as a sum involving integers and rational numbers, offering insight into their structure.
Using the transfer matrix method, these submonoids are shown to be enumerable under the max operation across finite commutative monoids.
Finitely many integers λ exist, allowing the expression of submonoid counts as a polynomial, directly addressing issues raised by Knuth.
This research has implications for understanding saturated transfer systems in equivariant infinite loop space theory.

Abstract

Given a finite commutative monoid M, we show that submonoids of M n - where n = \0, 1, , n\ is equipped with the max operation - 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 n is _λb_λλⁿ. 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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