Enumerating submonoids of finite commutative monoids

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.