Enumerating submonoids of finite commutative monoids

Given a finite commutative monoid $M$, we show that submonoids of $M\times [n]$ - where $[n]=\{0,1,\dots ,n\}$ 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.