The finite basis problem for flat semirings $S_c(W)$, $M_c(W)$ and $M(W)$

We study the finite basis problem for flat semirings of the forms $S_c(W)$, $M_c(W)$, and $M(W)$, where $W$ is a nonempty set of words in a free commutative semigroup, a free commutative monoid, and a free monoid, respectively. We completely classify such flat semirings with respect to the finite basis property, allowing $W$ to be infinite. We prove that $S_c(W)$ is finitely based if and only if every word in $W$ is either a cube of a letter or has length at most two, whereas $M_c(W)$ and $M(W)$ are finitely based if and only if $W$ consists solely of the empty word. As applications, we recover the nonfinite basability of $\mathrm{♭}(\mathbb{Z})$ and the max-plus semiring $(\mathbb{Z},\max ,+)$.