Equational theories of flat semirings $S(W)$

We study the finite basis problem for flat semirings of the form $S(W)$, where $W$ is a nonempty set of nonempty words, possibly infinite. We prove that $S(W)$ is finitely based if and only if the strict interior of every word in $W$ is linear; here the strict interior of a word is obtained by deleting its first and last letters. This completely resolves an open problem raised by Jackson et al.~(J. Algebra 611: 211–245, 2022).