The Finite Basis Problem for Multiplicatively Idempotent Semirings

The variety of semirings satisfying $x^2\approx x$ is hereditarily finitely based in the signature consisting of addition and multiplication, without designated constants. Multiplication may be noncommutative, and the semirings need not be finite. A subdirect decomposition reduces the proof to the subvarieties defined by $3x\approx x$ and $3x\approx2x$. In the latter case, we express each polynomial identity using a common doubled part and isolated word occurrences. Four fixed identities align the first- and last-occurrence words of these occurrences. Induction on their content extracts protected identities. The associated word identities define a band variety, to which the finite basis theorem for bands applies.