We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific $4$-element additively idempotent semirings, $S_{ (4,545)}$ and $S_{ (4,634)}$, whose additive reducts are chains, have no finite basis for their identities. Furthermore, we show that the interval $[\mathsf{ V}(S_{ (4,545)}),\mathsf{ V}(S_{ (4,634)})]$ in the lattice of semiring varieties contains \(2^{ \aleph_0}\) distinct varieties. Consequently, the join of two finitely based additively idempotent semiring varieties is not necessarily finitely based. Moreover, we obtain the smallest example of a finitely based additively idempotent semiring $S$ whose extension $S^0$ (obtained by adjoining a new element) is nonfinitely based.
Categories: math.RA math.GR