Two Nonfinitely Based Additively Idempotent Semirings Of Order Four

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, [Formula: see text] and [Formula: see text], whose addi- tive reducts are chains, have no finite basis for their identities. Furthermore, we show that the interval [Formula: see text] in the lattice of semiring varieties has the cardinality of the continuum. Consequently, the join of t- wo 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.