A nonfinitely based additively idempotent semiring of order four

We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a $4$-element semiring $S_{(4,124)}$ whose additive reduct has two minimal elements and two coatoms. Consequently, these semirings have no finite basis for their identities.