The varieties generated by a nil-ai-semiring

From an ai-semiring [Formula: see text], one can always obtain new ai-semirings [Formula: see text] (respectively, [Formula: see text]) by adjoining an extra top element and multiplicative zero [Formula: see text] (respectively, a bottom element and multiplicative zero [Formula: see text]). The aim of this paper is to study the varieties generated respectively by these ai-semirings [Formula: see text] and [Formula: see text]. It is proved that [Formula: see text] is finitely based if and only if [Formula: see text] consists of cubes of letters and words of length at most [Formula: see text]. Also, we provide a sufficient condition under which a finite nil-ai-semiring is nonfinitely based. As applications, we show that the ai-semiring [Formula: see text] is nonfinitely based for any finite set [Formula: see text] containing at least one nonempty word, and that the ai-semiring [Formula: see text] is nonfinitely based, where [Formula: see text] is a finite set of words in the free commutative semigroup [Formula: see text] over alphabet [Formula: see text], whenever the maximum of lengths of words in [Formula: see text] is [Formula: see text] and [Formula: see text] does not contain the [Formula: see text]th power of a letter. This affirmatively answers one problem raised by [M. Jackson, M. M. Ren and X. Z. Zhao, Nonfinitely based ai-semirings with finitely based semigroup reducts, J. Algebra 611 (2022) 211–245].