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 textth power of a letter. This affirmatively answers one problem raised by Jackson et al. 5.
Wu et al. (Fri,) studied this question.