Some nil-ai-semiring varieties

We study some nil-ai-semiring varieties. We establish a model for the free object in the variety FC generated by all commutative flat semirings. Also, we provide two sufficient conditions under which a finite ai-semiring is nonfinitely based. As a consequence, we show that the power semiring P S ˙ c ( W ) of the finite nil-semigroup S ˙ c ( W ) is nonfinitely based, where W is a finite set of words in the free commutative semigroup X c + over an alphabet X , whenever the maximum of lengths of words in W is k ≥ 3 and W does not contain the k th power of a letter. This partially answers a problem raised by Jackson et al. (J Algebr 611: 211–245, 2022).