The finite basis problem for ai-semirings obtained from a commutative flat semiring by adjoining a zero

The aim of this paper is to study some varieties of commutative ai-semirings. We provide a sufficient condition under which a commutative nil-semiring is nonfinitely based. As applications, we show that the ai-semiring Sc(W)0 is nonfinitely based, where W is a finite set of nonempty words with length less than or equal to k on the free commutative semigroup Xc+ over an alphabet X, if k >= 2 and W does not contain the k power of a letter, or k is an odd number greater than 2 and W contains k powers of letters. On the basis, it is proved that Sc(W)0 is finitely based if and only if W is a finite set of letters or words of length less than or equal to 2 on Xc+ which contains a square of a letter. This partially answers a problem raised by Jackson et al.