The lattice of ai-semiring varieties satisfying xn ≈ x and xy ≈ yx

We study the lattice L((n, 1)) of subvarieties of the ai-semiring variety (n, 1) defined by xn x and xy yx. We divide L((n, 1)) into five intervals and provide an explicit description of each member of these intervals except [(2, 1), (n, 1)]. Based on these results, we show that if n - 1 is square-free, then L( (n, 1)) is a distributive lattice of order 2 + 2r+1 + 3r, where r denotes the number of prime divisors of n - 1. Also, all members of L((n, 1)) are finitely based and finitely generated and so (n, 1) is a Cross variety. Moreover, the axiomatic rank of each member of L((n, 1)) is less than four.