On saturated varieties of posemigroups

We show that a permutative variety of posemigroups satisfying a permutation identity x 1 x 2 ⋯ x n = x i 1 x i 2 ⋯ x i n with i 1 ≠ 1 and i n - 1 ≠ n - 1 [ i n ≠ n and i 2 ≠ 2 ] is saturated if and only if it admits an identity I such that I is not a permutation identity and at least one side of I has no repeated variables. Then we show that the variety of po-rectangular bands is saturated. Finally, we show that a posemigroup S is saturated if the subposemigroup S n , the product of n copies of S , is saturated for some positive integer n .