On saturated varieties of posemigroups

Abstract

We show that a permutative variety of posemigroups satisfying a permutation identity $x_1x_2\cdots x_n=x_{i_1}x_{i_2}\cdots x_{i_n}$ with $i_1\ne 1~ \text{ and }~i_{n-1}\ne n-1 ~[i_n\ne n~\text{ and }~i_{2}\ne 2]$ 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.