On free semilattice-ordered semigroups satisfying x n =x

Let B(k,0,n) denote the group with k generators which is free in the group variety defined by the identity x (n) =1. Let B (slo) (k,1,n) denote the semilattice-ordered semigroup with k generators which is free in the semilattice-ordered semigroup variety defined by the identity x (n) =x. We prove a generalization of the Green-Rees theorem: B (slo) (k,1,n) is finite for all ka parts per thousand yen1 if and only if B(k,0,n-1) is finite for all ka parts per thousand yen1. We find a formula for card(B (slo) (1,1,n)). We construct B (slo) (k,1,n) for some concrete values of k and n.