On isomorphisms of R- and L-cross-sections of wreath products of finite inverse symmetric semigroups

We classify R- and L-cross-sections of wreath products of finite inverse symmetric semigroups ISm p ISn up to isomorphism. We show that every isomorphism of R (L-) cross-sections of ISm p ISn is a conjugacy. As an auxiliary result, we get that every isomorphism of R- (L-) cross-sections of ISn is also a conjugacy. We also compute the number of non-isomorphic R (L-) cross-sections of ISm p ISn.