The Gap Between Partial and Full: an Addendum

The paper 3] was devoted to an extensive study of the interval of the lattice of semigroup pseudovarieties between the pseudovarieties O and PO generated by all semigroups of full and respectively, partial, order preserving transformations of a nite chain. It was shown that not only the interval O; PO] itself but also many of its naturally arising subintervals are huge in the sense that they all contain a chain isomorphic to the chain of reals with the usual order as well as an antichain of the cardinality of the continuum. However, at least one important question concerning the relative location of the pseudovarieties O and PO was left open in 3]. To formulate the question, let us rst recall two observations rst made in 4]: the pseudovariety O is self-dual, that is, for each semigroup S 2 O, the dual semigroup S belongs to O as well 4, Theorem 2.4]; the pseudovariety PO is not self-dual, more precisely, there exists a nite R-trivial semigroup S 2 PO such that the dual semigroup S is not in PO 4, Result 4.1]. Denoting by PO the dual of the pseudovariety PO, we see that the self-duality of O and the inclusion O PO imply the \dual" inclusion O PO. Therefore we also have the inclusion