  This paper is a contribution to the theory of finite semigroups and their
classification in pseudovarieties, which is motivated by its connections with
computer science. The question addressed is what role can play the
consideration of an order compatible with the semigroup operation. In the case
of unions of groups, so-called completely regular semigroups, the problem of
which new pseudovarieties appear in the ordered context is solved. As
applications, it is shown that the lattice of pseudovarieties of ordered
completely regular semigroups is modular and that taking the intersection with
the pseudovariety of bands defines a complete endomorphism of the lattice of
all pseudovarieties of ordered semigroups.
