  We deal with involution ordered semigroups possessing a greatest element, we
introduce the concepts of $*$-regularity, $*$-intra-regularity, $*$-bi-ideal
element and $*$-quasi-ideal element in this type of semigroups and, using the
right and left ideal elements, we give relations between the regularity and
$*$-regularity, between intra-regularity and $*$-intra-regularity. Finally, we
prove that in an involution $*$-regular $\vee e$-semigroup every $*$-bi-ideal
element can be considered as a product of a right and a left ideal element, we
describe the form of the filter generated by an element of an involution
$*$-intra-regular $poe$-semigroup $S$, showing that every $\cal N$-class of $S$
has a greatest element.
