Endomorphism monoids of free acts and O-wreath products of monoids

In this part of the paper we consider the standard involution on the O-wreath products of monoids and Hermitian bilinear forms on free acts. It is a study of the representation of the continuous endomorphism monoid of a free act over a monoid with zero and involution as O-wreath with standard involution. We give necessary and sufficient conditions for the O-wreath product with standard involution to be a semigroup with a regular involution, a regular *-semigroup, a Rickart *-semigroup, and a Bear *-semigroup. We obtain as a consequence corresponding conditions for the continuous endomorphism monoid of a free act.