  We introduce and develop a language of semigroups over the braid groups for a
study of braid monodromy factorizations (bmf's) of plane algebraic curves and
other related objects. As an application we give a new proof of Orevkov's
theorem on realization of a bmf over a disc by algebraic curves and show that
the complexity of such a realization can not be bounded in terms of the types
of the factors of the bmf. Besides, we prove that the type of a bmf is
distinguishing Hurwitz curves with singularities of inseparable types up to
$H$-isotopy and $J$-holomorphic cuspidal curves in $\C P^2$ up to symplectic
isotopy.
