  We study diverse parametrized versions of the operad of associative algebra,
where the parameter are taken in an associative semigroup $\Omega$
(generalization of matching or family associative algebras) or in its cartesian
square (two-parameters associative algebras). We give a description of the free
algebras on these operads, study their formal series and prove that they are
Koszul when the set of parameters is finite. We also study operadic morphisms
between the operad of classical associative algebras and these objects.
