  We introduce a calculus for the class $ASM(X)$ of direct images of
semi-meromorphic currents on a reduded analytic space $X$, that extends the
classical calculus due to Coleff, Herrera and Passare. Our main result is that
each element in this class acts as a kind of multiplication on the sheaf
$\PM_X$ of pseudomeromorphic currents on $X$. We also prove that $ASM(X)$ as
well as $\PM_X$ and certain subsheaves are closed under the action of
holomorphic differential operators and interior multiplication by holomorphic
vector fields.
