Poincare series of multi-filtered algebras and partitivity

It is proved that if an algebra R over a field can be endowed with a pointe d and finite-dimensional N-n-filtration such that the associated N-n-graded algebra T is semi-commutative, then R is left and right finitely partitive . In order to do this, a multi-variable Poincare series for every finitely generated graded T-module is considered and it is shown that this Poincare series is a rational function. The methods apply to some iterated Ore exten sions such as quantum matrices and quantum Weyl algebras as well as to the quantized enveloping algebra of sl(nu + 1).