Valuation Extensions of Filtered and Graded Algebras
  In this note we relate the valuations of the algebras appearing in the non-commutative geometry of quantized algebras to properties of sub-lattices in some vector spaces. We consider the case of algebras with $PBW$-bases and prove that under some mild assumptions the valuations of the ground field extend to a non-commutative valuation. Later we introduce the notion of $F$-reductor and graded reductor and reduce the problem of finding an extending non-commutative valuation to finding a reductor in an associated graded ring having a domain for its reduction. 
C. Baetica and F. Van Oystaeyen