In this paper we further study the theory of standard bases in filtered structures as introduced by Robbiano. In particular, we study the situation in which a commutative ring A is filtered by commutative ordered monoids △ and τ in such a way that the τ filtration refines the △ filtration. We use these results to determine conditions under which a A standard basis can be lifted to a τ standard basis. Finally, we interpret these results in three situations where the refinements are given by changing the status of the variables in a polynomial ring, refining a partial order on the monomials in a polynomial ring, and refining a filtration by anI-adic filtration for an idealIin the ring A.