  We define a class of partial orders on a Coxeter group associated with sets
of reflections. In special cases, these lie between the left weak order and the
Bruhat order. We prove that these posets are graded by the length function and
that the projections on the right parabolic quotients are always order
preserving. We also introduce the notion of $k$-Bruhat graph, $k$-absolute
length and $k$-absolute order, proposing some related conjectures and problems.
