  This paper provides some evidence for conjectural relations between
extensions of (right) weak order on Coxeter groups, closure operators on root
systems, and Bruhat order. The conjecture focused upon here refines an earlier
question as to whether the set of initial sections of reflection orders,
ordered by inclusion, forms a complete lattice. Meet and join in weak order are
described in terms of a suitable closure operator. Galois connections are
defined from the power set of W to itself, under which maximal subgroups of
certain groupoids correspond to certain complete meet subsemilattices of weak
order. An analogue of weak order for standard parabolic subsets of any rank of
the root system is defined, reducing to the usual weak order in rank zero, and
having some analogous properties in rank one (and conjecturally in general).
