We consider quotients of finitely generated Coxeter groups under
the weak order. Our result is that the quotient of an affine Weyl
group by the corresponding finite Weyl group is a lattice, and that up
to isomorphism, these are the only quotients of infinite Coxeter
groups that are lattices. In this paper, we restrict our
attention to the non-affine case; the affine case appears elsewhere.
We reduce to the hyperbolic case by an argument using
induced subgraphs of Coxeter graphs. Within each quotient, we
produce a set of elements with no common upper bound, generated by
a Maple program. We are often able to use the same set repeatedly
because the sets satisfy the following conjecture: if a set of elements does not
have an upper bound in a particular Coxeter group, then it does
not have an upper bound in any Coxeter group whose graph can be
obtained from the graph of the original group by increasing edge
weights.
