  Dehornoy showed that the Artin braid groups $B_n$ are left-orderable. This
ordering is discrete, but we show that, for $n >2$ the Dehornoy ordering, when
restricted to certain natural subgroups, becomes a dense ordering. Among
subgroups which arise are the commutator subgroup and the kernel of the Burau
representation (for those $n$ for which the kernel is nontrivial). These
results follow from a characterization of least positive elements of any normal
subgroup of $B_n$ which is discretely ordered by the Dehornoy ordering.
