  Various partial orders related to the structures of dual canonical monoids
are investigated. It is shown that the nilpotent variety of a dual canonical
monoid is equidimensional; its dimension is found. It is shown in type A that
certain intervals of the Putcha poset of a dual canonical monoid are isomorphic
to the Renner monoids of matrices. The notion of a two-sided weak order on a
normal reductive monoid is introduced. A criterion, in terms of type maps, for
the covering relations in a two-sided weak order to have degree 2 is found. It
is shown that, for the unique equivariant divisor of a dual canonical monoid
(the asymptotic semigroup), the covering relations of the two-sided weak order
are always of degree 1. These computations provide new insights for the
two-sided weak orders on Coxeter groups. In type A, some enumerative results
for the covering relations are presented.
