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.