Asymptotic semigroups and two-sided weak orders

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