# Asymptotic Semigroups and Two-sided Weak Orders

## TL;DR

This paper explores the structure of dual canonical monoids, introduces a two-sided weak order, and provides new insights into their properties and related algebraic objects, with specific results in type A.

## Contribution

It introduces the notion of a two-sided weak order on normal reductive monoids and analyzes its properties, including covering relations and their degrees.

## Key findings

- Nilpotent variety of dual canonical monoid is equidimensional with computed dimension.
- Intervals of Putcha poset in type A are isomorphic to Renner monoids.
- Covering relations in the two-sided weak order have degree 1 for the asymptotic semigroup.

## Abstract

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.