Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If one specializes the Hecke algebra associated to W to q=0 one gets the monoid algebra of H(W) (replace generators by their negatives to see this). It is also the monoid generated by foldings across the walls of the fundamental chamber of the Coxeter complex of W. It has been studied by a number of people and is sometimes called the Springer-Richardson monoid.
Margolis and I came across the following construction of it and I wanted to know if it is known. Let P(W) be the power set of W. It is a monoid with usual set product: $AB=\lbrace ab\mid a\in A, b\in B\rbrace$. Let $I(w)$ be the principal Bruhat ideal generated by $w\in W$, e.g., $I(s)=\lbrace 1,s\rbrace$ for $s\in S$. Then the principal Bruhat ideals form a submonoid of P(W) isomorphic to H(W).
Question: Does this construction appear explicitly in the literature and what is a reference?
