  We prove some injectivity results: that a Coxeter monoid $\mathbb{Z}$-algebra
(or $0$-Hecke algebra) injects in the incidence $\mathbb{Z}$-algebra of the
corresponding Bruhat poset, for any Coxeter group; that the Hecke algebra of a
right-angled Coxeter group injects in the Coxeter monoid
$\mathbb{Z}[q,q^{-1}]$-algebra (and then in the incidence
$\mathbb{Z}[q,q^{-1}]$-algebra of the corresponding Bruhat poset); that a
right-angled Artin group injects in the group of invertible elements of the
Hecke algebra of the corresponding Coxeter group (and then in the group of
invertible elements of a Coxeter monoid algebra and in the one of an incidence
algebra).
