Artin group injection in the Hecke algebra for right-angled groups

We prove some injectivity results: that a Coxeter monoid Z-algebra (or 0-Hecke algebra) injects in the incidence 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 Z[q, q- 1] -algebra (and then in the incidence 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).