On Artin groups admitting retractions to parabolic subgroups

We generalize the retractions to standard parabolic subgroups for even Artin groups to FC-type Artin groups and other more general families. We prove that these retractions uniquely extend to any parabolic subgroup. We use retractions to generalize the results of Antolín and Foniqi that reduce the problem of intersection of parabolic subgroups to weaker conditions. As a corollary, we characterize coherence for the FC case.