Admissible submonoids of Artin–Tits monoids

We show the analogue of Mühlherr’s [B. Mühlherr, Coxeter groups in Coxeter groups, in: Finite Geometry and Combinatorics, Cambridge University Press, 1993, pp. 277–287] for Artin–Tits monoids and for Artin–Tits groups of spherical type. That is, the submonoid (resp. subgroup) of an Artin–Tits monoid (resp. group of spherical type) induced by an admissible partition of the Coxeter graph is an Artin–Tits monoid (resp. group). This generalizes and unifies the situation of the submonoid (resp. subgroup) of fixed elements of an Artin–Tits monoid (resp. group of spherical type) under the action of graph automorphisms, and the notion of LCM-homomorphisms defined by Crisp in [J. Crisp, Injective maps between Artin groups, in: Geom. Group Theory Down Under (Canberra 1996), de Gruyter, Berlin, 1999, pp. 119–137] and generalized by Godelle in [E. Godelle, Morphismes injectifs entre groupes d’Artin-Tits, Algebr. Geom. Topol. 2 (2002) 519–536]. We then complete the classification of the admissible partitions for which the Coxeter graphs involved have no infinite label, started by Mühlherr in [B. Mühlherr, Some contributions to the theory of buildings based on the gate property, Dissertation, Tübingen, 1994]. This leads us to the classification of Crisp’s LCM-homomorphisms.