2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107373We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Gamma is any graph not containing a four-cycle without chords, then the group G(Gamma) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Gamma) does not have a subgroup isomorphic to a direct product of non-abelian free groups. We also obtain corresponding and more general results for monoids.14 pages, 1 figure, typos fixed, introduction improved slightly, results unchangedGroup Theory20F36, 20M05text