  Divisibility monoids (resp. Garside monoids) are a natural algebraic
generalization of Mazurkiewicz trace monoids (resp. spherical Artin monoids),
namely monoids in which the distributivity of the underlying lattices (resp.
the existence of common multiples) is kept as an hypothesis, but the relations
between the generators are not supposed to necessarily be commutations (resp.
be of Coxeter type). Here, we show that the quasi-center of these monoids can
be studied and described similarly, and then we exhibit the intersection
between the two classes of monoids.
