  The notion of a generalized scale emerged in recent joint work with
Afsar-Brownlowe-Larsen on equilibrium states on C*-algebras of right LCM
monoids, where it features as the key datum for the dynamics under
investigation. This work provides the structure theory for such monoidal
homomorphisms. We establish uniqueness of the generalized scale and
characterize its existence in terms of a simplicial graph arising from a new
notion of irreducibility inside right LCM monoids. In addition, the method
yields an explicit construction of the generalized scale if existent. We
discuss applications for graph products as well as algebraic dynamical systems,
and reveal a striking connection to Saito's degree map.
