The nature of generalized scales

TL;DR

This paper develops a comprehensive structure theory for generalized scales in right LCM monoids, establishing their uniqueness, existence criteria, and explicit construction, with applications to algebraic dynamical systems and connections to Saito's degree map.

Contribution

It introduces a new framework for understanding generalized scales, including their uniqueness, existence conditions, and explicit construction methods, within the context of right LCM monoids.

Findings

Established the uniqueness of the generalized scale.

Characterized the existence of the generalized scale via a simplicial graph.

Provided an explicit construction method for the generalized scale.

Abstract

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.