Relative Generalized Boolean Dynamical System Algebras

TL;DR

This paper explores algebraic structures related to generalized Boolean dynamical systems, establishing their connections to Cuntz-Pimsner algebras and skew group rings, and characterizing their ideal structure through a graded uniqueness theorem.

Contribution

It introduces the notion of relative generalized Boolean dynamical systems and extends the graded uniqueness theorem to these systems, providing new insights into their algebraic properties.

Findings

Proves these algebras are Cuntz-Pimsner algebras and partial skew group rings.

Characterizes graded ideals via the underlying dynamical system.

Shows Morita equivalence to systems without singularities.

Abstract

We study an algebraic analog of a C*-algebra associated to a generalized Boolean dynamical system which parallels the relation between graph C*-algebras and Leavitt path algebras. We prove that such algebras are Cuntz-Pimsner algebras and partial skew group rings and use these facts to prove a graded uniqueness theorem. We then describe the notion of a relative generalized Boolean dynamical system and generalize the graded uniqueness theorem to relative generalized Boolean dynamical system algebras. We use the graded uniqueness theorem to characterize the graded ideals of a relative generalized Boolean dynamical system algebra in terms of the underlying dynamical system. We prove that every generalized Boolean dynamical system algebra is Morita equivalent to an algebra associated to a generalized Boolean dynamical system with no singularities. Finally, we give an alternate characterization of the class of generalized Boolean dynamical system algebras that uses an underlying graph structure derived from Stone duality.