On the Structure of Covers of Sofic Shifts

TL;DR

This paper introduces a canonical cover for sofic shifts, analyzes their layered structures, and explores implications for flow invariants and C*-algebra ideals, advancing understanding of sofic shift dynamics.

Contribution

It presents a new canonical cover for arbitrary sofic shifts and reveals layered structures of existing covers, enhancing the analysis of their invariants and algebraic properties.

Findings

The left Krieger and past set covers can be divided into natural layers.

The range of a flow-invariant is characterized.

Insights into the ideal structure of associated C*-algebras are provided.

Abstract

A canonical cover generalizing the left Fischer cover to arbitrary sofic shifts is introduced and used to prove that the left Krieger cover and the past set cover of a sofic shift can be divided into natural layers. These results are used to find the range of a flow-invariant and to investigate the ideal structure of the universal C*-algebra associated to a sofic shift space.