Reducibility of Covers of AFT shifts

TL;DR

This paper investigates the reducibility properties of covers of sofic shifts, establishing flow invariance, and explores implications for the structure of associated algebras, with new examples and theoretical results.

Contribution

It proves flow invariance of reducibility structures and shows that certain covers are reducible for specific classes of shifts, providing new insights and examples.

Findings

Reducibility structure is a flow invariant for several covers of sofic shifts.

Left Krieger and past set covers are reducible for irreducible subshifts of almost finite type.

Matsumoto algebra of certain sofic shifts is not simple.

Abstract

In this paper we show that the reducibility structure of several covers of sofic shifts is a flow invariant. In addition, we prove that for an irreducible subshift of almost finite type the left Krieger cover and the past set cover are reducible. We provide an example which shows that there are non almost finite type shifts which have reducible left Krieger covers. As an application we show that the Matsumoto algebra of an irreducible, strictly sofic shift of almost finite type is not simple.