Reducibility of covers of AFT shifts

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.