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.