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 investigate the ideal structure of the universal C*-algebra
associated to a sofic shift space and to find the range of a flow-invariant.
Finally, it is proved that the condition (*) introduced by Carlsen and
Matsumoto holds if and only if the left Krieger cover is the maximal essential
subgraph of the past set cover.