A conjugacy invariant for reducible sofic shifts and its semigroup characterizations

We introduce a notion of magic words and, through them, we present a lattice of sub-synchronizing subshifts which describes the synchronizing parts of a sofic shiftS. We show that topological conjugacy maps subsynchronizing subshifts onto sub-synchronizing subshifts, it preserves their mutual relationship (i.e. the corresponding lattices are isomorphic) and the corresponding covers within the Krieger covers are topologically conjugate. Using the magic words, a full characterization of the syntactic monoid of a shift of finite type is given. We show that a synchronizing deterministic presentation of every sub-synchronizing subshift ofS can be seen within a two-sided ideal of the syntactic monoid ofS.