It is studied how taking the inverse image
by a sliding block code affects the syntactic semigroup of a sofic
subshift. The main tool are ζ-semigroups, considered as
recognition structures for sofic subshifts.
A new algebraic invariant is obtained for
weak equivalence of sofic subshifts, by
determining which classes of sofic subshifts
naturally defined by pseudovarieties of finite semigroups are closed
under weak equivalence. Among such classes are the classes of almost
finite type subshifts and aperiodic subshifts.
The algebraic invariant is compared with other robust conjugacy