It is studied how taking the inverse image by a sliding block code
affects the syntactic semigroup of a sofic subshift. Two independent approaches are
used: ζ-semigroups as recognition structures for sofic subshifts, and relatively free
profinite semigroups. 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 invariants.Research programme AutoMathA of ESF; Pessoa bilateral project Egide/Grices 11113YM "Automata, profinite semigroups and symbolic dynamics"; FCT, grant SFRH/BD/24200/2005; POCI 2010; FS
