Centrally labeled Rauzy graphs are Rauzy graphs of subshifts such that the vertices are factors of even length, and such that the edges are labeled by its middle letter (this makes sense, since the edges are factors of odd length). As automata devices, centrally labeled Rauzy graphs have a transition semigroup. We exhibit fundamental properties of these semigroups, many of them valid for arbitrary finite local automata. Results are obtained that explain how, for an arbitrarily given irreducible subshift, the transition semigroups of different centrally labeled Rauzy graphs are related. It is very easy to prove that these semigroups have a 0-minimum J-class (except for the full shift case). One proves that, for a given minimal subshift X, these J-classes form a projective limit of partial compact semigroups that is isomorphic to the J-class associated to X in the free profinite aperiodic semigroup, thus giving some insight about its structure. A weaker result for irreducible subshifts is obtained in the process.

Marginally to these results, the following question is investigated: when does the transition semigroup of a centrally labeled Rauzy graph coincide with the syntactic semigroup of the language recognized by it? Although in general such coincidence does not happen, even for minimal subshifts, for Sturmian subshifts that coincidence holds.

(Joint work with Jorge Almeida)