Communicated by M. Sapir MSC:Primary: 20M05 20M07 secondary: 37B10 20M50 a b s t r a c t Some fundamental questions about infinite-vertex (free) profinite semigroupoids are clarified, putting in evidence differences with the finite-vertex case. This is done with examples of free profinite semigroupoids generated by the graph of a subshift. It is also proved that for minimal subshifts, the infinite edges of such free profinite semigroupoids form a connected compact groupoid.

It is given a structural conjugacy invariant in the set of pseudowords whose finite factors are factors of a given subshift. Some profinite semigroup tools are developed for this purpose. With these tools a shift equivalence invariant of sofic subshifts is obtained, improving an invariant introduced by Béal, Fiorenzi and Perrin using different techniques. This new invariant is used to prove that some almost finite type subshifts with the same zeta function are not shift equivalent. Int. J. Algebra Comput. 2006.16:629-655. Downloaded from www.worldscientific.com by UNIVERSITY OF MICHIGAN on 02/03/15. For personal use only.Proof. By Lemma 2.12 and sinceδ −1 X (δ X (L(X ))) = L(X ), every element of δ X (L(X )) † is the J -class of the image byδ X of an idempotent-bound element of L(X ). Let u be an element of L(X ) such that u = euf for some idempotents e and f of A + . Suppose that w is such thatδ X (u)≤ JδX (w). Then there are elements p and q of A * such thatδ X (u) =δ X (pwq). By Corollary 4.4,δ Y (ḡ(u)) =δ Y (ḡ(epwqf )), thusδ Y (ḡ(u))≤ JδY (ḡ(w)), which proves that G s is order preserving. In particular, if u J v thenδ Y (ḡ(u))Jδ Y (ḡ(v)), thus G s is a well-defined function.The following is an immediate corollary of Proposition 3.7. Proposition 4.6. Let G : X → Y and H : Y → Z be conjugacies between sofic subshifts. Then H s • G s = (H • G) s . Moreover, (Id X ) s = Id δX (L(X )) † .Corollary 4.7. The poset δ X (L(X )) † is a conjugacy invariant of sofic subshifts.Proposition 4.8. Let G : X ⊆ A Z → Y ⊆ B Z be a conjugacy between sofic subshifts. If K is a J -class of δ X (L(X )) then K is regular if and only if G s (K) is regular.Proof. By Lemma 2.12, if K is a regular J -class of δ X (L(X )) then there is a regular J -class J of A + such that K = [δ X (J)] J . We have J ⊆ L(X ). Since G † preserves the regularity of a J -class, the J -class G s (K) = [δ Y (G † (J))] J is regular. Hence G s sends regular J -classes to regular J -classes. The same is true with (G −1 ) s , which is the map (G s ) −1 , by Proposition 4.6.Proposition 4.9. Let G : X ⊆ A Z → Y ⊆ B Z be a conjugacy between sofic subshifts. If K is an idempotent-bound J -class of δ X (L(X )) then the Schützenberger groups of K and G s (K) are isomorphic.

Abstract. In previous work, the first author established a natural bijection between minimal subshifts and maximal regular J -classes of free profinite semigroups. In this paper, the Schützenberger groups of such J -classes are investigated in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for several types of minimal subshifts associated with substitutions. The Schützenberger subgroup of the J -class corresponding to the Prouhet-Thue-Morse subshift is shown to admit a somewhat simpler presentation, from which it follows that it satisfies the conjecture, that it has rank three, and that it is non-free relatively to any pseudovariety of groups.

For a pseudovariety V of ordered semigroups, let S (V) be the class of sofic subshifts whose syntactic semigroup lies in V. It is proved that if V contains Sl − then S (V * D) is closed under taking shift equivalent subshifts, and conversely, if S (V) is closed under taking conjugate subshifts then V contains LSl − and S (V) = S (V * D). Almost finite type subshifts are characterized as the irreducible elements of S (LInv), which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.

We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. We also establish some other analogies between the kernel of the free profinite semigroup and the J -class associated to an irreducible sofic shift.