Abstract
We study a correspondence associating to each subshift of
a subcategory of the Karoubi envelope of the free profinite semigroup generated by A. The objects of this category are the idempotents in the mirage of
that is, in the set of pseudowords whose finite factors are blocks of
The natural equivalence class of the category is shown to be invariant under flow equivalence. As a corollary of our proof, we deduce the flow invariance of the profinite group that Almeida associated to each irreducible subshift. We also show, in a functorial manner, that the isomorphism class of the category is invariant under conjugacy. Finally, we see that the zeta function of
is naturally encoded in the category. These results hold, with obvious translations, for relatively free profinite semigroups over many pseudovarieties, including all of the form
with
a pseudovariety of groups.
Notes
1 For every Green relation and every
one has
if
is an equivalence. Here we are applying this property in the special case
2 We give [Citation24, Section 2.3] as a reference for this property of for the sake of better readability, but the property was known before: in the language of pseudowords, it is implicit in [Citation1, Section 3.7], and in fact it amounts to the fact that
is the join of
and its dual
a fact already appearing in [Citation25].
3 The arguments used in the proof of this implication are basically the same that were used in the proof of [Citation11, Lemma 2.2], but there one finds the assumption that contains
to guarantee that
does contain nontrivial monoids and therefore is according to the statement in [Citation1, Theorem 10.6.12]. As seen in our recapitulation of those arguments, such assumption is unnecessary.