CONJUGACY INVARIANTS OF SUBSHIFTS: AN APPROACH FROM PROFINITE SEMIGROUP THEORY

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.

SUBWORD COMPLEXITY OF PROFINITE WORDS AND SUBGROUPS OF FREE PROFINITE SEMIGROUPS

We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated continuous endomorphisms, subword complexity, and the associated entropy. Main results include a general scheme to produce such subgroups and a proof that the complement of the minimal ideal in a free profinite semigroup on more than one generator is closed under all implicit operations that do not lie in the minimal ideal and even under their infinite iteration.

PROFINITE SEMIGROUPS, VARIETIES, EXPANSIONS AND THE STRUCTURE OF RELATIVELY FREE PROFINITE SEMIGROUPS

Building on the now generally accepted thesis that profinite semigroups are important to the study of finite semigroups, this paper proposes to apply various of the techniques, already used in studying algebraic semigroups, to profinite semigroups. The goal in mind is to understand free profinite semigroups on a finite set. To do this we define profinite varieties. We then introduce expansions of profinite semigroups, giving examples of several classes of such expansions. These expansions will then be useful in studying various structural properties of relatively free profinite semigroups, since these semigroups will be fixed points of certain expansions. This study also requires a look at profinite categories, semigroupoids, and Cayley graphs, all of which we handle in turn. We also study the structure of the minimal ideal of relatively free profinite semigroups showing, in particular, that the minimal ideal of the free profinite semigroup on a finite set with more than two generators is not a relatively free profinite completely simple semigroup, as well as some generalizations to related pseudovarieties.