General Topology-I

Mathematica--A System for Doing Mathematics by Computer.

The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

The english translation

This chapter presents some of the links between automata theory and symbolic dynamics The emphasis is on two particular points The first one is the interplay between some particular classes of automata, such as local automata and results on embeddings of shifts of finite type The second one is the connection between syntactic semigroups and the classification of sofic shifts up to conjugacy

Symbolic dynamics

https://estudogeral.sib.uc.pt/bitstream/10316/10043/1/ficheiro.pdf

Infinite-vertex free profinite semigroupoids and symbolic dynamics

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 Beal, 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.

/pdf/conjugacy-invariants-of-subshifts-an-approach-from-profinite-4gimdusfqn.pdf

Conjugacy invariants of subshifts: an approach from profinite semigroup theory

We prove that the Karoubi envelope of a shift — defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift — is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants — some new and some old — are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov-Dyck and Markov-Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property (A) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property (A) is decidable for sofic shifts.

/pdf/a-categorical-invariant-of-flow-equivalence-of-shifts-681su0shcs.pdf

A categorical invariant of flow equivalence of shifts

Research programme AutoMathA of ESF; FCT project PTDC/MAT/65481/2006 and FCT post-doctoral grant
SFRH/BPD/46415/2008; NSERC; DFG

/pdf/profinite-groups-associated-to-sofic-shifts-are-free-17v81zle54.pdf

Profinite groups associated to sofic shifts are free

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 Schutzenberger 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 all non-periodic minimal subshifts associated with substitutions. It entails that it is decidable whether a finite group is a quotient of such a profinite group. As a further application, the Schutzenberger group 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 has rank three, and that it is non-free relatively to any pseudovariety of groups.

/pdf/presentations-of-schutzenberger-groups-of-minimal-subshifts-596yvbyk4k.pdf

Alfredo Costa

Papers

Infinite-vertex free profinite semigroupoids and symbolic dynamics

Conjugacy invariants of subshifts: an approach from profinite semigroup theory

A categorical invariant of flow equivalence of shifts

Profinite groups associated to sofic shifts are free

Presentations of Schützenberger groups of minimal subshifts