Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts. (Q869222)

Let \(A\) be a finite alphabet and \(A^\mathbb{Z}\) be the set of sequences of letters of \(A\) indexed by \(\mathbb{Z}\). A non-empty compact subset of \(A^\mathbb{Z}\) which is closed under the shift operation and its inverse is called a `subshift'. Subshifts are in one-to-one correspondence with non-empty factorial prolongable languages. A subshift is called `sofic' if its corresponding language is rational. Via Pin's correspondence theorem, given a pseudovariety \(V\) of ordered semigroups, there exists a pseudovariety \({\mathcal L}(V)\) of sofic subshifts whose (ordered) syntactic semigroup is in \(V\) (here the syntactic semigroup of a subshift is the syntactic semigroup of the corresponding factorial prolongable language). Using the profinite approach in the study of pseudovarieties of semigroups, the author proves that if \(V\) contains the pseudovariety \(SI^-\) of commutative ordered monoids such that every element is idempotent and greater than or equal to the neutral element, then \({\mathcal L}(V*D)\), where \(V*D\) denotes the semidirect product of the pseudovariety \(V\) and the pseudovariety of definite semigroups \(D\), is closed under taking `conjugate' subshifts (subshifts are conjugate if there is a shift commuting homeomorphism between them) and is also closed under taking `shift equivalent' subshifts (a notion stricly weaker than the former one). Conversely, he proves that if \({\mathcal L}(V*D)\) is closed under taking conjugate subshifts, then \(V\) contains the pseudovariety of ordered semigroups whose ordered submonoids are in \(SI^-\) and \({\mathcal L}(V)={\mathcal L}(V*D)\). In addition, the author gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.