Closures of regular languages for profinite topologies

Abstract

Abstract. The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology.A similar descriptive procedure is shown to hold for the pseudovariety A of aperiodic semigroups, where the closure is taken in the free aperiodic ω-semigroup. It is inherited by a subpseudovariety of a given pseudovariety if both of them enjoy the property of being full. The pseudovariety A, as well as some of its subpseudovarieties are shown to be full.The interest in such descriptions stems from the fact that, for each of the main pseudovarieties V in our examples, the closures of two regular languages are disjoint if and only if the languages can be separated by a language whose syntactic semigroup lies in V. In the cases of A and of the pseudovariety DA of semigroups in which all regular elements are idempotents, this is a new result.

Factoriality and the Pin-Reutenauer procedure

“…We now show that (3.2) holds over the pseudovariety S, for every pure implicit signature σ containing κ and every rational language L ⊆ X + . It is easy to see that the inclusion from right to left in (3.2) always holds, see Almeida et al (2014). The rest of this subsection is devoted to the proof of the other inclusion.…”

“…It has been established recently by Almeida et al (2014) that The Pin-Reutenauer procedure holds for a number of pseudovarieties. However, the results of this paper rely on independent, technically nontrivial results for the pseudovariety A of aperiodic semigroups: first, it was proved that the Pin-Reutenauer procedure is valid for A using the solution of the word problem for the free aperiodic κ-algebra given by McCammond (2001); Huschenbett and Kufleitner (2014); Almeida et al (2015).…”

“…This result is obtained by first investigating a property named factoriality. Factoriality of V with respect to, say, the signature κ means that Ω κ X V is closed under taking factors in Ω X V. It was shown by Almeida et al (2014) that if V is factorial, then the Pin-Reutenauer procedure holds with respect to concatenation, that is, KL = K L for arbitrary K, L (not just for rational ones). However, it was also noted that the pseudovariety S cannot be factorial for nontrivial countable signatures, such as κ.…”

“…Besides the independent interest of such results, the technical tool used to prove them, named factorization history, is also the key to establish that the Pin-Reutenauer is valid for S. We further characterize pseudovarieties in which the Pin-Reutenauer procedure holds in terms of an abstract property named fullness, introduced by Almeida and Steinberg (2000a). The main idea is that the validity of the Pin-Reutenauer procedure for a pseudovariety V is inherited by a subpseudovariety W, as established by Almeida et al (2014), provided both V and W are full. Conversely, we prove that if the Pin-Reutenauer procedure works for V, then V is full.…”

McCammond’s normal forms for free aperiodic semigroups revisited

“…It plays, in particular, an important role in establishing the main result of that paper, namely a characterization of pseudowords over A which are given by ω-terms. Other applications of Theorem 7.4 and of properties of the languages L n [α], such as an algorithm to compute the closure cl A (L) of a regular language L have been published in [8].…”