Positive Varieties and Infinite Words

Carrying on the work of Arnold, Pécuchet and Perrin, Wilke has obtained a counterpart of Eilenberg’s variety theorem for finite and infinite words. In this paper, we extend this theory for classes of languages that are closed under union and intersection, but not necessarily under complement. As an example, we give a purely algebraic characterization of various classes of recognizable sets defined by topological properties (open, closed, FΣ and Gδ) or by combinatorial properties Carrying on the work of Arnold [1], Pécuchet [7, 8] and Perrin [9–11], Wilke [25, 26] has obtained a counterpart of Eilenberg’s variety theorem for finite and infinite words. The word “and” is emphasized in the last sentence, because it is really important to work simultaneously with finite and infinite words. The fitness of this approach was corroborated by recent contributions [13, 14, 18, 25, 27]. A variant of the notion of syntactic semigroup was recently proposed by the author [17]. The key idea is to define a partial order on syntactic semigroups, leading to the notion of ordered syntactic semigroups. The resulting extension of Eilenberg’s variety theory permits to treat classes of languages that are closed under union and intersection, but not necessarily under complement, a major difference with the original theory. The aim of this paper is to extend this theory to infinite words, thus completing the table below. In the setting proposed by Wilke, semigroups are not suitable any more. They can be replaced by ω-semigroups, which are, roughly speaking, semigroups equipped with an infinite product [13, 25, 27]. Eilenberg [4] Pin [17] Finite words Varieties of semigroups Var. of ordered semigroups ⇔ +-varieties ⇔ Positive +-varieties Wilke [25, 26] This paper Finite or Varieties of ω-semigroups Var. of ordered ω-semigroups infinite words ⇔ ∞-varieties ⇔ Positive ∞-varieties The “ordered” approach has many interesting consequences but leads to a complete rewriting of the theory. We have selected three examples, of rather different nature, to convince the reader of the power of this new approach. In Sect. 5, we give a purely algebraic characterization of four classes of recognizable sets defined by topological properties. This includes in particular the class of deterministic ω-languages (i.e. recognized by a deterministic Büchi automaton). Similar results were known only for topological classes closed under complement. Note that all these characterizations are effective, since a simple algorithm to compute the syntactic ω-semigroup of a recognizable ω-language was given in [13]. In Sect. 6 we address a question originally considered by Pécuchet [7, 8]. Since every recognizable ω-language can be written as a finite union of languages of the form XY , where X and Y are recognizable languages, the question arose to know whether this result could be “relativized” to varieties. Theorem 3, which extends the results of Pécuchet, gives such a result for varieties of ordered semigroups. It requires the concept of weak recognizability, introduced by Perrin [11], and the notion of ordered Büchi automaton, which might be interesting in itself. Our last example, developed in Sect. 7, is a good illustration of the problems that arise when trying to generalize to infinite words a given class of recognizable languages. Indeed, as was observed by Pécuchet, there are at least three natural ways to associate a class of ω-languages to a variety of finite semigroups V. One can consider the sets recognized by a finite ω-semigroup S whose semigroup part belongs to V, or those weakly recognized by a semigroup of V, or finally, inspired by McNaughton’s theorem, the boolean combinations of sets of the form −→ L where L is recognized by a semigroup of V. We will see how these three classes relate to each other in our case study, the shuffle ideals. Due to the lack of place, no proofs are given, but they can be found in [14]. For more details, the reader is referred to [15, 16, 20] for the variety theory for finite words and to [13, 14, 19, 24–26] for the theory of ω-languages. 1 Notations and basic definitions Let A be a finite alphabet. The free monoid on A is denoted A and the free semigroup,A. The set of infinite words on A is denoted A . Finally, A denotes the set of finite or infinite words. In this paper, a subset X of A will be systematically identified with the pair (X+, Xω), where X+ = X ∩ A + and Xω = X ∩A . We now briefly review the standard definition of a Büchi automaton and introduce the notion of an ordered Büchi automaton.