On fixed points of the lower set operator

Abstract

International audienceLower subsets of an ordered semigroup form in a natural way an ordered semigroup. This lower set operator gives an analogue of the power operator already studied in semigroup theory. We present a complete description of the lower set operator applied to varieties of ordered semigroups. We also obtain large families of fixed points for this operator applied to pseudovarieties of ordered semigroups, including all examples found in the literature. This is achieved by constructing six types of inequalities that are preserved by the lower set operator. These types of inequalities are shown to be independent in a certain sense. Several applications are also presented, including the preservation of the period for a pseudovariety of ordered semigroups whose image under the lower set operator is proper.Les sections commençantes d'un semigroupe ordonné forment de façon naturelle un semigroupe ordonné. L'opérateur qui associe à un semigroupe ordonné le semigroupe ordonné de ses sections commençantes définit un opérateur analogue à l'opérateur qui associe à un semigroupe son semigroupe des parties. Nous présentons une description complète de cet opérateur étendu aux variétés de semigroupes ordonnés. Nous obtenons également de grandes familles de points fixes de cet opérateur appliqué aux pseudovariétés de semigroupes ordonnés, qui recouvrent en particulier tous les exemples connus dans la littérature. Ce résultat est obtenu en construisant six types d'inégalités qui sont conservés par notre opérateur. Nous montrons que ces types d'inégalités sont dans un certain sens indépendants. Nous présentons également plusieurs applications, notamment la préservation de la période pour une pseudovariété de semigroupes ordonnés dont l'image par l'opérateur des sections commençantes est propre

On the insertion of n-powers

“…Consider the pseudovarieties J + = 1 x and LI + = x ω x ω yx ω , respectively of ordered monoids and of ordered semigroups. By (Pin and Weil, 1997, Theorem 5.9), for a pseudovariety of monoids V, the polynomial closure (ii) of V is the pseudovariety of ordered monoids Pol V = LI + m V. As was proved by Pin and Weil (1996), LI + m V is defined by the inequalities of the form u ω u ω vu ω such that the pseudoidentities u = v = v 2 hold in V. In particular, in case V is a pseudovariety of groups, one may (i) In the literature, one often finds the syntactic quasiorder defined to be the reverse quasiorder (see Almeida et al (2015) for historical details). (ii) meaning the pseudovariety of ordered monoids Pol V corresponding to the positive variety of languages generated by the class of languages which, for a finite alphabet A, consists of the products of the form L 0 a 1 L 1 • • • anLn, where the a i ∈ A and the L i are V-languages.…”

Commutative Positive Varieties of Languages

“…Given a variety of ordered monoids V, let P ↓ V [P ↓ 0 V] denote the variety of ordered monoids generated by the monoids of the form P ↓ (M ) [P ↓ 0 (M )], where M ∈ V. The operator P ↓ was intensively studied in [4]. In particular, it is known that both P ↓ and P ↓ 0 are idempotent operators.…”

“…Let V ′ be the closure of V under shuffle, or equivalently, under product over one-letter alphabets. We claim that V ′ (a * ) consists of the empty set and the languages of the form a n (F + a 5 a * ) (7.5) where n 0 and F is a subset of (1 + a) 4 . First of all, the languages of the form (7.5) and the empty set form a lattice closed under product, since if 0 n m and F and G are subsets of (1 + a) 4 , then a n (F + a 5 a * ) + a m (G + a 5 a * ) = a n (F + a m−n G + a 5 a * ) a n (F + a 5 a * ) ∩ a m (G + a 5 a * ) = a m (a m−n ) −1 (F + a 5 a * ) ∩ G + a 5 a * a n (F + a 5 a * )a m (G + a 5 a * ) = a n+m (F G + a 5 a * )…”

The ω-inequality problem for concatenation hierarchies of star-free languages

“…Equivalently, the syntactic ordered monoid Synt(L) of L belongs to V, where Synt(L) is the quotient of A * by the congruence L ∩ L , ordered by the partial order induced by the quasi-order L , where u L v is defined by the following condition: for every x, y ∈ A * , xuy ∈ L implies xvy ∈ L. The natural homomorphism ϕ : A * → Synt(L) as well as its extension φ are both called syntactic homomorphisms. It should be noted that, in several papers by Pin and coauthors, such as [24], the syntactic order L is defined to be the dual of the order considered here; see [6] for an explanation as to why our choice should be preferred.…”