José Carlos Costa

Free profinite locally idempotent and locally commutative semigroups

Abstract This paper is concerned with the structure of semigroups of implicit operations on the pseudovariety L Sl of finite locally idempotent and locally commutative semigroups. We depart from a general result of Almeida and Weil to give two descriptions of these semigroups: the first in terms of infinite words, and the second in terms of infinite and bi-infinite words. We then derive some applications.

Reducibility of joins involving some locally trivial pseudovarieties

Abstract In this paper, we show that σ-reducibility is preserved under joins with K, where K is the pseudovariety of semigroups in which idempotents are left zeros. Reducibility of joins with D, the pseudovariety of semigroups in which idempotents are right zeros, is also considered. In this case, we were able to prove that σ- reducibility is preserved for joins with pseudovarieties verifying a certain property of cancellation. As an example involving the semidirect product, we prove that Sl*K is κ-tame, where Sl stands for the pseudovariety of semilattices.

Biinfinite words with maximal recurrent unbordered factors

A finite non-empty word z is said to be a border of a finite non-empty word w if w=uz=zv for some non-empty words u and v. A finite non-empty word is said to be bordered if it admits a border, and it is said to be unbordered otherwise. In this paper, we give two characterizations of the biinfinite words of the form ?uvu?, where u and v are finite words, in terms of its unbordered factors.The main result of the paper states that the words of the form ?uvu? are precisely the biinfinite words w=?a?2a?1a0a1a2? for which there exists a pair (l0,r0) of integers with l0

Complete reducibility of pseudovarieties

The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results establishing this and the stronger property of complete reducibility for specic pseudovarieties.

Free profinite R-trivial, locally idempotent and locally commutative semigroups

This paper is concerned with the structure of implicit operations on R ∩ LJ1, the pseudovariety of all R-trivial, locally idempotent and locally commutative semigroups. We give a unique factorization statement, in terms of component projections and idempotent elements, for the implicit operations on R ∩ LJ1. As an application we give a combinatorial description of the languages that are both R-trivial and locally testable. A similar study is conducted for the pseudovariety DA ∩ LJ1 of locally idempotent and locally commutative semigroups in which each regulär D-class is a rectangular band.

On bases of identities for the ω-variety generated by locally testable semigroups

Let LSl be the pseudovariety of local semilattices. It is well known that this pseudovariety coincides with the class of locally testable semigroups. In this paper, we exhibit an infinite basis of @w-identities for the @w-variety generated by LSl and we show that this @w-variety is not finitely based.

Some variations on the notion of locally testable language

The aim of this paper is to complete the characterization of the languages that are Boolean combinations (of a subset) of languages of the form wA∗, A∗w or L(w, r, t, n), where A is an alphabet, w ∈ A, r, t ≥ 0, n ≥ 1 and L(w, r, t, n) denotes the set of all words u in A such that the number of occurrences of the factor w in u is congruent to r threshold t mod n. For each class C of languages such that A+C is a Boolean algebra generated by some of the following types of languages: wA∗, A∗w, A∗wA∗ = L(w, 1, 1, 1) or L(w, r, t, 1), and such that C does not constitute a variety of languages, we compute the smallest variety of languages containing C and the largest variety of languages contained in C.

ʿOlam ha-ze/ʿolam ha-ba, al-dunyā/al-āḫira : étude comparée de deux couples de termes dans la littérature talmudique et le Coran1

La litterature rabbinique et le Coran croient en l’existence de deux mondes : le monde present et le monde futur (ʿolam ha-ze/ʿolam ha-ba en hebreu, dunyā/āḫira en arabe : deux couples de mots sans relation linguistique). Les etudes sur l’eschatologie rabbinique ont toujours reconnu l’importance de ce theme. A l’inverse, il a ete relativement neglige dans le champ des etudes coraniques et necessite encore de nouvelles recherches. Nous avons compare trois types de textes dans les deux corpus : ceux qui ne mentionnent que le monde present ou futur et ceux qui traitent simultanement des deux mondes. A premiere vue, le Coran montre plus d’affinites avec les apocalypses juives ou la litterature chretienne (particulierement syriaque). Cependant, la comparaison souligne egalement quelques correlations significatives avec les donnees rabbiniques, comme l’usage frequent du couple « monde present/monde futur » et plusieurs aspects de la doctrine de la retribution. Finalement, notre analyse confirme que l’Arabie pre-islamique se caracterisait par une importante diversite et complexite religieuses, a l’interieur du judaisme et en dehors de lui.

TAMENESS OF JOINS INVOLVING THE PSEUDOVARIETY OF LOCAL SEMILATTICES

In this paper we prove that, if V is a -tame pseudovariety which satisfies the pseudoidentity xy !+1 z = xyz, then the pseudovariety join LSl_V is also -tame. Here, LSl denotes the pseudovariety of local semilattices and denotes the implicit signature consisting of the multiplication and the (! 1)-power. As a consequence, we deduce that LSl_ V is decidable. In particular the joins LSl_ Ab, LSl_ G, LSl_ OCR and LSl_ CR are decidable.

SOME PSEUDOVARIETY JOINS INVOLVING GROUPS AND LOCALLY TRIVIAL SEMIGROUPS

In this paper, we present the computation of some pseudovariety joins of the form LI∨H∨V where LI is the pseudovariety of locally trivial semigroups and H is any pseudovariety of groups. Similar results are obtained for the pseudovarieties K, of semigroups in which idempotents are left zeros, and its dual D, in the place of LI.