International Journal of Algebra and Computation

We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-V topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extension-closed. Turning to practical computations, we modify Ribes and Zalesskiĭ's algorithm to compute the pro-p closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups.

Résumé: Nous établissons un lien entre le problème du calcul de l'adhéerence d'un sous-groupe finiment engendré du groupe libre dans la topologie pro-V, oú V est une pseudovariété de groupes finis, et un probléme d'extension pour les automates inversifs qui peut être énoncé de la faç con suivante: étant données des transformations partielles injectives d'un ensemble fini, peuvent-elles être étendues en des permutations qui engendrent un groupe dans V? Les deux problèmes sont équivalents si V est fermée par extensions. Nous intéressant ensuite aux calculs pratiques, nous modifions l'algorithme de Ribes et Zalesskiĭ pour calculer l'adhérence pro-p d'un sous-groupe finiment engendré du groupe libre en temps polynomial et pour calculer effectivement sa clôture pro-nilpotente. Enfin nous appliquons nos résultats à un problème de théorie des monoïdes finis, celui de de l'appartenance dans les pseudovariétés de monoïdes inversifs qui sont des produits de Mal'cev de demi-treillis et d'une pseudovariété de groupes.

In this paper, we study the geometric equivalence of algebras in several varieties of algebras. We solve some of the problems formulated in [2], in particular, that of geometric equivalence for real-closed fields and finitely generated commutative groups.

In this paper we consider monoids M with an idempotent cross-section Λ of regular -classes such that for all idempotents e, f of M with e≥f, there exists x in the unit group such that x^-1ex, x^-1fx∈Λ. Such monoids arise frequently in connection with group representations. We show that Λ is then necessarily a semilattice and that the idempotent structure of M can for the most part, be determined within the unit group.

We identify a particular class of "canonical monoids" inspired by some projective compactifications of algebraic groups. We show that these canonical monoids are 0-semilattice unions of completely 0-simple semigroups.

We describe two practical algorithms for computing with word-hyperbolic groups, both of which we have implemented. The first is a method for estimating the maximum width, if it exists, of geodesic bigons in the Cayley graph of a finitely presented group G. Our procedure will terminate if and only this maximum width exists, and it has been proved by Papasoglu that this is the case if and only if G is word-hyperbolic. So the algorithm amounts to a method of verifying the property of word-hyperbolicity of G. The aim of the second algorithm is to compute the thinness constant for geodesic triangles in the Cayley graph of G. This seems to be a much more difficult problem, but our implementation does succeed with straightforward examples. Both algorithms involve substantial computations with finite state automata.

We construct a finitely based congruence-distributive variety of algebras in a finite language whose set of subalgebras of finite simple algebras is non-recursive.