International Journal of Algebra and Computation

A subgroup H of a free product of groups G_α, α∈ I, is termed factor free if for every and β∈I one has SHS^-1∩G_β= {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote , where r(K) is the rank of K. It has earlier been proved by the author that if H, K are finitely generated factor free subgroups of then . It is proved in the article that this estimate is sharp and cannot be improved, that is, there are factor free subgroups H, K in so that and . It is also proved that if the factors G_α, α∈ I, are linearly ordered groups and H, K are finitely generated factor free subgroups of then .

Let U be a word in letters , m >2, and a group G be given by presentation G=< x₁,…, x_m‖ U x_i U^-1= x_i+1, i=1,…, m-1>. It is proven that this presentation is aspherical provided the word U does not have the form U₂ U₁, where U₁ is a word in letters and U₂ is a word in letters . It is also proven that the (images of) x₁,…, x_m-1 freely generate a free subgroup of G if and only if the word U does not have the foregoing form U₂ U₁.

Résumé: Cet article présente une étude combinatoire du monoïde Chinois, un monoïde ternaire proche du monoïde plaxique, fondé sur le schéma cba≡bca≡cab. Un algorithme proche de l'algorithme de Schensted nous permet de caractériser les classes d'équivalence et d'exhiber une section du monoïde. Nous énonçons également une correspondance de Robinson–Schensted pour le monoïde Chinois avant de nous intéresser au calcul du cardinal de certaines classes. Ce travail a permis de développer de nouveaux outils combinatoires. Entre autres, nous avons trouvé un plongement de chacune des classes d'équivalence dans la plus grande classe. Quant à la dernière partie de cet article, elle présente l'étude des relations de conjugaison.

This paper presents a combinatorial study of the Chinese monoid, a ternary monoid related to the plactic monoid and based on the relation scheme cba≡bca≡cab. An algorithm similar to Schensted's algorithm yields a characterization of the equivalence classes and a cross-section theorem. We also establish a Robinson–Schensted correspondence for the Chinese monoid before computing the order of specific Chinese classes. For this work, we had to develop some new combinatorial tools. Among other things we discovered an embedding of every equivalence class in the largest one. Finally, the end of this paper is devoted to the study of conjugacy classes.

A universal three-state three-neighbor cellular automaton will be constructed. The space selected for this cellular automaton is a hexagonal tiling where the cells are in the vertices and the neighbors are the three nearest cells. We define the local transition rule as well as the basic elements that will aid to build digital circuits and, by the way, prove the universality of this cellular automaton. The local transition rule is defined to be isotropic.

This paper addresses the question of simultaneously solving a set of equations in one variable over torsion-free groups.

In this paper we show that the braid groups B_n, n≤6, have proper torsion-free non-abelian quotients. We also show that for n≤6 the homotopy braid groups are torsion-free.

Let h₁, h₂,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w₁, w₂, …, where each w_i is a word in h_i and possibly other h_j, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that , where is the subgroup generated by the roots of the elements in H. If H₀ is finitely generated and the sequence of subgroups H₀, H₁, H₂, … satisfies then the sequence stabilizes, i.e. for some m, H_i=H_i+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" the exponents a_i satisfy gcd(a₁,…,a_n)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.

An example of a finitely presented monoid is given that does not satisfy the homotopical finiteness condition , although it satisfies both the homological finiteness conditions left and right .