International Journal of Algebra and Computation

Let $F(p)$, $p\ge 2$ be the family of generalized Thompson’s groups. Here, $F(2)$ is the famous Richard Thompson’s group usually denoted by $F$. We find the growth rate of the monoid of positive words in $F(p)$ and show that it does not exceed $p+1/2$. Also, we describe new normal forms for elements of $F(p)$ and, using these forms, we find a lower bound for the growth rate of $F(p)$ in its natural generators. This lower bound asymptotically equals $(p-1/2){log}_{2}e+1/2$ for large values of $p$.

This paper deals with the reducibility property of semidirect products of the form $V\ast D$ relatively to graph equation systems, where D denotes the pseudovariety of definite semigroups. We show that if the pseudovariety $V$ is reducible with respect to the canonical signature $\kappa$ consisting of the multiplication and the $(\omega -1)$-power, then $V\ast D$ is also reducible with respect to $\kappa$.

We classify the finitely generated prosupersolvable groups that satisfy Schreier’s formula for the number of generators of open subgroups.

We consider groups $G$ which have generating sets consisting of pairwise conjugate elements. We introduce a natural condition on such sets, namely path-connectedness, which has strong consequences when $G$ acts on a real tree. The main result of the paper states that the group of type-preserving automorphisms of a regular simplicial tree has a generating set with this property.

Braids can be represented geometrically as laminations of punctured disks. The geometric complexity of a braid is the minimal complexity of a lamination that represents it, and tight laminations are representatives of minimal complexity. These laminations give rise to a normal form of braids, via a relaxation algorithm. We study here this relaxation algorithm and the associated normal form. We prove that this normal form is regular and prefix-closed. We provide an effective construction of a deterministic automaton that recognizes this normal form.

A homogeneous pattern is a linear multivariate polynomial without constant term. Bras-Amorós and García-Sánchez introduced the notion of pattern for numerical semigroups, which generalizes the definition of Arf numerical semigroups. The notion of pattern for numerical semigroups is extended in this paper into a family of homogeneous patterns ${𝒫}$. A numerical semigroup admitting a family of homogeneous patterns ${𝒫}$ at $(r,d)$-level is characterized. We pay our attention in this paper to the families with the action of some permutation groups, especially those consisting of certain partition stabilizers. Stable or sensitive patterns are focused on and we characterize them for several specific permutation groups.

It is proved that the fundamental groups of boolean representable simplicial complexes (BRSC) are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension $\ge 2$. In the case of dimension 2, it is shown that BRSC have the homotopy type of a wedge of spheres of dimensions 1 and 2. Also, in the case of dimension 2, necessary and sufficient conditions for shellability and being sequentially Cohen–Macaulay are determined. Complexity bounds are provided for all the algorithms involved.