The concept of generalized digroup was proposed by Salazar-Díaz, Velásquez and Wills-Toro in their paper “Generalized digroups” as a non trivial extension of groups. In this way, many concepts and results given in the category of groups can be extended in a natural form to the category of generalized digroups. The aim of this paper is to present the construction of the free generalized digroup and study its properties. Although this construction is vastly different from the one given for the case of groups, we will use this concept, the classical construction for groups and the semidirect product to construct the tensor generalized digroup as well as the semidirect product of generalized digroups. Additionally, we give a new structural result for generalized digroups using compatible actions of groups and an equivariant map from a group set to the group corresponding to notions of associative dialgebras and augmented racks.

Brauer configuration algebras are path algebras induced by appropriated multiset systems. Since their structures underlie combinatorial data, the general description of some of their algebraic invariants (e.g., their dimensions or the dimensions of their centers) is a hard problem. Integer partitions and compositions of a given integer number are examples of multiset systems which can be used to define Brauer configuration algebras. This paper gives formulas for the dimensions of Brauer configuration algebras (and their centers) induced by some integer partitions. As an application of these results, we give examples of Brauer configurations, which can be realized as branch data of suitable branched coverings over different surfaces.

Este texto analiza algunos aspectos que rodean la incorporación de la innovación y de la investigación en el mundo escolar como parte de la práctica pedagógica y sus implicaciones para los maestros, centrando la atención en los sujetos y no en los procesos o en los resultados de las innovaciones y de las investigaciones mismas. Las reflexiones que aquí se presentan surgen de la experiencia de los autores, quienes por más de una década han trabajado en procesos de formación de profesores en servicio, investigación sobre vidas de maestros y acompañamiento a procesos de innovación e investigación escolar.

We study g-digroup actions by proving some results that are well known in group actions, for example: the orbit stabilizer theorem, the class equation, Burnside’s formula, and we relate g-digroup actions with trioids. We introduce the definition of a representation of a g-digroup into dialgebras and establish a strong connection with g-digroup action, like in the group case.