Augmented, free and tensor generalized digroups

Abstract

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.

The structure of g-digroup actions and representation theory

“…In this section we give a short review of some definitions and results about g-digroups, for a deeper study see [14] and [17].…”

“…Theorem 4. ( [14]) Let X be a countable set. Then FD(X) is a free g-digroup, with i : X ֒→ FD(X) given by i(x) = x.…”

“…Due to the nature of the extension of groups to g-digroups, one might think that many definitions and results on group theory can be directly extended to g-digroups, see [14] and [15] in which we extend the concepts of free groups and order and we introduce the concept of tensor product, however, unexpected results could decline the balance, for example, in [15] we proved that Lagrange's theorem is not always true for g-digroups, so we have proposed some variants of Sylows's theorems. Following this line of extending results, in the current paper we introduce, the concept of classical action of g-digroups from two points of view, one is by considering a connection with the representation theory of g-digroups stated in [14], the other one is by extending the definition of action for digroups given in [5]. The latter way is not natural and the orbits are not well defined, therefore a theorem like Burnside's formula is not achieved yet.…”

“…This paper is organized as follows. In section 2, we recall the concepts of g-digroups, g-subdigroups, free g-digroups and the symmetric g-digroup following papers [17], [14] and [15]. In section 3 we introduce the concept of classical action of g-digroups which is a natural extension of the one given in group theory, we prove Burnside's formula for g-digroup actions and a version of Cayley's theorem, weaker than the one given in [15], orbit spaces are also introduced.…”

A geometrical approach to g-digroups

“…Mostovoy (categorical) [11], and M. Bordemann and F. Wagemann (Augmented Leibniz algebras) [12]. Due to the nature of the extension of groups to g-digroups, several results from group theory have been directly extended to g-digroups, see [13] and [14].…”