Abstract
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.
Highlights
The digroup structure is introduced by M
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
We introduce the notions of generalized tensor digroups and generating sets and we nish with the concept of the semidirect product of generalized digroups
Summary
The digroup structure is introduced by M. A slightly di erent structure studied in [1] is called generalized digroup. It doesn’t request bilateral inverses for its elements. Smith shows that any digroup with bilateral inverses is equivalent to what he calls a ( + )-diquasigroup (Theorem 10.8). His proof uses digroups generated by two groups that act in a commutative way over a set. This idea is similar to a work developed in [9] which leads to express associative dialgebras in terms of bimodules over associative algebras and equivariant maps
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