The concept of digroup has been proposed as a generalization of continuous groups whose tangent space is a Leibniz algebra. We study a further generalization of the digroup structure in which we do not demand that the inverses are necessarily bilateral. We characterize a generalized digroup as a set and as a product. We show an analogous to the first isomorphism theorem. Additionally, we explore the concept of dialgebra digroup.

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.

We investigate some product structures in R. Thompson’s group $V$ , primarily by studying the topological dynamics associated with $V$ ’s action on the Cantor set $\mathfrak {C}$ . We draw attention to the class $\mathcal {D}_{(V,\mathfrak {C})}$ of groups which have embeddings as demonstrative subgroups of $V$ whose class can be used to assist in forming various products. Note that $\mathcal {D}_{(V,\mathfrak {C})}$ contains all finite groups, the free group on two generators, and $\mathbf {Q}/\mathbf {Z}$ , and is closed under passing to subgroups and under taking direct products of any member by any finite member. If $G\leq V$ and $H\in \mathcal {D}_{(V,\mathfrak {C})}$ , then $G\wr H$ embeds into $V$ . Finally, if $G$ , $H\in \mathcal {D}_{(V,\mathfrak {C})}$ , then $G*H$ embeds in $V$ . Using a dynamical approach, we also show the perhaps surprising result that $Z^2*Z$ does not embed in $V$ , even though $V$ has many embedded copies of $Z^2$ and has many embedded copies of free products of various pairs of its subgroups.

The varieties of dialgebras (also known as Loday-type algebras) over a given type of algebra have been the subject of multiple recent developments. We provide here a construction of such dialgebra varieties via bimodules over an algebra and a surjective equivariant map. Our construction is equivalent to the KP construction (Kolesnikov–Pozhidaev construction) when departing from the set of linearized identities of the algebra variety. The novel construction simplifies the obtention of the dialgebra equations without forcing a complete linearization of the algebra identities. We illustrate the use of the novel construction providing the dialgebras associated to several varieties of algebras, including those over diverse Lie admissible algebras. We provide some novel explorations on the structure of the dialgebras which are easily articulated through our construction.