Digroup

In the mathematical area of algebra, a digroup is a generalization of a group that has two one-sided product operations, $\vdash$ and $\dashv$ , instead of the single operation in a group. Digroups were introduced independently by Liu (2004), Felipe (2006), and Kinyon (2007), inspired by a question about Leibniz algebras.

To explain digroups, consider a group. In a group there is one operation, such as addition in the set of integers; there is a single "unit" element, like 0 in the integers, and there are inverses, like $-x$ in the integers, for which both the following equations hold: $(-x)+x=0$ and $x+(-x)=0$ . A digroup replaces the one operation by two operations that interact in a complicated way, as stated below. A digroup may also have more than one "unit", and an element $x$ may have different inverses for each "unit". This makes a digroup vastly more complicated than a group. Despite that complexity, there are reasons to consider digroups, for which see the references.

Definition

A digroup is a set D with two binary operations, $\vdash$ and $\dashv$ , that satisfy the following laws (e.g., Ongay 2010):

Associativity:

\vdash

and

\dashv

are associative,

(x\vdash y)\vdash z=(x\dashv y)\vdash z,

x\dashv (y\dashv z)=x\dashv (y\vdash z),

(x\vdash y)\dashv z=x\vdash (y\dashv z).

Bar units: There is at least one bar unit, an $e\in D$ , such that for every $x\in D,$

e\vdash x=x\dashv e=x.

The set of bar units is called the halo of D.

Inverse: For each bar unit e, each $x\in D$ has a unique e-inverse, $x_{e}^{-1}\in D$ , such that

x\vdash x_{e}^{-1}=x_{e}^{-1}\dashv x=e.

Generalized digroup

In a generalized digroup or g-digroup, a generalization due to Salazar-Díaz, Velásquez, and Wills-Toro (2016), each element has a left inverse and a right inverse instead of one two-sided inverse.

One reason for this generalization is that it permits analogs of the isomorphism theorems of group theory that cannot be formulated within digroups.

References

Raúl Felipe (2006), Digroups and their linear representations, East-West Journal of Mathematics Vol. 8, No. 1, 27–48.
Michael K. Kinyon (2007), Leibniz algebras, Lie racks, and digroups, Journal of Lie Theory, Vol. 17, No. 4, 99–114.
Keqin Liu (2004), Transformation digroups, unpublished manuscript, arXiv:GR/0409256.
Fausto Ongay (2010), On the notion of digroup, Comunicación del CIMAT, No. I-10-04/17-05-2010.
O.P. Salazar-Díaz, R. Velásquez, and L. A. Wills-Toro (2016), Generalized digroups, Communications in Algebra, Vol. 44, 2760–2785.