Digroup

In the mathematical area of algebra, a digroup is a generalization of a group that has two one-sided product operations, ${\displaystyle \vdash }$ and ${\displaystyle \dashv }$, 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 ${\displaystyle -x}$ in the integers, for which both the following equations hold: ${\displaystyle (-x)+x=0}$ and ${\displaystyle x+(-x)=0}$. 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 ${\displaystyle x}$ 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, ${\displaystyle \vdash }$ and ${\displaystyle \dashv }$, that satisfy the following laws (e.g., Ongay 2010):

• Associativity:

${\displaystyle \vdash }$ and ${\displaystyle \dashv }$ are associative,

${\displaystyle (x\vdash y)\vdash z=(x\dashv y)\vdash z,}$

${\displaystyle x\dashv (y\dashv z)=x\dashv (y\vdash z),}$

${\displaystyle (x\vdash y)\dashv z=x\vdash (y\dashv z).}$

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

${\displaystyle e\vdash x=x\dashv e=x.}$

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

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

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

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.