Digroup

In the mathematical area of algebra, a digroup is a generalization of a group that has two one-sided product operations, $⊢$ and $⊣$, 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, $⊢$ and $⊣$, that satisfy the following laws (e.g., Ongay 2010):

• Associativity:

$⊢$ and $⊣$ are associative,

$(x⊢y)⊢z=(x⊣y)⊢z,$

$x⊣(y⊣z)=x⊣(y⊢z),$

$(x⊢y)⊣z=x⊢(y⊣z).$

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

$e⊢x=x⊣e=x.$

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

$x⊢x_e^{-1}=x_e^{-1}⊣x=e.$

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• Keqin Liu (2004), Transformation digroups, unpublished manuscript, arXiv:GR/0409256.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.