Digroup

In the mathematical subject of algebra, a digroup is a generalization of a group that has two one-sided product operations, [math]\displaystyle{ \vdash }[/math] and [math]\displaystyle{ \dashv }[/math], instead of the single operation in a group. Digroups were introduced independently by Felipe (2006), Kinyon (2007), and Liu (2004).

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 [math]\displaystyle{ -x }[/math] in the integers, for which both the following equations hold: [math]\displaystyle{ (-x)+x=0 }[/math] and [math]\displaystyle{ x+(-x)=0 }[/math]. 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 [math]\displaystyle{ x }[/math] 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, [math]\displaystyle{ \vdash }[/math] and [math]\displaystyle{ \dashv }[/math], that satisfy the following laws (e.g., Ongay 2010):

• Associativity:

[math]\displaystyle{ \vdash }[/math] and [math]\displaystyle{ \dashv }[/math] are associative,

[math]\displaystyle{ (x \vdash y) \vdash z = (x \dashv y) \vdash z, }[/math]

[math]\displaystyle{ x \dashv (y \dashv z) = x \dashv (y \vdash z), }[/math]

[math]\displaystyle{ (x \vdash y) \dashv z = x \vdash (y \dashv z). }[/math]

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

[math]\displaystyle{ e \vdash x = x \dashv e = x. }[/math]

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

• Inverse: For each bar unit e, each [math]\displaystyle{ x \in D }[/math] has a unique e-inverse, [math]\displaystyle{ x_e^{-1} \in D }[/math], such that

[math]\displaystyle{ x \vdash x_e^{-1} = x_e^{-1} \dashv x = e. }[/math]

Generalization

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

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,[5.pdf] 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.