Digroup

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

• Associativity:

<math>\vdash</math> and <math>\dashv</math> are associative,

<math>(x \vdash y) \vdash z = (x \dashv y) \vdash z,</math>

<math>x \dashv (y \dashv z) = x \dashv (y \vdash z),</math>

<math>(x \vdash y) \dashv z = x \vdash (y \dashv z).</math>

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

<math>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> x \in D</math> has a unique e-inverse, <math>x_e^{-1} \in D</math>, such that

<math>x \vdash x_e^{-1} = x_e^{-1} \dashv x = e.</math>

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.