Digroup Explained
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
in the integers, for which both the following equations hold:
and
. 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
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):
and
are associative,
(x\vdashy)\vdashz=(x\dashvy)\vdashz,
x\dashv(y\dashvz)=x\dashv(y\vdashz),
(x\vdashy)\dashvz=x\vdash(y\dashvz).
- Bar units: There is at least one bar unit, an
, such that for every
The set of bar units is called the halo of D.
- Inverse: For each bar unit e, each
has a unique
e-inverse,
, such that
Generalized digroup
In a generalized digroup or g-digroup, a generalization due to Salazar-DÃaz, 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- Keqin Liu (2004), Transformation digroups, unpublished manuscript, arXiv:GR/0409256.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-DÃaz, R. Velásquez, and L. A. Wills-Toro (2016), Generalized digroups, Communications in Algebra, Vol. 44, 2760–2785.
