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Digroup

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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

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A digroup is a set D with two binary operations, and , that satisfy the following laws (e.g., Ongay 2010):

  • Associativity:
and are associative,
  • 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

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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

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  • Keqin Liu (2004), Transformation digroups, unpublished manuscript, arXiv:GR/0409256.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