Editing Digroup

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In the mathematical area of [[algebra]], a '''digroup''' is a generalization of a [[Group (mathematics)|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 algebra]]s.

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 operation]]s,  <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==
{{Reflist}}
*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), [https://www.cimat.mx/BiblioAdmin/RTAdmin/reportes/enlinea/I-10-04.pdf 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.

[[Category:Abstract algebra]]