Digroup

In the mathematical subject of algebra, a digroup is a generalization of a group that has two one-sided product operations, $$\vdash$$ and $$\dashv$$, 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 $$-x$$ in the integers, for which both the following equations hold: $$(-x)+x=0$$ and $$x+(-x)=0$$. 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 $$x$$ 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, $$\vdash$$ and $$\dashv$$, that satisfy the following laws (e.g., Ongay 2010):
 * Associativity:
 * $$\vdash$$ and $$\dashv$$ are associative,
 * $$(x \vdash y) \vdash z = (x \dashv y) \vdash z,$$
 * $$x \dashv (y \dashv z) = x \dashv (y \vdash z),$$
 * $$(x \vdash y) \dashv z = x \vdash (y \dashv z).$$


 * Bar units: There is at least one bar unit, an $$e \in D$$, such that for every $$ x \in D,$$
 * $$e \vdash x = x \dashv e = x.$$
 * The set of bar units is called the halo of D.


 * Inverse: For each bar unit e, each $$ x \in D$$ has a unique e-inverse, $$x_e^{-1} \in D$$, such that
 * $$x \vdash x_e^{-1} = x_e^{-1} \dashv x = e.$$

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.