Digroup


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, Felipe, and Kinyon, 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 :
  • Associativity:
  • Bar units: There is at least one bar unit, an, such that for every
  • 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-Díaz, Velásquez, and Wills-Toro, 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.