alternative definition of a quasigroup

In the parent entry, a quasigroup is defined as a set, together with a binary operationMathworldPlanetmath on it satisfying two formulasMathworldPlanetmathPlanetmath, both of which using existential quantifiersMathworldPlanetmath. In this entry, we give an alternative, but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, definition of a quasigroup using only universally quantified formulas. In other words, the class of quasigroups is an equational class.

Definition. A quasigroup is a set Q with three binary operations (multiplicationPlanetmathPlanetmath), \ (left division), and / (right division), such that the following are satisfied:

  • (Q,) is a groupoid (not in the category theoretic sense)

  • (left division identitiesPlanetmathPlanetmathPlanetmath) for all a,bQ, a\(ab)=b and a(a\b)=b

  • (right division identities) for all a,bQ, (ab)/b=a and (a/b)b=a

Proposition 1.

The two definitions of a quasigroup are equivalent.


Suppose Q is a quasigroup using the definition given in the parent entry ( Define \ on Q as follows: for a,bQ, set a\b:=c where c is the unique element such that ac=b. Because c is unique, \ is well-defined. Now, let x=ab and y=a\x. Since ay=x=ab, and y is uniquely determined, this forces y=b. Next, let x=a\b, then ax=b, or a(a\b)=b. Similarly, define / on Q so that a/b is the unique element d such that db=a. The verification of the two right division identities is left for the reader.

Conversely, let Q be a quasigroup as defined in this entry. For any a,bQ, let c=a\b and d=b/a. Then ac=a(a\b)=b and da=(b/a)a=b. ∎

Title alternative definition of a quasigroup
Canonical name AlternativeDefinitionOfAQuasigroup
Date of creation 2013-03-22 18:28:56
Last modified on 2013-03-22 18:28:56
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 20N05
Related topic SupercategoryPlanetmathPlanetmath
Defines left division
Defines right division