alternative definition of a quasigroup
In the parent entry, a quasigroup is defined as a set, together with a binary operation^{} on it satisfying two formulas^{}, both of which using existential quantifiers^{}. In this entry, we give an alternative, but equivalent^{}, 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 $\cdot $ (multiplication^{}), $\backslash $ (left division), and $/$ (right division), such that the following are satisfied:

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

•
(left division identities^{}) for all $a,b\in Q$, $a\backslash (a\cdot b)=b$ and $a\cdot (a\backslash b)=b$

•
(right division identities) for all $a,b\in Q$, $(a\cdot b)/b=a$ and $(a/b)\cdot b=a$
Proposition 1.
The two definitions of a quasigroup are equivalent.
Proof.
Suppose $Q$ is a quasigroup using the definition given in the parent entry (http://planetmath.org/LoopAndQuasigroup). Define $\backslash $ on $Q$ as follows: for $a,b\in Q$, set $a\backslash b:=c$ where $c$ is the unique element such that $a\cdot c=b$. Because $c$ is unique, $\backslash $ is welldefined. Now, let $x=a\cdot b$ and $y=a\backslash x$. Since $a\cdot y=x=a\cdot b$, and $y$ is uniquely determined, this forces $y=b$. Next, let $x=a\backslash b$, then $a\cdot x=b$, or $a\cdot (a\backslash b)=b$. Similarly, define $/$ on $Q$ so that $a/b$ is the unique element $d$ such that $d\cdot b=a$. 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 $a,b\in Q$, let $c=a\backslash b$ and $d=b/a$. Then $a\cdot c=a\cdot (a\backslash b)=b$ and $d\cdot a=(b/a)\cdot a=b$. ∎
Title  alternative definition of a quasigroup 

Canonical name  AlternativeDefinitionOfAQuasigroup 
Date of creation  20130322 18:28:56 
Last modified on  20130322 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  Supercategory^{} 
Defines  left division 
Defines  right division 