# alternative definition of a quasigroup

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:

###### 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 well-defined. 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 AlternativeDefinitionOfAQuasigroup 2013-03-22 18:28:56 2013-03-22 18:28:56 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 20N05 Supercategory  left division right division