A commutative loop is a commutativeunital magma$(G,(-)\cdot(-):G \times G \to G,1:G)$ equipped with a binary operation $(-)/(-):G \times G \to G$ called division such that $(x/y) \cdot y = x$ and $(x \cdot y)/y = x$.

With division and identity

A commutative loop is a pointedmagma$(G,/,1)$ such that:

For all $a$ in $G$, $a/a=1$

For all $a$ in $G$, $1/(1/a)=a$

For all $a$ and $b$ in $G$, $a/(1/b) = b/(1/a)$

with multiplication defined as $a \cdot b= a/(1/b)$.

With multiplication, inverses, and identity

A commutative loop is a commutativeunital magma$(G, (-)\cdot(-):G\times G\to G,1:G)$ equipped with a inverse$(-)^{-1}:G \to G$ such that $(x \cdot y^{-1}) \cdot y = x$ and $(x \cdot y) \cdot y^{-1} = x$.