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Loop
Definition
$X$ |
$ \langle X,* \rangle \in \text{Loop}(X)$ |
$\langle X,* \rangle \in \mathrm{Quasigroup}(X)$ |
$a,a^{-1}\in X$ |
$\forall a.\ \exists a^{-1}.\ (a*a^{-1}=a^{-1}*a=e)$ |
Here we used infix notation for “$*$”.
Ramifications
Discussion
The binary operation is often called multiplication.
The axioms $*\in \mathrm{binaryOp}(X)$ above means that a monoid is closed with respect to the multiplication.
One generally calls $X$ the loop, i.e. the set where the operation “$*$” is defined on.
Reference
Wikipedia: Quasigroup