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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

Context

Subset of

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