Loop

Set

context $X$
postulate $ \langle X,* \rangle \in \text{Loop}(X)$
context $\langle X,* \rangle \in \mathrm{Quasigroup}(X)$
range $e,a\in X$
postulate $\exists e.\ \forall a.\ (a*e=e*a=a) $

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

Parents

Subset of

Quasigroup