Quasigroup

Set

context $X$
postulate $ \langle X,* \rangle \in \text{Quasigroup}(X)$
context $\langle X,* \rangle \in \mathrm{Magma}(X)$
range $a,b,x,y\in X$
postulate $ \forall a.\ \forall b.\ \exists x.\ a*x=b $
postulate $ \forall a.\ \forall b.\ \exists y.\ y*a=b $

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 quasigroup, i.e. the set where the operation “$*$” is defined on.

Reference

Wikipedia: Quasigroup

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