Left Quasigroup if (2-3) Parastrophe of Magma is Magma

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Theorem

Let $\left({S, \circ}\right)$ be a magma.

If the $(2-3)$parastrophe of $\left({S, \circ}\right)$ is a magma then $\left({S, \circ}\right)$ is a left quasigroup.


Proof

By the definition of a left quasigroup it must be shown that:

$\forall a, b \in S: \exists ! x \in S: a \circ x = b$

Suppose there exists $a, b \in S$ such that $a \circ x = b$ does not have a unique solution for $x$.

Then in the $(2-3)$parastrophe of $\left({S, \circ}\right)$ we see that $\circ$ as a mapping either fails to be left-total or many-to-one for $a \circ b$.

So $\left({S, \circ}\right)$ is not a magma which contradicts our assumption.

$\blacksquare$