Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops

Theorem
Let $S$ be a finite set of cardinality $n$.

Let $e \in S$

Let there be $m$ commutative operations $\oplus$ on $S$ such that $\struct {S, \oplus}$ is an algebra loop whose identity is $e$.

Then there are $n! m$ commutative binary operations $\otimes$ on $S$ such that $\struct {S, \otimes}$ is a quasigroup.