Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops/Examples/Order 3

Examples of Use of Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops
Let $S$ have exactly $3$ elements.

There are $6$ quasigroups $\struct {S, \otimes}$ on $S$ such that $\otimes$ is a commutative operation.

Proof
Let $e \in S$ be a distinguished element of $S$.

Let $N$ be the number of quasigroups $\struct {S, \otimes}$ on $S$ such that $\otimes$ is a commutative operation.

From Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops we have that:
 * $N = n! m$

where:
 * $n$ is the cardinality of $S$
 * $m$ is the number of commutative operations $\oplus$ on $S$ such that $\struct {S, \oplus}$ is an algebra loop whose identity is $e$.

In this case, we have:
 * $n = 3$
 * $m = 1$ from Algebra Loops of Order 3

Hence:
 * $N = 3! \times 1 = 6$