Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops/Examples/Order 4
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Examples of Use of Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops
Let $S$ have exactly $4$ elements.
There are $96$ 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:
- $n = 4$ by hypothesis
- $m = 4$ from Algebra Loops of Order 4
Hence:
- $N = 4! \times 4 = 96$
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.9 \ \text {(c)}$