Operation Induced by Permutation on Magma is Closed

Theorem

Let $\struct {S, \circ}$ be a magma.

Let $\sigma: S \to S$ be a permutation on $S$.

Let $\circ_\sigma$ be the operation on $S$ induced by $\sigma$:

$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$

Then $\circ_\sigma$ is closed on $S$

Proof

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Suppose $S$ is the empty set.

Let $\sigma: S \to S$ be a permutation on $S$.

Since $S$ is empty, by definition, $\sigma$ is the empty map.

Since $\circ_\sigma$ is the operation on $S$ induced by $\sigma$, it follows that $\circ_\sigma$ is the empty map.

It is vacuously true that $\circ_\sigma$ is closed on $S$, as required.

Suppose $S$ is non-empty.

Let $a, b \in S$.

By definition of magma, $\circ$ is closed on $S$.

Thus:

$a \circ b \in S$

Hence:

$a \circ b \in \Dom \sigma$

As $\sigma$ is a permutation on $S$, it follows directly that:

$\map \sigma {a \circ b} \in S$

The operation on $S$ induced by $\sigma$ is well-defined. In other words, $\circ_\sigma$ is well-defined.

Hence:

$a \circ_\sigma b \in S$

$\blacksquare$