Magma Subset Product with Self

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Theorem

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

Let $T \subseteq S$.


Then $\left({T, \circ}\right)$ is a magma if and only if $T \circ T \subseteq T$, where $T \circ T$ is the subset product of $T$ with itself.


Proof

By definition:

$T \circ T = \left\{{x = a \circ b: a, b \in T}\right\}$


Necessary Condition

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

Then $T$ is closed.

That is:

$\forall x, y \in T: x \circ y \in T$

Thus:

$x \circ y \in T \circ T \implies x \circ y \in T$

$\Box$


Sufficient Condition

Let $T \circ T \subseteq T$.

Then:

$x \circ y \in T \circ T \implies x \circ y \in T$

That is, $T$ is closed.

Therefore $\left({T, \circ}\right)$ is a magma by definition.

$\blacksquare$


Sources