Idempotent Magma Element forms Singleton Submagma

Theorem

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

Let $x \in S$ be an idempotent element of $\struct {S, \circ}$.

Then $\struct {\set x, \circ}$ is a submagma of $\struct {S, \circ}$.

Proof

By Singleton of Element is Subset:

$x \in S \iff \set x \subseteq S$

By the definition of idempotence:

$x \circ x = x \in \set x$

Thus $\set x$ is a subset of $S$ which is closed under $\circ$.

By the definition of submagma, the result follows.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Example $8.3$