Idempotent Magma Element forms Singleton Submagma
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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 $8.3$