Identity of Submagma containing Identity of Magma is Same Identity
Jump to navigation
Jump to search
Theorem
Let $\struct {S, \circ}$ be a magma which has an identity $e$.
Let $\struct {T, \circ}$ be a submagma of $\struct {S, \circ}$ such that $e \in T$.
Then $e$ is the identity of $T$.
Proof
From Identity is Unique, there can be only one identity $e$ of $\struct {S, \circ}$.
We have that:
\(\ds x\) | \(\in\) | \(\ds T\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds S\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x \circ e\) | \(=\) | \(\ds x = e \circ x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall x \in T: \, \) | \(\ds x \circ e\) | \(=\) | \(\ds x = e \circ x\) |
That is, $e$ is the (unique} identity of $T$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets