Identity is Unique/Proof 1
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure that has an identity element $e \in S$.
Then $e$ is unique.
Proof
Suppose $e_1$ and $e_2$ are both identity elements of $\struct {S, \circ}$.
Then by the definition of identity element:
- $\forall s \in S: s \circ e_1 = s = e_2 \circ s$
Then:
- $e_1 = e_2 \circ e_1 = e_2$
So:
- $e_1 = e_2$
and there is only one identity element after all.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.3$. Units and zeros
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Theorem $4.1$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups: Theorem $1.2$