Let $\left(X,\tau\right)$ an algebraic structure, where $X\neq\emptyset$ and $\tau: X^2 \rightarrow X$.
Proposition: $\exists ! \iota \in X : \forall x \in X, x \tau \iota = \iota \tau x = x$.
Proof: Suppose another identity $\iota' \in X$.
So if $\iota \tau \iota' = \iota' \tau \iota = \iota$
and $\iota' \tau \iota = \iota \tau \iota' = \iota'$
then $\iota = \iota'. \qquad \Box$
4.11.2013
On the uniqueness of the identity element
Pubblicato da Klein Bottles and other amenities alle 23:43
Etichette: algebra, group theory, identity, identity element
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