4.11.2013

On the uniqueness of the identity element

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 $\iota = \iota'. \qquad \Box$