Left Operation has no Left Identities
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Theorem
Let $S$ be a set with more than $1$ element.
Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\gets$ is the left operation.
Then $\struct {S, \gets}$ has no left identities.
Proof
From Element under Left Operation is Right Identity, every element of $\struct {S, \gets}$ is a right identity.
Because there are at least $2$ elements in $\struct {S, \gets}$, it follows that $\struct {S, \gets}$ has more than one right identity.
From More than one Right Identity then no Left Identity, it follows that $\struct {S, \gets}$ has no left identity.
$\blacksquare$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.3 \ \text{(b)}$