Right Operation has no Right Identities

Theorem

Let $S$ be a set with more than $1$ element.

Let $\struct {S, \to}$ be an algebraic structure in which the operation $\to$ is the right operation.

Then $\struct {S, \to}$ has no right identities.

Proof

From Element under Right Operation is Left Identity, every element of $\struct {S, \to}$ is a left identity.

Because there are at least $2$ elements in $\struct {S, \to}$, it follows that $\struct {S, \to}$ has more than one left identity.

From More than one Left Identity then no Right Identity, it follows that $\struct {S, \to}$ has no right identity.

$\blacksquare$

Also see

• Left Operation has no Left Identities

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.3 \ \text{(b)}$