Element under Left Operation is Right Identity

Theorem

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

Then all of the elements of $\struct {S, \gets}$ are right identities.

Proof

From Structure under Left Operation is Semigroup, $\struct {S, \gets}$ is a semigroup.

From the definition of left operation:

$\forall x, y \in S: x \gets y = x$

from which it is apparent that all elements of $S$ are right identities.

$\blacksquare$

Also see

• Element under Right Operation is Left Identity
• All Elements of Left Operation are Left Zeroes

Sources

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