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.

Also see

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