Element under Right Operation is Left Identity

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

Then all of the elements of $\struct {S, \to}$ left identities.

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

From the definition of right operation:
 * $\forall x, y \in S: x \to y = y$

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

Also see

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