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 $\struct {S, \to}$ is a semigroup all of whose elements are left identities.

Proof
From Right Operation is Associative, $\rightarrow$ is associative.

By the nature of the right operation, $\struct {S, \to}$ is closed:
 * $\forall x, y \in S: x \to y = y \in S$

whatever $S$ may be.

So $\struct {S, \to}$ 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