Element under Left Operation is Right Identity

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

Then no matter what $S$ is, $\struct {S, \leftarrow}$ is a semigroup all of whose elements are right identities.

Thus it can be seen that any right identity in a semigroup is not necessarily unique.

Proof
From Left Operation is Associative, $\leftarrow$ is associative.

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

whatever $S$ may be.

So $\struct {S, \leftarrow}$ is a semigroup.

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

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

From More than one Right Identity then no Left Identity, it also follows that there is no left identity.

Also see

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