Element under Right Operation is Left Identity

Theorem
Let $\left({S, \rightarrow}\right)$ be an algebraic structure in which the operation $\rightarrow$ is the right operation.

Then no matter what $S$ is, $\left({S, \rightarrow}\right)$ is a semigroup all of whose elements are left identities.

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

Proof

 * It has been established that $\rightarrow$ is associative.


 * It is also immediately apparent that $\left({S, \rightarrow}\right)$ is closed, from the nature of the right operation:
 * $\forall x, y \in S: x \rightarrow y = y \in S$

whatever $S$ may be.

So $\left({S, \rightarrow}\right)$ is definitely a semigroup.


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

from which it can immediately be seen that all elements of $S$ are indeed left identities.

From More than One Left Identity then No Right Identity, it also follows that there is no right identity.

Also see

 * Left Operation All Elements Right Identities
 * Right Operation All Elements Right Zeroes