Element under Left Operation is Right Identity

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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.

$\blacksquare$


Also see


Sources