# Element under Left Operation is Right Identity

## Contents

## 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

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Exercise $4.3 \ \text{(b)}$