# Element under Right Operation is Left Identity

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## Contents

## Theorem

Let $\struct {S, \rightarrow}$ be an algebraic structure in which the operation $\rightarrow$ is the right operation.

Then $\struct {S, \rightarrow}$ 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

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

By the nature of the right operation, $\struct {S, \rightarrow}$ is closed:

- $\forall x, y \in S: x \rightarrow y = y \in S$

whatever $S$ may be.

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

From the definition of right operation:

- $\forall x, y \in S: x \rightarrow y = y$

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

$\blacksquare$

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

## Also see

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $4$. Groups: Exercise $4$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Exercise $4.3 \ \text{(b)}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Semigroups: Exercise $3 \ \text{(i)}$