# Algebraic Structures formed by Left and Right Operations are not Isomorphic for Cardinality Greater than 1

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## Theorem

Let $S$ be a set.

Let $\gets$ and $\to$ denote the left operation and right operation respectively.

Let $\card S > 1$.

The algebraic structures $\struct {S, \gets}$ and $\struct {S, \to}$ are not isomorphic.

## Proof

Aiming for a contradiction, suppose there exists an isomorphism $\phi$ from $\struct {S, \gets}$ to $\struct {S, \to}$.

Because $\card S > 1$ we have that there exist at least $2$ distinct elements of $S$.

Let these be $x$ and $y$.

Hence:

\(\ds \map \phi {x \gets y}\) | \(=\) | \(\ds \map \phi x\) | Definition of Left Operation | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi x \to \map \phi y\) | Definition of Isomorphism (Abstract Algebra) | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi y\) | Definition of Right Operation |

Hence $\map \phi x = \map \phi y$ and so $\phi$ is not an injection.

Hence $\phi$ is not a bijection and so not an isomorphism.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.11 \ \text {(b)}$