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 $6.11 \ \text {(b)}$