Left Regular Representation of Subset Product
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Theorem
Let $\struct {S, \circ}$ be a magma.
Let $T \subseteq S$ be a subset of $S$.
Let $\lambda_a: S \to S$ be the left regular representation of $S$ with respect to $a$.
Then:
- $\lambda_a \sqbrk T = \set a \circ T = a \circ T$
where $a \circ T$ denotes subset product with a singleton.
Proof
\(\ds \lambda_a \sqbrk T\) | \(=\) | \(\ds \set {s \in S: \exists t \in T: s = \map {\lambda_a} t}\) | Definition of Image of Subset under Mapping | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {s \in S: \exists t \in T: s = a \circ t}\) | Definition of Left Regular Representation | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {a \circ t: t \in T}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a \circ T\) | Definition of Subset Product |
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 41$. Multiplication of subsets of a group