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