Right Regular Representation of Subset Product

From ProofWiki
Jump to navigation Jump to search


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 $a$.


$\rho_a \sqbrk T = T \circ \set a = T \circ a$

where $T \circ a$ denotes subset product with a singleton.


\(\ds \rho_a \sqbrk T\) \(=\) \(\ds \set {s \in S: \exists t \in T: s = \map {\rho_a} t}\) 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


Also see