Right Regular Representation of Subset Product

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

• Left Regular Representation of Subset Product

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 41$. Multiplication of subsets of a group