# Left Regular Representation of Subset Product

## 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

 $\displaystyle \lambda_a \sqbrk T$ $=$ $\displaystyle \set {s \in S: \exists t \in T: s = \map {\lambda_a} t}$ Definition of Image of Subset under Mapping $\displaystyle$ $=$ $\displaystyle \set {s \in S: \exists t \in T: s = a \circ t}$ Definition of Left Regular Representation $\displaystyle$ $=$ $\displaystyle \set {a \circ t: t \in T}$ $\displaystyle$ $=$ $\displaystyle a \circ T$ Definition of Subset Product

$\blacksquare$