Left Regular Representation of Subset Product

From ProofWiki
Jump to navigation Jump to search

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

$\blacksquare$


Sources