Regular Representations of Subset Product

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Theorem

Left Regular Representation of Subset Product

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.


Right Regular Representation of Subset Product

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.