Subset Product with Identity

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Let $\struct {S, \circ}$ be a magma.

Let $\struct {S, \circ}$ have an identity element $e$.

Then $e \circ S = S \circ e = S$, where $\circ$ is understood to be the subset product with singleton.


\(\ds e \circ S\) \(=\) \(\ds \set e \circ S\) Definition of Subset Product with Singleton
\(\ds \) \(=\) \(\ds \set {x \circ y: x \in \set e, \, y \in S}\) Definition of Subset Product
\(\ds \) \(=\) \(\ds \set {e \circ y: y \in S}\)
\(\ds \) \(=\) \(\ds \set {y: y \in S}\) Definition of Identity Element
\(\ds \) \(=\) \(\ds S\)


$e \circ S = S$

A similar argument shows that:

$S \circ e = S$