Structure Induced by Absorbing Operations is Absorbing

Theorem

Let $\struct {T, \circ, *}$ be an algebraic structure, and let $S$ be a set.

Let $\struct {T^S, \oplus, \otimes}$ be the structure on $T^S$ induced by $\circ$ and $*$.

Let $\circ$ and $*$ satisfy the absorption law:

$\forall a, b \in S: a \circ \paren {a * b} = a$

Then the pointwise operations $\oplus$ and $\otimes$ on $T^S$ also satisfy the absorption law:

$\forall f, g \in T^S: f \oplus \paren {f \otimes g} = f$

Proof

Let $f, g \in T^S$.

Then:

\(\ds \forall x \in S: \, \)

\(\ds \map {\paren {f \oplus \paren {f \otimes g} } } x\)

\(=\)

\(\ds \map f x \circ \map {\paren {f \otimes g} } x\)

Definition of Pointwise Operation

\(\ds \)

\(=\)

\(\ds \map f x \circ \paren {\map f x * \map g x}\)

Definition of Pointwise Operation

\(\ds \)

\(=\)

\(\ds \map f x\)

$\circ$ absorbs $*$

From Equality of Mappings:

$f \oplus \paren {f \otimes g} = f$

Since $f, g$ were arbitrary:

$\forall f, g \in T^S : f \oplus \paren {f \otimes g} = f$

Hence $\oplus$ absorbs $\otimes$ by definition.

$\blacksquare$