Restriction of Operation Distributivity

Theorem

Let $\struct {S, *, \circ}$ be an algebraic structure.

Let $T \subseteq S$.

If the operation $\circ$ is distributive over $*$ in $\struct {S, *, \circ}$, then it is also distributive over $*$ on a restriction $\struct {T, * \restriction_T, \circ \restriction_T}$.

Proof

\(\ds \)

\(\)

\(\ds T \subseteq S\)

\(\ds \leadsto \ \ \)

\(\ds \)

\(\)

\(\ds \forall a, b, c \in T: a, b, c \in S\)

Definition of Subset

\(\ds \leadsto \ \ \)

\(\ds \)

\(\)

\(\ds a \circ_T \paren {b *_T c}\)

\(\ds \)

\(=\)

\(\ds a \circ \paren {b * c}\)

\(\ds \)

\(=\)

\(\ds \paren {a \circ b} * \paren {a \circ c}\)

as $\circ$ is distributive over $*$

\(\ds \)

\(=\)

\(\ds \paren {a \circ_T b} *_T \paren {a \circ_T c}\)

Similarly for $\paren {a *_T b} \circ_T c = \paren {a \circ_T c} *_T \paren {b \circ_T c}$.

$\blacksquare$