Substructure of Entropic Structure is Entropic

Theorem

Let $\struct {S, \odot}$ be an entropic structure:

$\forall a, b, c, d \in S: \paren {a \odot b} \odot \paren {c \odot d} = \paren {a \odot c} \odot \paren {b \odot d}$

Let $\struct {T, \odot_T}$ be a substructure of $\struct {S, \odot}$.

Then $\struct {T, \odot_T}$ is also an entropic structure.

Proof

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

\(\ds \)

\(\)

\(\ds \paren {a \odot_T b} \odot_T \paren {c \odot_T d}\)

\(\ds \)

\(=\)

\(\ds \paren {a \odot b} \odot \paren {c \odot d}\)

Definition of Operation Induced by Restriction

\(\ds \)

\(=\)

\(\ds \paren {a \odot c} \odot \paren {b \odot d}\)

as $S$ is an entropic structure

\(\ds \)

\(=\)

\(\ds \paren {a \odot_T c} \odot_T \paren {b \odot_T d}\)

Definition of Operation Induced by Restriction

$\blacksquare$