Commutative B-Algebra is Entropic Structure

Theorem

Let $\struct {G, *}$ be a commutative $B$-algebra.

Then $\struct {G, *}$ is an entropic structure.

Proof

From Commutative $B$-Algebra Induces Abelian Group we have that there exists an abelian group $\struct {G, \circ}$ such that:

$\forall a, b \in G: a \circ b^{-1} = a * b$

where $a * b$ is defined by the binary operation in $\struct {G, *}$.

From Abelian Group Induces Entropic Structure, we have directly that $\struct {G, *}$ is an entropic structure.

$\blacksquare$