Commutative B-Algebra is Entropic Structure

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Theorem

Let $\left({G, *}\right)$ be a commutative $B$-algebra.


Then $\left({G, *}\right)$ is an entropic structure.


Proof

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

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

where $a * b$ is defined by the binary operation in $\left({G, *}\right)$.

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

$\blacksquare$