Commutative Semigroup is Entropic Structure

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Theorem

A commutative semigroup is an entropic structure.


Proof

Let $\left({S, \circ}\right)$ be a commutative semigroup.

Let $a, b, c, d \in S$.

Then:

\(\displaystyle \left({a \circ b}\right) \circ \left({c \circ d}\right)\) \(=\) \(\displaystyle a \circ \left({b \circ \left({c \circ d}\right)}\right)\) Associativity of $\circ$
\(\displaystyle \) \(=\) \(\displaystyle a \circ \left({\left({b \circ c}\right) \circ d}\right)\) Associativity of $\circ$
\(\displaystyle \) \(=\) \(\displaystyle a \circ \left({\left({c \circ b}\right) \circ d}\right)\) Commutativity of $\circ$
\(\displaystyle \) \(=\) \(\displaystyle a \circ \left({c \circ \left({b \circ d}\right)}\right)\) Associativity of $\circ$
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ c}\right) \circ \left({b \circ d}\right)\) Associativity of $\circ$

$\blacksquare$


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