Commutative Semigroup is Entropic Structure

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Theorem

A commutative semigroup is an entropic structure.


Proof

Let $\struct {S, \circ}$ be a commutative semigroup.

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

Then:

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

$\blacksquare$


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