# Commutative Semigroup is 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 $\circ$

$\blacksquare$