Commutative Semigroup is Entropic Structure

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 $\circ$

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $13.12 \ \text{(a)}$