Commutative Semigroup is Entropic Structure

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 $\circ$

$\blacksquare$

Sources

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