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:

\(\ds \paren {a \circ b} \circ \paren {c \circ d}\) \(=\) \(\ds a \circ \paren {b \circ \paren {c \circ d} }\) Semigroup Axiom $\text S 1$: Associativity
\(\ds \) \(=\) \(\ds a \circ \paren {\paren {b \circ c} \circ d}\) Semigroup Axiom $\text S 1$: Associativity
\(\ds \) \(=\) \(\ds a \circ \paren {\paren {c \circ b} \circ d}\) Commutativity of $\circ$
\(\ds \) \(=\) \(\ds a \circ \paren {c \circ \paren {b \circ d} }\) Semigroup Axiom $\text S 1$: Associativity
\(\ds \) \(=\) \(\ds \paren {a \circ c} \circ \paren {b \circ d}\) Semigroup Axiom $\text S 1$: Associativity

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.12 \ \text{(a)}$