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 $13.12 \ \text{(a)}$