Commutative Semigroup is Entropic Structure
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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 $13.12 \ \text{(a)}$