Entropic Structure with Identity is Commutative Monoid
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Theorem
Let $\struct {S, \odot}$ be a magma.
Let $\struct {S, \odot}$ be an entropic structure:
- $\forall a, b, c, d \in S: \paren {a \odot b} \odot \paren {c \odot d} = \paren {a \odot c} \odot \paren {b \odot d}$
Let $\struct {S, \odot}$ have an identity element $e$.
Then $\struct {S, \odot}$ is a commutative monoid.
Proof
We have that $\struct {S, \odot}$ is a magma.
Thus a fortiori $\struct {S, \odot}$ is closed, and Monoid Axiom $\text S 0$: Closure is fulfilled.
Then:
\(\ds \forall a, b, c \in S: \, \) | \(\ds \paren {a \odot b} \odot c\) | \(=\) | \(\ds \paren {a \odot b} \odot \paren {e \odot c}\) | Definition of Identity Element | ||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \odot e} \odot \paren {b \odot c}\) | Definition of Entropic Structure | |||||||||||
\(\ds \) | \(=\) | \(\ds a \odot \paren {b \odot c}\) | Definition of Identity Element |
Thus $\odot$ is an associative operation and Monoid Axiom $\text S 1$: Associativity is fulfilled.
We are given that $e$ is an identity element for $\struct {S, \odot}$.
Hence Monoid Axiom $\text S 2$: Identity is fulfilled by hypothesis.
Thus we have that $\struct {S, \odot}$ is a monoid.
Then:
\(\ds \forall a, b \in S: \, \) | \(\ds a \odot b\) | \(=\) | \(\ds \paren {e \odot a} \odot \paren {b \odot e}\) | Definition of Identity Element | ||||||||||
\(\ds \) | \(=\) | \(\ds \paren {e \odot b} \odot \paren {a \odot e}\) | Definition of Entropic Structure | |||||||||||
\(\ds \) | \(=\) | \(\ds b \odot a\) | Definition of Identity Element |
Thus $\struct {S, \odot}$ is a commutative monoid.
$\blacksquare$
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 $13.12 \ \text{(e)}$