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 $13.12 \ \text{(e)}$