Operation which is Right Distributive over Every Commutative Associative Operation is Left Operation
Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ have the property that:
- for every arbitrary operation $*$ on $S$ which is both commutative and associative, $\circ$ is right distributive over $*$.
Then $\circ$ is the left operation $\gets$:
- $\forall a, b \in S: a \gets b = a$
Proof
First recall from Left Operation is Right Distributive over All Operations that the left operation is indeed right distributive over all operations, whether commutative or associative.
Let $*$ be an arbitrary operation on $S$ which is both commutative and associative.
As asserted, let $\circ$ be right distributive over $*$.
Let $c \in S$ be arbitrary.
Consider the constant operation $\sqbrk c$:
- $\forall a, b \in S: a \sqbrk c b := c$
We have from Constant Operation is Commutative and Constant Operation is Associative that $\sqbrk c$ is both commutative and associative.
Hence $\circ$ must be right distributive over $\sqbrk c$.
But then from Condition for Operation to be Right Distributive over Constant Operation:
- $c \circ x = c$
As $c$ is arbitrary:
- $\forall a, b \in S: a \circ b = a$
so $\circ$ has to be the left operation.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.25$