Extension Theorem for Distributive Operations/Associativity
Theorem
Let $\struct {R, *}$ be a commutative semigroup, all of whose elements are cancellable.
Let $\struct {T, *}$ be an inverse completion of $\struct {R, *}$.
Let $\circ$ be an operation on $R$ which distributes over $*$.
Let $\circ'$ be the unique operation on $T$ which distributes over $*$ in $T$ and induces on $R$ the operation $\circ$.
Then:
- If $\circ$ is associative, then so is $\circ'$.
Proof
We have that $\circ'$ exists and is unique by Extension Theorem for Distributive Operations: Existence and Uniqueness.
Suppose $\circ$ is associative.
As $\circ'$ distributes over $*$, for all $n, p \in R$, the mappings:
\(\ds x \mapsto \paren {x \circ' n} \circ' p\) | \(,\) | \(\ds x \in T\) | ||||||||||||
\(\ds x \mapsto x \circ' \paren {n \circ' p}\) | \(,\) | \(\ds x \in T\) |
are endomorphisms of $\struct {T, *}$ that coincide on $R$ by the associativity of $\circ$ and hence are the same mapping.
Therefore:
- $\forall x \in T, n, p \in R: \paren {x \circ' n} \circ' p = x \circ' \paren {n \circ' p}$
Similarly, for all $x \in T, p \in R$, the mappings:
\(\ds y \mapsto \paren {x \circ' y} \circ' p\) | \(,\) | \(\ds y \in T\) | ||||||||||||
\(\ds y \mapsto x \circ' \paren {y \circ' p}\) | \(,\) | \(\ds y \in T\) |
are endomorphisms of $\struct {T, *}$ that coincide on $R$ by what we have proved and hence are the same mapping.
Therefore:
- $\forall x, y \in T, p \in R: \paren {x \circ' y} \circ' p = x \circ' \paren {y \circ' p}$
Finally, for all $x, y \in T$, the mappings:
\(\ds z \mapsto \paren {x \circ' y} \circ' z\) | \(,\) | \(\ds z \in T\) | ||||||||||||
\(\ds z \mapsto x \circ' \paren {y \circ' z}\) | \(,\) | \(\ds z \in T\) |
are endomorphisms of $\struct {T, *}$ that coincide on $R$ by what we have proved and hence are the same mapping.
Therefore $\circ'$ is associative.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $20$. The Integers: Theorem $20.8: 1^\circ$