Extension Theorem for Distributive Operations
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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 $*$.
Then the following hold:
Existence and Uniqueness
- There exists a unique operation $\circ'$ on $T$ which distributes over $*$ in $T$ and induces on $R$ the operation $\circ$.
Associativity
- If $\circ$ is associative, then so is $\circ'$.
Commutativity
- If $\circ$ is commutative, then so is $\circ'$
Identity
Cancellability
- Every element of $R$ cancellable for $\circ$ is also cancellable for $\circ'$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $20$. The Integers: Theorem $20.8$