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

If $e$ is an identity for $\circ$, then $e$ is also an identity for $\circ'$.


Cancellability

Every element of $R$ cancellable for $\circ$ is also cancellable for $\circ'$.


Sources