Multiplication on R (via Dedekind Cuts)

Multiplication on

R

Let $A, B \in R$ . If $A \geq O$ and $B \geq O$ (see Ordering of R (via Dedekind Cuts) if you're confused) then:

A B = {a b : a \in A, b \in B, w/ a, b \geq 0} \cup O

See Addition on R (via Dedekind Cuts) for the definition of $O$ . The reason for the $\cup O$ would force at least having all the negatives here when $A, B$ are both "positive" in the sense. For the other cases like when $A < 0$ and $B > 0$ :

A B = - [(- A) B]

In general:

A B = {\begin{cases} as given & if A \geq O, B \geq O \\ - [A (- B)] & if A \geq O, B < O \\ - [(- A) B] & if A < O, B \geq O \\ (- A) (- B) & if A < O, B < O \end{cases}

Notice if either $A = O$ or $B = O$ then $A B = O$ since the non- $O$ part in the definition becomes the empty set $\emptyset$ .

We'd have to show:

Left Distributivity
Associativity, Commutativity
The multiplicative identity $I$