Definition:Multiplication of Cuts

Definition
Let $0^*$ denote the rational cut associated with the (rational) number $0$.

Let $\alpha$ and $\beta$ be cuts.

The operation of multiplication is defined on $\alpha$ and $\beta$ as:
 * $\alpha \beta := \begin {cases}

\size \alpha \, \size \beta & : \alpha \ge 0^*, \beta \ge 0^* \\ -\paren {\size \alpha \, \size \beta} & : \alpha < 0^*, \beta \ge 0^* \\ -\paren {\size \alpha \, \size \beta} & : \alpha \ge 0^*, \beta < 0^* \\ \size \alpha \, \size \beta & : \alpha < 0^*, \beta < 0^* \end {cases}$

where:
 * $\size \alpha$ denotes the absolute value of $\alpha$
 * $\size \alpha \, \size \beta$ is defined as in Multiplication of Positive Cuts
 * $\ge$ denotes the ordering on cuts.

In this context, $\alpha \beta$ is known as the product of $\alpha$ and $\beta$.

Also see

 * Definition:Multiplication of Positive Cuts, which is subsumed by this definition