Definition:Strict Weak Ordering

A strict weak ordering on a set $$S$$ is a relation $$\mathcal{R}$$ such that:


 * $$\mathcal{R}$$ is a strict partial ordering;


 * The incomparability relation $$\mathcal{R}'$$ defined as:


 * $$a \mathcal{R}' b \ \stackrel {\mathbf {def}} {=\!=} \ \neg a \mathcal{R} b \and \neg b \mathcal{R} a$$

is transitive.