Strict weak order ($a\sim b⟺\neg (a<b)\wedge \neg (b<a)$)

An irreflexive transitive comparison whose induced incomparability relation is transitive. Incomparability is then an equivalence relation, and its classes are totally ordered by the comparison. A comparison-based sorting algorithm can treat incomparable elements as ties; a merge algorithm that takes from the left input on ties preserves stable sorting. Ordinary ordering on real numbers is an example; machine NaN values require separate handling.