Comparability

Comparability is sometimes denoted by the symbol ${\displaystyle {\overset {<}{\underset {>}{=}}}}$, and incomparability by the symbol ${\displaystyle {\cancel {\overset {<}{\underset {>}{=}}}}\!}$.[1] Thus, for any pair of elements x and y of a partially ordered set, exactly one of ${\displaystyle x\ {\overset {<}{\underset {>}{=}}}\ y}$ and ${\displaystyle x{\cancel {\overset {<}{\underset {>}{=}}}}y}$ is true.

The comparability graph of a partially ordered set P has as vertices the elements of P and has as edges precisely those pairs {x, y} of elements for which ${\displaystyle x\ {\overset {<}{\underset {>}{=}}}\ y}$.[2]

When classifying mathematical objects (e.g., topological spaces), two criteria are said to be comparable when the objects that obey one criterion constitute a subset of the objects that obey the other, which is to say when they are comparable under the partial order ⊂. For example, the T₁ and T₂ criteria are comparable, while the T₁ and sobriety criteria are not.

• Strict weak ordering, a partial ordering in which incomparability is a transitive relation

"PlanetMath: partial order". Retrieved 6 April 2010.