Comparability

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In mathematics, any two elements x and y of a set P that is partially ordered by a binary relation ≤ are comparable when either x ≤ y or y ≤ x. If it is not the case that x and y are comparable, then they are called incomparable.

A totally ordered set is exactly a partially ordered set in which every pair of elements is comparable.

It follows immediately from the definitions of comparability and incomparability that both relations are symmetric, that is x is comparable to y if and only if y is comparable to x, and likewise for incomparability.

Notation

Comparability is denoted by the symbol ⊥, and incomparability by the symbol ||.^[1] Thus, for any pair of elements x and y of a partially ordered set, exactly one of

• x ⊥ y and
• x || y

is true.

Comparability graphs

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 x ⊥ y.^[2]

Classification

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.

See also

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

References

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