# Comparability

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**.

Look up in Wiktionary, the free dictionary.comparability |

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

- and

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 .[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 T_{1} and T_{2} criteria are comparable, while the T_{1} 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.

- Trotter, William T. (1992),
*Combinatorics and Partially Ordered Sets:Dimension Theory*, Johns Hopkins Univ. Press, p. 3 - Gilmore, P. C.; Hoffman, A. J. (1964), "A characterization of comparability graphs and of interval graphs",
*Canadian Journal of Mathematics*,**16**: 539–548, doi:10.4153/CJM-1964-055-5.