Comparability

Comparability
See also: Comparison

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.

Contents

Notation

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

  • x ⊥ y, or
  • x || y.

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.[1]

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. For example, the T1 and T2 criteria are comparable, while the T1 and sobriety criteria are not.

References

"PlanetMath: partial order". http://planetmath.org/encyclopedia/PartialOrder.html. Retrieved 6 April 2010. 

  1. ^ Gilmore, P. C.; Hoffman, A. J. (1964), "A characterization of comparability graphs and of interval graphs", Canadian Journal of Mathematics 16: 539–548, http://www.cms.math.ca/cjm/v16/p539 .

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