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{{Wiktionary|comparability}}{{See also|Comparison (mathematics)}}File:Infinite lattice of divisors.svg|thumb|Hasse diagram of the natural numbers, partially ordered by "xâ‰¤y if x dividesdividesIn mathematics, any two elements x and y of a set P that is partially ordered by a binary relation â‰¤ are comparable when either x â‰¤ y or y â‰¤ 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 overset{}{=}}, and incomparability by the symbol cancel{overset{}{=}}}.{{citation|title=Combinatorics and Partially Ordered Sets:Dimension Theory|first=William T.|last=Trotter|publisher=Johns Hopkins Univ. Press|year=1992|pages=3}}Thus, for any pair of elements x and y of a partially ordered set, exactly one of
• x overset{}{=}}y and
• x cancel{overset{}{=}}}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 xoverset{}{=}}y .{{citation|title=A characterization of comparability graphs and of interval graphs|first1=P. C.|last1=Gilmore|first2=A. J.|last2=Hoffman|author2-link=Alan Hoffman (mathematician)|url=http://www.cms.math.ca/cjm/v16/p539|journal=Canadian Journal of Mathematics|volume=16|year=1964|pages=539â€“548|doi=10.4153/CJM-1964-055-5}}.

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

References

WEB,weblink PlanetMath: partial order, 6 April 2010, {{reflist}}

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