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Comparability
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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 "x≤y if x dividesdividesIn 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 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=539–548|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 ⊂. For example, the T1 and T2 criteria are comparable, while the T1 and sobriety criteria are not.

See also

References

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

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