Comparability

See also:Comparison (mathematics)

Hasse diagramof thenatural numbers,partially ordered by "x≤yifxdividesy".The numbers 4 and 6 are incomparable, since neither divides the other.

Inmathematics,two elementsxandyof a setPare said to becomparablewith respect to abinary relation≤ if at least one ofx≤yory≤xis true. They are calledincomparableif they are not comparable.

Rigorous definition

Abinary relationon a set$P$is by definition any subset$R$of$P\times P.$Given$x,y\in P,$$xRy$is written if and only if$(x,y)\in R,$in which case$x$is said to berelatedto$y$by$R.$ An element$x\in P$is said to be$R$-comparable,orcomparable(with respect to$R$), to an element$y\in P$if$xRy$or$yRx.$ Often, a symbol indicating comparison, such as$\,<\,$(or$\,\leq \,,$$\,>,\,$$\geq,$and many others) is used instead of$R,$in which case$x<y$is written in place of$xRy,$which is why the term "comparable" is used.

Comparability with respect to$R$induces a canonical binary relation on$P$;specifically, thecomparability relationinduced by$R$is defined to be the set of all pairs$(x,y)\in P\times P$such that$x$is comparable to$y$;that is, such that at least one of$xRy$and$yRx$is true. Similarly, theincomparability relationon$P$induced by$R$is defined to be the set of all pairs$(x,y)\in P\times P$such that$x$is incomparable to$y;$that is, such that neither$xRy$nor$yRx$is true.

If the symbol$\,<\,$is used in place of$\,\leq \,$then comparability with respect to$\,<\,$is sometimes denoted by the symbol${\overset {<}{\underset {>}{=}}}$,and incomparability by the symbol${\cancel {\overset {<}{\underset {>}{=}}}}\!$. Thus, for any two elements$x$and$y$of a partially ordered set, exactly one of$x\ {\overset {<}{\underset {>}{=}}}\ y$and$x{\cancel {\overset {<}{\underset {>}{=}}}}y$is true.

Example

Atotally orderedset is apartially ordered setin which any two elements are comparable. TheSzpilrajn extension theoremstates that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable.

Properties

Both of the relationscomparabilityandincomparabilityaresymmetric,that is$x$is comparable to$y$if and only if$y$is comparable to$x,$and likewise for incomparability.

Comparability graphs

Main article:Comparability graph

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$x\ {\overset {<}{\underset {>}{=}}}\ y$.

Classification

Whenclassifyingmathematical objects (e.g.,topological spaces), twocriteriaare 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, theT₁andT₂criteria are comparable, while the T₁andsobrietycriteria are not.

See also

• Strict weak ordering– Mathematical ranking of a setPages displaying short descriptions of redirect targets,a partial ordering in which incomparability is atransitive relation

External links

• "PlanetMath: partial order".Archived fromthe originalon 11 July 2012.Retrieved6 April2010.