Comparability

 ( Binary Relations ) 

In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true. They are called incomparable if they are not comparable.

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Rigorous definition A binary relation on a set $P$ is by definition any subset $R$ of $P\; \backslash times\; P.$ Given $x,\; y\; \backslash in\; P,$ $x\; R\; y$ is written if and only if $(x,\; y)\; \backslash in\; R,$ in which case $x$ is said to be to $y$ by $R.$ An element $x\; \backslash in\; P$ is said to be ', or ' (), to an element $y\; \backslash in\; P$ if $x\; R\; y$ or $y\; R\; x.$ Often, a symbol indicating comparison, such as (or $\backslash ,\backslash leq\backslash ,,$ $\backslash ,>,\backslash ,$ $\backslash geq,$ and many others) is used instead of $R,$ in which case is written in place of $x\; R\; y,$ which is why the term "comparable" is used.

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

If the symbol is used in place of $\backslash ,\backslash leq\backslash ,$ then comparability with respect to is sometimes denoted by the symbol , and incomparability by the symbol . Thus, for any two elements $x$ and $y$ of a partially ordered set, exactly one of and is true.

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Example A Total order set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states 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.

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Properties Both of the relations and are symmetric, that is $x$ is comparable to $y$ if and only if $y$ is comparable to $x,$ and likewise for incomparability.

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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 $\backslash \{\; x,\; y\; \backslash \}$ of elements for which ..

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Classification When classifying mathematical objects (e.g., topological spaces), two 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 Hausdorff space criteria are comparable, while the T₁ and Sober space criteria are not.

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See also

• , a partial ordering in which incomparability is a transitive relation

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External links

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