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

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

If the symbol ${\displaystyle \,<\,}$  is used in place of ${\displaystyle \,\leq \,}$  then comparability with respect to ${\displaystyle \,<\,}$  is sometimes denoted by the symbol ${\displaystyle {\overset {<}{\underset {>}{=}}}}$ , and incomparability by the symbol ${\displaystyle {\cancel {\overset {<}{\underset {>}{=}}}}\!}$ .^[1] Thus, for any two elements ${\displaystyle x}$  and ${\displaystyle y}$  of a partially ordered set, exactly one of ${\displaystyle x\ {\overset {<}{\underset {>}{=}}}\ y}$  and ${\displaystyle x{\cancel {\overset {<}{\underset {>}{=}}}}y}$  is true.

A totally ordered 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.

Both of the relations comparability and incomparability are symmetric, that is ${\displaystyle x}$  is comparable to ${\displaystyle y}$  if and only if ${\displaystyle y}$  is comparable to ${\displaystyle x,}$  and likewise for incomparability.

The comparability graph of a partially ordered set ${\displaystyle P}$  has as vertices the elements of ${\displaystyle P}$  and has as edges precisely those pairs ${\displaystyle \{x,y\}}$  of elements for which ${\displaystyle x\ {\overset {<}{\underset {>}{=}}}\ y}$ .^[2]

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.

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

• "PlanetMath: partial order". Archived from the original on 11 July 2012. Retrieved 6 April 2010.