Comparability

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.

Rigorous definition
A binary relation on a set $$P$$ is by definition any subset $$R$$ of $$P \times P.$$ Given $$x, y \in P,$$ $$x R y$$ is written if and only if $$(x, y) \in R,$$ in which case $$x$$ is said to be  to $$y$$ by $$R.$$ An element $$x \in P$$ is said to be ', or ', to an element $$y \in P$$ if $$x R y$$ or $$y R x.$$ 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 $$x R y,$$ 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 $$(x, y) \in P \times P$$ 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 $$(x, y) \in P \times P$$ 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 $$\,\leq\,$$ 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
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.

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.

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

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 T2 criteria are comparable, while the T1 and sobriety criteria are not.