Editing Order theory (section)

=== Visualizing a poset ===
[[File:Lattice of the divisibility of 60.svg|thumb|right|250px|[[Hasse diagram]] of the set of all divisors of 60, partially ordered by divisibility]]
[[Hasse diagram]]s can visually represent the elements and relations of a partial ordering. These are [[graph drawing]]s where the [[vertex (graph theory)|vertices]] are the elements of the poset and the ordering relation is indicated by both the [[graph theory|edges]] and the relative positioning of the vertices. Orders are drawn bottom-up: if an element ''x'' is smaller than (precedes) ''y'' then there exists a path from ''x'' to ''y'' that is directed upwards. It is often necessary for the edges connecting elements to cross each other, but elements must never be located within an edge. An instructive exercise is to draw the Hasse diagram for the set of natural numbers that are smaller than or equal to 13, ordered by | (the ''[[Divisor|divides]]'' relation).

Even some infinite sets can be diagrammed by superimposing an [[ellipsis]] (...) on a finite sub-order. This works well for the natural numbers, but it fails for the reals, where there is no immediate successor above 0; however, quite often one can obtain an intuition related to diagrams of a similar kind{{vague|date=January 2017}}.