Editing Hasse diagram (section)

==Diagram design==

Although Hasse diagrams are simple, as well as intuitive, tools for dealing with finite [[Partially ordered set|posets]], it turns out to be rather difficult to draw "good" diagrams. The reason is that, in general, there are many different possible ways to draw a Hasse diagram for a given poset. The simple technique of just starting with the [[minimal element]]s of an order and then drawing greater elements incrementally often produces quite poor results: symmetries and internal structure of the order are easily lost.

The following example demonstrates the issue. Consider the [[power set]] of a 4-element set ordered by inclusion <math>\subseteq</math>. Below are four different Hasse diagrams for this partial order. Each subset has a node labelled with a binary encoding that shows whether a certain element is in the subset (1) or not (0):

{| style="margin: 0 auto;"
| [[File:Hypercubeorder binary.svg|230px|]] ||&nbsp;&nbsp;&nbsp;|| [[File:Hypercubecubes binary.svg|260px|]] 
|-
|[[File:Hypercubestar binary.svg|240px|]]
|&nbsp;&nbsp;&nbsp;
|[[File:Hypercubematrix_binary.svg|center|229x229px]]
|}

The first diagram makes clear that the power set is a [[graded poset]]. The second diagram has the same graded structure, but by making some edges longer than others, it emphasizes that the [[tesseract|4-dimensional cube]] is a combinatorial union of two 3-dimensional cubes, and that a tetrahedron ([[abstract polytope|abstract 3-polytope]]) likewise merges two triangles ([[abstract polytope|abstract 2-polytopes]]).  The third diagram shows some of the internal symmetry of the structure. In the fourth diagram the vertices are arranged in a 4×4 grid.