Hasse diagram

In order theory, a Hasse diagram (; German: [ˈhasə]) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set (S, ≤) one represents each element of S as a vertex in the plane and draws a line segment or curve that goes upward from x to y whenever y covers x (that is, whenever xy and there is no z such that xzy). These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order.

The diagrams are named after Helmut Hasse (1898–1979); according to Garrett Birkhoff (1948), they are so called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams. One example that predates Hasse can be found in Henri Gustav Vogt (1895). Although Hasse diagrams were originally devised as a technique for making drawings of partially ordered sets by hand, they have more recently been created automatically using graph drawing techniques.[1]

The phrase "Hasse diagram" may also refer to the transitive reduction as an abstract directed acyclic graph, independently of any drawing of that graph, but this usage is eschewed here.[2][3][4]

Although Hasse diagrams are simple as well as intuitive tools for dealing with finite posets, it turns out to be rather difficult to draw "good" diagrams. The reason is that there will in general be many possible ways to draw a Hasse diagram for a given poset. The simple technique of just starting with the minimal elements 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 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 4-dimensional cube is a combinatorial union of two 3-dimensional cubes, and that a tetrahedron (abstract 3-polytope) likewise merges two triangles (abstract 2-polytopes). The third diagram shows some of the internal symmetry of the structure. In the fourth diagram the vertices are arranged like the elements of a 4×4 matrix.

If a partial order can be drawn as a Hasse diagram in which no two edges cross, its covering graph is said to be upward planar. A number of results on upward planarity and on crossing-free Hasse diagram construction are known:

Expressing the example by standard UML inheritance connectors. Each set is a distinct object (standard UML boxes are rectangular).

The standard diagrama for a chain of inclusions is the UML class, connecting sets by the inheritance relation. The illustration shows a nested set collection, C: