Hasse diagram

From Academic Kids

In the mathematical area of order theory, a Hasse diagram (pronounced HAHS uh, named after Helmut Hasse (1898–1979)) is a simple picture of a finite partially ordered set. Concretely, one represents each member of S as a vertex on the page and draws a line that goes upward from x to y if x < y, and there is no z such that x < z < y. In this case, we say y covers x, or y is an immediate successor of x. Furthermore it is required that the vertices are positioned in such a way that each line meets exactly two vertices: its two endpoints. Any such diagram (given that the vertices are labeled) uniquely determines a partial order, but there are many possible diagrams for specifying a given order.

Contents

Examples

  • The set A = { 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 } of all divisors of 60 is partially ordered by divisibility has the Hasse diagram:
Missing image
Lattice_of_the_divisibility_of_60.png


  • The set of all 15 partitions of the set { 1, 2, 3, 4 } is partially ordered by "refinement", i.e. a finer partition is "less than" a coarser partition, has the Hasse diagram:
Missing image
Lattice_of_partitions_of_an_order_4_set.png


Motivation

If we were to try to create some visual representation of a partially ordered set (S, ≤), how would we proceed? We could begin by first creating a graph, where every node on the graph is an element in S, and every edge (u, v) in that graph would represent the relation u ≤ v.

Doing this, and trying to draw the graph, would result in a graph that would be very "busy". In fact, we carry a lot of redundant information in such a graph. Recall the requirements on a partial order:

  • aa (reflexivity)
  • if ab and bc then ac (transitivity)
  • if ab and ba then a = b (antisymmetry)

Now, in our original graph, we have a number of edges - loops, on each node in the graph - in the form (u, u), because reflexivity means that uu. This must be true for every element in S (otherwise it would not be a partial order).

Say we were now to create a diagram, as above now, without loops, of the partially ordered set ({1,2,3,4}, ≤), where a finer partition of that set is "less than" a coarser partition. We would obtain the following graph:

Missing image
Lattice_of_partitions_of_an_order_4_set_redundant.png


However, in this graph, we still carry redundant information. Referring back to the requirements of a partial order, we see the requirement of transitivity. In the above graph, we are including edges (a,c), (a,b), and (b,c). We do not need to carry the extra edge (a,c) because the other two edges imply the third exists.

This means we need only include an edge between a member of the set, and its immediate predecessor. We do not need the edges to the other predecessors because we have transitivity, nor do we need to draw loops at each edge because we have reflexivity.

If we were to stop here and draw the diagram again according to these new requirements, we obtain the first image above, in the Example section. We can stop here, but it may be useful to define the Hasse diagram in terms of another relation which automatically excludes these cases.

Cover relation

Symbolically, all edges in the Hasse diagram should be of the form (x, y) where x < y (this stricter relation means we exclude cases of loops as before), and that there exists no element z in the set such that x < z < y (this is another way of excluding drawing extra edges because of transitivity).

This relation is known as the cover relation, and the Hasse diagram is often defined in terms of this relation. An element y is said to cover x if y is an immediate successor of x. The (strict) partial ordering is then just the transitive closure of this cover relation.

The Hasse diagram of S may then be defined abstractly as the set of all ordered pairs (x, y) such that y covers x, i.e., the Hasse diagram may be identified with the inverse of the cover relation.

Finding a "good" Hasse diagram

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 are in general many (in fact: infinitely many) possible ways to draw a Hasse diagram for a given poset. Yet, the simple technique of just starting with the minimal elements of an order and then adding greater elements incrementally often produces quite poor results: symmetries and internal structure of the order are easily lost.

The following example demonstrates the problem. Consider the powerset of the set S = {a, b, c, d}, i.e. the set of all subsets of S, written as <math>\mathcal{P}(S)<math>. This powerset can easily be ordered via subset inclusion <math>\subseteq<math>. Below there are three different principal ways to draw a Hasse diagram for this order:

Missing image
Hypercube_order.png


    Missing image
Hypercube_star.png


    Missing image
Hypercube_cubes.png


The labels in these diagrams have been left out to save space, but it should be an easy exercise to assign the subsets of S to the given vertices. The leftmost version is probably closest to the naive way of constructing diagrams: the five layers of the graph represent the numbers of elements that the subsets at each level contain. Note that there are many different ways to assign concrete one-element sets to the second layer, but that this assignment will determine the labels of all other elements. The circumstance that more than one labeling of each of the diagrams is possible reflects the fact that the poset in this example is automorphic — even in many different ways.

The above example demonstrates how different Hasse diagrams for the same order can be, and how each representation can reflect different aspects of the underlying mathematical structure. The first diagram relates the number of elements to the level of each vertex. The second drawing strongly emphasizes the internal symmetry of the structure. Finally, the third one constructs the picture from two cubes such that the relationship between the powerset 2S and the product order 2 × 2{a, b, c} is emphasized.

Various algorithms for drawing better diagrams have been proposed, but today good diagrams still heavily rely on human assistance. However, even humans need quite some practice to draw instructive diagrams.

See also

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