Hasse diagram

In mathematics, a Hasse diagram is (also ordering or simply line graph called ) a certain graphical representation of finite partially ordered sets. Such diagrams are named after the mathematician Helmut Hasse.

The Hasse diagram for a partial order arises as a representation of a directed graph, where are the elements of the node. Two nodes and are connected by an edge if and only if and there is no node with (Here as and understand ). The direction of the edge is thus expressed that the node is above. Such an arrangement can be achieved, since the Hasse diagram is cycle-free. Loops at reflexivity be omitted.

Sometimes Hasse diagrams are also used to represent Strict orders ( order relations of the second kind ).

Examples

Divider Association

The divisor of a natural number can be represented by a Hasse diagram, since they form with respect to the divisibility half ordered set ( " divider Association "). The following figure shows the Hasse diagram of the divisors of 60

Partitions

The set of partitions of the set { 1, 2, 3, 4} with the " fineness " as a partial order.

Power set

The -element power set of an n- element set with the set inclusion can be represented as a Hasse diagram. Here are the elements of the power set of the nodes and two elements are connected by an edge if they are in a subset relation. The empty set represented by the lowest node is a subset of all elements; the universe represented by the top node is a superset of all elements.

Particularly clear and widespread is the arrangement of the volumes that contain the same number of elements in the same plane of the Hasse diagram - just as it is common and recommended to arrange the quantities in the plains from left to right lexicographically.

A small example of a Hasse diagram of a power set gives the crowd:

A slightly more elaborate diagram obtained with the sechzehnelementige power set of a four element amount, this is of particular importance for propositional logic and set theory. Your most obvious in the manner described representation is the left of the three graphs showing the rhombendodekaedrischen three-dimensional shadow of a four-dimensional cube. The other two graphs on the right show the rhombendodekaedrischen also possible Hasse diagrams of the power set of a four element amount, which may be more suitable for some purposes than the stratification according to the number of elements. Graphical representations that are ideal for all purposes, there is not. You must take appropriate Hasse diagrams in dealing with a specific topic are only found often.