hassediagram

A Hasse diagram is a graphical representation of a finite partially ordered set (poset). In such a diagram, each element is depicted as a point, and there is an edge from x to y whenever x < y and there is no element z with x < z < y; that is, the edge represents a cover relation. Edges are drawn as unobstructed line segments, and the diagram is arranged so that higher elements lie above lower ones. Transitive relations are not shown as separate edges; the diagram is the transitive reduction of the order.

Construction and conventions: The diagram is often arranged by rank or level, with minimal elements at the bottom and maximal elements at the top. The exact placement is not unique, and different drawings can represent the same poset. The orientation is conventional but can vary; the essential point is that if x < y then y appears above x in the diagram. Edges typically have no arrows since the vertical relation already encodes the direction of the order.

Properties and use: Hasse diagrams convey the structure of a poset visually and are especially helpful for finite lattices and other order-theoretic notions. They form directed acyclic graphs (DAGs) when edges are oriented upward. Reachability in the diagram corresponds to the order relation. They can aid in identifying meets (greatest lower bounds) and joins (least upper bounds) when these operations exist, though the existence and computation of meets and joins may require additional information about the poset.

Examples: Classic examples include the divisibility poset of the positive divisors of a number, ordered by divisibility; the boolean lattice of the power set of a finite set, ordered by inclusion; and the lattice of partitions of a finite set, ordered by refinement. The concept is named after Helmut Hasse, who introduced this visualization in order theory.