Hasse diagram

A Hasse diagram is the standard way to draw a finite partially ordered set so you can see its structure at a glance. The trick is to draw only the bare minimum of edges: you connect two elements just when one sits immediately above the other, with nothing in between, and let transitivity fill in the rest implicitly. Reading upward means ‘greater’, so the picture compresses all the order information into a clean ranked graph.

Formally, in a poset (P, ≤) one says b covers a, written a ⋖ b, if a < b and there is no c with a < c < b. The Hasse diagram is the graph whose vertices are the elements of P and whose edges are exactly the covering pairs, drawn so that b is positioned higher than a whenever a ⋖ b. One then recovers the full order ≤ as the reflexive-transitive closure: a ≤ b iff there is an upward path from a to b. Reflexive loops and all edges implied by transitivity are deliberately omitted.

Hasse diagrams are indispensable for visualizing lattices and small posets — meets and joins become ‘lowest common ancestor below’ and ‘highest common descendant above’, and special lattices acquire recognizable silhouettes. The diamond M_3 and the pentagon N_5 are literally named for their Hasse-diagram shapes, and the forbidden-sublattice tests for distributive and modular lattices are usually stated and checked diagrammatically. Caveat: the diagram requires the covering relation to make sense, so it is meaningful chiefly for finite (or at least well-founded, locally finite) posets.