Partial orders, Hasse diagrams, and lattices (intro)

A Hasse diagram is the standard way to visualize a finite poset. The idea is elegantly simple: draw each element as a node, position smaller elements lower and larger elements higher, and connect two nodes with an edge if and only if one covers the other. Crucially, we omit edges implied by transitivity and omit self-loops implied by reflexivity. The result is a clean, minimal diagram from which the full ordering can be recovered by following paths upward.

Reading the diagram is straightforward: a ≤ b in the poset if and only if there is a path going upward from a to b. For instance, 2 ≤ 12 because we can travel 2 → 4 → 12 (or 2 → 6 → 12). Meanwhile, 4 and 6 sit at the same level with no upward path between them, confirming they are incomparable under divisibility. This compact representation strips away all redundant information—every edge in the diagram corresponds to a covering relation, and every other relation can be deduced by following chains of coverings upward. The construction rules are therefore: (1) draw a node for each element, (2) position a below b whenever a < b, (3) draw an edge between a and b only when b covers a, and (4) omit all arrowheads since direction is conveyed by vertical position.