Hasse diagram

A Hasse diagram represents a finite partially ordered set by drawing its elements as points and its cover relations as edges, usually placing larger elements above smaller ones.

Partially ordered set: A set equipped with a reflexive, antisymmetric, and transitive relation. A Hasse diagram is a compact drawing of a finite partially ordered set.
Order ideal: A subset of a partially ordered set containing every element below each of its members. Its elements form a downward-closed region in a Hasse diagram.
Divisibility: The relation in which one integer divides another without remainder. Divisibility among the divisors of an integer produces a classic Hasse diagram.
Directed graph: A graph whose edges have assigned directions. Unlike a general directed graph, a Hasse diagram suppresses transitive edges and encodes a partial order.
Cover relation: A relation in a partial order where one element is directly above another, with no intermediate element. Precisely the relations drawn as edges in a Hasse diagram.
Maximal element: An element of a partially ordered set with no strictly greater element. Maximal elements appear at the top boundary, though several may occur.
Boolean lattice: The partially ordered set of all subsets of a finite set, ordered by inclusion. Its diagram connects subsets that differ by adding one element.
Lattice diagram: A drawing of a lattice that displays its elements and order relations. Lattice diagrams use Hasse-style edges but represent the more specific structure of a lattice.
Partial order: A reflexive, antisymmetric, and transitive relation on a set. The underlying relation determines which elements are comparable in the diagram.
Minimal element: An element of a partially ordered set with no strictly smaller element. Minimal elements appear at the bottom boundary, though several may occur.

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