Hasse diagram

In order theory, a Hasse diagram (; German: [ˈhasə]) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ( S , ≤ ) {\displaystyle (S,\leq )} one represents each element of S {\displaystyle S} as a vertex in the plane and draws a line segment or curve that goes upward from one vertex x {\displaystyle x} to another vertex y {\displaystyle y} whenever y {\displaystyle y} covers x {\displaystyle x} (that is, whenever x ≠ y {\displaystyle x\neq y} , x ≤ y {\displaystyle x\leq y} and there is no z {\displaystyle z} distinct from x {\displaystyle x} and y {\displaystyle y} with x ≤ z ≤ y {\displaystyle x\leq z\leq y} ). These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. Hasse diagrams are named after Helmut Hasse (1898–1979); according to Garrett Birkhoff, they are so called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams. One example that predates Hasse can be found in an 1895 work by Henri Gustave Vogt. Although Hasse diagrams were originally devised as a technique for making drawings of partially ordered sets by hand, they have more recently been created automatically using graph drawing techniques. In some sources, the phrase "Hasse diagram" has a different meaning: the directed acyclic graph obtained from the covering relation of a partially ordered set, independently of any drawing of that graph.