In order theory, a Hasse diagram ( /ˈhæsə/; 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, ≤) one represents each element of S as a vertex in the plane and draws a line segment or curve that goes upward from x to y whenever y covers x (that is, whenever x < y and there is no z such that x < z < 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 Birkhoff (1948), they are so-called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams; they appear, e.g., in Vogt (1895). 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.
Other articles related to "hasse diagram":
... If a partial order can be drawn as a Hasse diagram in which no two edges cross, its covering graph is said to be upward planar ... A number of results on upward planarity and on crossing-free Hasse diagram construction are known If the partial order to be drawn is a lattice, then it can be drawn without crossings if and only ... element, then it may be tested in linear time whether it has a non-crossing Hasse diagram ...
Famous quotes containing the word diagram:
“If a fish is the movement of water embodied, given shape, then cat is a diagram and pattern of subtle air.”
—Doris Lessing (b. 1919)