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Hasse diagram
Visual depiction of a partially ordered set / From Wikipedia, the free encyclopedia
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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 one represents each element of
as a vertex in the plane and draws a line segment or curve that goes upward from one vertex
to another vertex
whenever
covers
(that is, whenever
,
and there is no
distinct from
and
with
). 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.
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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.[1] 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.[2][3] 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.[4]
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.[5]