Hasse Diagram
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In order theory, a branch of mathematics, a Hasse diagram is a simple picture of a finite partially ordered set, forming a drawing of the transitive reduction of the partial order. Concretely, for a partially ordered set (S, ) one represents each element of S as a vertex on the page and draws a line segment or curve that goes upward from x to y if x y, and there is no z such that x z y (here, is obtained from by removing elements (x,x) for all x). In this case, we say y covers x, or y is an immediate successor of x. Furthermore it is required that the vertices are positioned in such a way that...
In order theory, a branch of mathematics, a Hasse diagram is a simple picture of a finite partially ordered set, forming a drawing of the transitive reduction of the partial order. Concretely, for a partially ordered set (S, ) one represents each element of S as a vertex on the page and draws a line segment or curve that goes upward from x to y if x y, and there is no z such that x z y (here, is obtained from by removing elements (x,x) for all x). In this case, we say y covers x, or y is an immediate successor of x. Furthermore it is required that the vertices are positioned in such a way that each curve meets exactly two vertices: its two endpoints. Any such diagram (given that the vertices are labeled) uniquely determines a partial order, and any finite partial order has a unique transitive reduction, but there are many possible placements of elements in the plane, resulting in different Hasse diagrams for a given order that may have widely varying appearances. 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.
