Helmut Hasse in the context of "Jacques Herbrand"

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 ${\displaystyle (S,\leq )}$ one represents each element of ${\displaystyle S}$ as a vertex in the plane and draws a line segment or curve that goes upward from one vertex ${\displaystyle x}$ to another vertex ${\displaystyle y}$ whenever ${\displaystyle y}$ covers ${\displaystyle x}$ (that is, whenever ${\displaystyle x\neq y}$, ${\displaystyle x\leq y}$ and there is no ${\displaystyle z}$ distinct from ${\displaystyle x}$ and ${\displaystyle y}$ with ${\displaystyle x\leq z\leq y}$). 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 mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.

In mathematics, local class field theory (LCFT), introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Q_p (where p is any prime number), or the field of formal Laurent series F_q((T)) over a finite field F_q.