Duality (mathematics)
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In mathematics, duality has numerous meanings. They are interconnected, mostly, without there being a single master duality. Generally speaking, dualities translate concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion. Often duality is associated with some sort of general operation, where finding the "dual" of an object twice retrieves the original object (hence the "duality"). This does not preclude the possibility that an object is its own dual, but there should be at least some objects which are distinct from their duals.
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[edit] Geometric dualities
In one group of dualities, the concepts and theorems of a certain mathematical theory are mechanically translated into other concepts and theorems of the same theory. The prototypical example here is the duality in projective geometry: given any theorem in plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem. Other examples include:
- dual polyhedron
- dual graph of a planar graph
- dual problem in optimization theory
- De Morgan dual in logic
- duality in order theory
[edit] Contravariant dualities
In another group of dualities, the objects of one theory are translated into objects of another theory and the morphisms between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed. For the general notion in category theory that underlies these dualities, see opposite category. Examples include:
- dual spaces in linear algebra
- Pontryagin duality, relating certain abelian groups to other abelian groups and the background to Fourier analysis
- Tannaka-Krein duality, a non-commutative analogue of Pontryagin duality
- Stone duality, relating Boolean algebras to certain topological spaces
[edit] Poincaré-style dualities
Theorems showing that certain objects of interest are the dual spaces (in the sense of linear algebra) of other objects of interest are also often called dualities. Examples:
[edit] See also
- Dual numbers, a certain associative algebra; the term "dual" here is synonymous with double, and is unrelated to the notions given above.
- Dual graph in graph theory
- Hodge dual
- Duality (electrical engineering)
- Lagrange duality
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