Duality (mathematics)


In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often by means of an involution operation: if the dual of is, then the dual of is. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original. Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in mathematics" and "an important general theme that has manifestations in almost every area of mathematics".
Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.
From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow its dual.

Introductory examples

In the words of Michael Atiyah,
The following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case.

Complement of a subset

A simple duality arises from considering subsets of a fixed set. To any subset, the complement consists of all those elements in that are not contained in. It is again a subset of. Taking the complement has the following properties:
  • Applying it twice gives back the original set, i.e.,. This is referred to by saying that the operation of taking the complement is an involution.
  • An inclusion of sets is turned into an inclusion in the opposite direction.
  • Given two subsets and of, is contained in if and only if is contained in.
This duality appears in topology as a duality between open and closed subsets of some fixed topological space : a subset of is closed if and only if its complement in is open. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any intersection of closed sets is closed. The interior of a set is the largest open set contained in it, and the closure of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any set is equal to the closure of the complement of.

Dual cone

A duality in geometry is provided by the dual cone construction. Given a set of points in the plane
The other two properties carry over without change:
  • It is still true that an inclusion is turned into an inclusion in the opposite direction.
  • Given two subsets and of the plane, is contained in if and only if is contained in.

    Dual vector space

A very important example of a duality arises in linear algebra by associating to any vector space its dual vector space. Its elements are the linear functionals, where is the field over which is defined.
The three properties of the dual cone carry over to this type of duality by replacing subsets of by vector space and inclusions of such subsets by linear maps. That is:
  • Applying the operation of taking the dual vector space twice gives another vector space. There is always a map. For some, namely precisely the finite-dimensional vector spaces, this map is an isomorphism.
  • A linear map gives rise to a map in the opposite direction.
  • Given two vector spaces and, the maps from to correspond to the maps from to.
A particular feature of this duality is that and are isomorphic for certain objects, namely finite-dimensional vector spaces. However, this is in a sense a lucky coincidence, for giving such an isomorphism requires a certain choice, for example the choice of a basis of. This is also true in the case if is a Hilbert space, via the Riesz representation theorem.

Galois theory

In all the dualities discussed before, the dual of an object is of the same kind as the object itself. For example, the dual of a vector space is again a vector space. Many duality statements are not of this kind. Instead, such dualities reveal a close relation between objects of seemingly different nature. One example of such a more general duality is from Galois theory. For a fixed Galois extension, one may associate the Galois group to any intermediate field . This group is a subgroup of the Galois group. Conversely, to any such subgroup there is the fixed field consisting of elements fixed by the elements in.
Compared to the [|above], this duality has the following features:
  • An extension of intermediate fields gives rise to an inclusion of Galois groups in the opposite direction:.
  • Associating to and to are inverse to each other. This is the content of the fundamental theorem of Galois theory.

    Order-reversing dualities

Given a poset , the dual poset comprises the same ground set but the converse relation. Familiar examples of dual partial orders include
  • the subset and superset relations and on any collection of sets, such as the subsets of a fixed set. This gives rise to the first example of a duality mentioned above.
  • the divides and multiple-of relations on the integers.
  • the descendant-of and ancestor-of relations on the set of humans.
A duality transform is an involutive antiautomorphism of a partially ordered set, that is, an order-reversing involution. In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, if, are two duality transforms then their composition is an order automorphism of ; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a power set are induced by permutations of.
A concept defined for a partial order will correspond to a dual concept on the dual poset. For instance, a minimal element of will be a maximal element of : minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are upper and lower bounds, lower sets and upper sets, and ideals and filters.
In topology, open sets and closed sets are dual concepts: the complement of an open set is closed, and vice versa. In matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid.

Dimension-reversing dualities

There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of the objects. A classical example of this is the duality of the Platonic solids, in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with the faces of the primal. Similarly, each edge of the dual corresponds to an edge of the primal, and each face of the dual corresponds to a vertex of the primal. These correspondences are incidence-preserving: if two parts of the primal polyhedron touch each other, so do the corresponding two parts of the dual polyhedron. More generally, using the concept of polar reciprocation, any convex polyhedron, or more generally any convex polytope, corresponds to a dual polyhedron or dual polytope, with an -dimensional feature of an -dimensional polytope corresponding to an -dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the face lattices of the primal and dual polyhedra or polytopes are themselves [|order-theoretic duals]. Duality of polytopes and order-theoretic duality are both involutions: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure.
File:Duals graphs.svg|right|thumb|240px|A planar graph in blue, and its dual graph in red.
From any three-dimensional polyhedron, one can form a planar graph, the graph of its vertices and edges. The dual polyhedron has a dual graph, a graph with one vertex for each face of the polyhedron and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by a cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge. An important example of this type comes from computational geometry: the duality for any finite set of points in the plane between the Delaunay triangulation of and the Voronoi diagram of. As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual. The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality, in the sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph.
A kind of geometric duality also occurs in optimization theory, but not one that reverses dimensions. A linear program may be specified by a system of real variables, and a linear function. Every linear program has a dual problem with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa.