has numerous meanings. Generally speaking, duality is a metamathematical involution
. Some duality concepts are closely related and there are explicit theorems governing their relationships. Others are more intuitively related, with no precise correspondence. Nonetheless, "duality is a very pervasive and important concept in (modern) mathematics". Generally speaking, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution
operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have fixed points
, the dual of A is sometimes A itself. For example, Desargues' theorem
in projective geometry
is self-dual in this sense.
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:
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:
In analysis, frequently problems are solved by passing to the dual description of functions and operators.
In Boolean algebra a self-dual function is one such that
Negation is self-dual.
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