Definitions

Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for function. Thus, for example, a partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, with the same meaning.

In many branches of mathematics, the term is qualified with a property specific to that branch, such as a continuous function in topology, a linear map in linear algebra, etc.

Some authors such as Serge Lang use map as a general term for an association of an element in the range with every element in the domain, and function only to refer to maps in which the range is a field.

Sets of maps with special properties are the subjects of many important theories: see for instance Lie group, mapping class group, permutation group.

In formal logic, the term is sometimes used for a functional predicate, whereas a function is a model of such a predicate in set theory.

In graph theory, a map is a drawing of a graph on a surface without intersecting edges (a planar graph).

In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.