# 42 Mappings

A mapping is an object that maps each element of its source to a value in its range.

Precisely, a mapping is a triple. The source is a set of objects. The range is another set of objects. The relation is a subset S of the cartesian product of the source with the range, such that for each element elm of the source there is exactly one element img of the range, so that the pair (elm,img) lies in S. This img is called the image of elm under the mapping, and we say that the mapping maps elm to img.

A multi valued mapping is an object that maps each element of its source to a set of values in its range.

Precisely, a multi valued mapping is a triple. The source is a set of objects. The range is another set of objects. The relation is a subset S of the cartesian product of the source with the range. For each element elm of the source the set img such that the pair (elm,img) lies in S is called the set of images of elm under the mapping, and we say that the mapping maps elm to this set.

Thus a mapping is a special case of a multi valued mapping where the set of images of each element of the source contains exactly one element, which is then called the image of the element under the mapping.

Mappings are created by mapping constructors such as `MappingByFunction` (see MappingByFunction) or `NaturalHomomorphism` (see NaturalHomomorphism).

This chapter contains sections that describe the functions that test whether an object is a mapping (see IsGeneralMapping), whether a mapping is single valued (see IsMapping), and the various functions that test if such a mapping has a certain property (see IsInjective, IsSurjective, IsBijection, IsHomomorphism, IsMonomorphism, IsEpimorphism, IsIsomorphism, IsEpimorphism, and IsAutomorphism).

Next this chapter contains functions that describe how mappings are compared (see Comparisons of Mappings) and the operations that are applicable to mappings (see Operations for Mappings).

Next this chapter contains sections that describe the functions that deal with the images and preimages of elements under mappings (see Image, Images, ImagesRepresentative, PreImage, PreImages, and PreImagesRepresentative).

Next this chapter contains sections that describe the functions that compute the composition of two mappings, the power of a mapping, the inverse of a mapping, and the identity mapping on a certain domain (see CompositionMapping, PowerMapping, InverseMapping, and IdentityMapping).

Finally this chapter also contains a section that describes how mappings are represented internally (see Mapping Records).

The functions described in this chapter are in the file `libname/"mapping.g"`.

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GAP 3.4.4
April 1997