[ref] 31 Mappings

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# 31 Mappings

### Sections

1. Creating Mappings
2. Properties and Attributes of (General) Mappings
3. Images under Mappings
4. Preimages under Mappings
5. Arithmetic Operations for General Mappings
6. Mappings which are Compatible with Algebraic Structures
7. Magma Homomorphisms
8. Mappings that Respect Multiplication
9. Mappings that Respect Addition
10. Linear Mappings
11. Ring Homomorphisms
12. General Mappings
13. Technical Matters Concerning General Mappings

A mapping in GAP is what is called a ``function'' in mathematics. GAP also implements generalized mappings in which one element might have several images, these can be imagined as subsets of the cartesian product and are often called ``relations''.

Most operations are declared for general mappings and therefore this manual often refers to ``(general) mappings'', unless you deliberately need the generalization you can ignore the ``general'' bit and just read it as ``mappings''.

A general mapping F in GAP is described by its source S, its range R, and a subset Rel of the direct product S ×R, which is called the underlying relation of F. S, R, and Rel are generalized domains (see Chapter Domains). The corresponding attributes for general mappings are `Source`, `Range`, and `UnderlyingRelation`.

Note that general mappings themselves are not domains. One reason for this is that two general mappings with same underlying relation are regarded as equal only if also the sources are equal and the ranges are equal. Other, more technical, reasons are that general mappings and domains have different basic operations, and that general mappings are arithmetic objects (see Arithmetic Operations for General Mappings); both should not apply to domains.

Each element of an underlying relation of a general mapping lies in the category of tuples (see IsTuple).

For each s Î S, the set { r Î R | (s,r) Î Rel } is called the set of images of s. Analogously, for r Î R, the set { s Î S | (s,r) Î Rel } is called the set of preimages of r.

The ordering of general mappings via `<` is defined by the ordering of source, range, and underlying relation. Specifically, if the Source and Range domains of map1 and map2 are the same, then one considers the union of the preimages of map1 and map2 as a strictly ordered set. The smaller of map1 and map2 is the one whose image is smaller on the first point of this sequence where they differ.

For mappings which preserve an algebraic structure a kernel is defined. Depending on the structure preserved the operation to compute this kernel is called differently, see section Mappings which are Compatible with Algebraic Structures.

Some technical details of general mappings are described in section General Mappings.

- `IsTuple(` obj`) C`

  `IsTuple` is a subcategory of the meet of `IsDenseList` (see IsDenseList), `IsMultiplicativeElementWithInverse` (see IsMultiplicativeElementWithInverse), and `IsAdditiveElementWithInverse` (see IsAdditiveElementWithInverse), where the arithmetic operations (addition, zero, additive inverse, multiplication, powering, one, inverse) are defined componentwise.

  Note that each of these operations will cause an error message if its result for at least one component cannot be formed.

  The sum and the product of a tuple and a list in `IsListDefault` is the list of sums and products, respectively. The sum and the product of a tuple and a non-list is the tuple of componentwise sums and products, respectively.

  ## 31.1 Creating Mappings

- `GeneralMappingByElements(` S`,` R`,` elms`) F`

  is the general mapping with source S and range R, and with underlying relation consisting of the tuples collection elms.

- `MappingByFunction(` S`,` R`,` fun`) F`
- `MappingByFunction(` S`,` R`,` fun`,` invfun`) F`
- `MappingByFunction(` S`,` R`,` fun`, `false`, prefun ) F

  `MappingByFunction` returns a mapping map with source S and range R, such that each element s of S is mapped to the element ``fun`(` s`)`, where fun is a GAP function.

  If the argument invfun is bound then map is a bijection between S and R, and the preimage of each element r of R is given by ``invfun`(` r`)`, where invfun is a GAP function.

  In the third variant, a function prefun is given that can be used to compute a single preimage. In this case, the third entry must be `false`.

  `MappingByFunction` creates a mapping which `IsNonSPGeneralMapping`

- `InverseGeneralMapping(` map`) A`

  The inverse general mapping of a general mapping map is the general mapping whose underlying relation (see UnderlyingRelation) contains a pair (r,s) if and only if the underlying relation of map contains the pair (s,r).

  See the introduction to Chapter Mappings for the subtleties concerning the difference between `InverseGeneralMapping` and `Inverse`.

  Note that the inverse general mapping of a mapping map is in general only a general mapping. If map knows to be bijective its inverse general mapping will know to be a mapping. In this case also `Inverse(` map`)` works.

- `CompositionMapping(` map1`,` map2`, ... ) F`

  `CompositionMapping` allows one to compose arbitrarily many general mappings, and delegates each step to `CompositionMapping2`.

  Additionally, the properties `IsInjective` and `IsSingleValued` are maintained; if the source of the i+1-th general mapping is identical to the range of the i-th general mapping, also `IsTotal` and `IsSurjective` are maintained. (So one should not call `CompositionMapping2` directly if one wants to maintain these properties.)

  Depending on the types of map1 and map2, the returned mapping might be constructed completely new (for example by giving domain generators and their images, this is for example the case if both mappings preserve the same alagebraic structures and GAP can decompose elements of the source of map2 into generators) or as an (iterated) composition (see IsCompositionMappingRep).

- `CompositionMapping2(` map2`,` map1`) O`

  `CompositionMapping2` returns the composition of map2 and map1, this is the general mapping that maps an element first under map1, and then maps the images under map2.

  (Note the reverse ordering of arguments in the composition via `*`.

- `IsCompositionMappingRep(` map`) R`

  Mappings in this representation are stored as composition of two mappings, (pre)images of elements are computed in a two-step process. The constituent mappings of the composition can be obtained via `ConstituentsCompositionMapping`.

- `ConstituentsCompositionMapping(` map`) F`

  If map is stored in the representation `IsCompositionMappingRep` as composition of two mappings map1 and map2, this function returns the two constituent mappings in a list [map1,map2].

- `ZeroMapping(` S`,` R`) O`

  A zero mapping is a total general mapping that maps each element of its source to the zero element of its range.

  (Each mapping with empty source is a zero mapping.)

- `IdentityMapping(` D`) A`

  is the bijective mapping with source and range equal to the collection D, which maps each element of D to itself.

- `Embedding(` S`,` T`) O`
- `Embedding(` S`,` i`) O`

  returns the embedding of the domain S in the domain T, or in the second form, some domain indexed by the positive integer i. The precise natures of the various methods are described elsewhere: for Lie algebras, see `LieFamily` (LieFamily); for group products, see Embeddings and Projections for Group Products for a general description, or for examples see Direct Products for direct products, Semidirect Products for semidirect products, or Wreath Products for wreath products; or for magma rings see Natural Embeddings related to Magma Rings.

- `Projection(` S`,` T`) O`
- `Projection(` S`,` i`) O`
- `Projection(` S`) O`

  returns the projection of the domain S onto the domain T, or in the second form, some domain indexed by the positive integer i, or in the third form some natural subdomain of S. Various methods are defined for group products; see Embeddings and Projections for Group Products for a general description, or for examples see Direct Products for direct products, Semidirect Products for semidirect products, Subdirect Products for subdirect products, or Wreath Products for wreath products.

- `RestrictedMapping(` map`,` subdom`) O`

  If subdom is a subdomain of the source of the general mapping map, this operation returns the restriction of map to subdom.

  ## 31.2 Properties and Attributes of (General) Mappings

- `IsTotal(` map`) P`

  is `true` if each element in the source S of the general mapping map has images, i.e., smap ¹ Æ for all s Î S, and `false` otherwise.

- `IsSingleValued(` map`) P`

  is `true` if each element in the source S of the general mapping map has at most one image, i.e., |smap | £ 1 for all s Î S, and `false` otherwise.

  Equivalently, `IsSingleValued(` map`)` is `true` if and only if the preimages of different elements in R are disjoint.

- `IsMapping(` map`) P`

  A mapping map is a general mapping that assigns to each element `elm` of its source a unique element `Image(` map`,` elm`)` of its range.

  Equivalently, the general mapping map is a mapping if and only if it is total and single-valued (see IsTotal, IsSingleValued).

- `IsInjective(` map`) P`

  is `true` if the images of different elements in the source S of the general mapping map are disjoint, i.e., xmap Çymap = Æ for x ¹ y Î S, and `false` otherwise.

  Equivalently, `IsInjective(` map`)` is `true` if and only if each element in the range of map has at most one preimage in S.

- `IsSurjective(` map`) P`

  is `true` if each element in the range R of the general mapping map has preimages in the source S of map, i.e., { s Î S | x Î smap } ¹ Æ for all x Î R, and `false` otherwise.

- `IsBijective(` map`) P`

  A general mapping map is bijective if and only if it is an injective and surjective mapping (see IsMapping, IsInjective, IsSurjective).

- `Range(` map`) A`

- `Source(` map`) A`

- `UnderlyingRelation(` map`) A`

  The underlying relation of a general mapping map is the domain of pairs (s,r), with s in the source and r in the range of map (see Source, Range), and r Î `ImagesElm(` map`, s )`.

  Each element of the underlying relation is a tuple (see IsTuple).

- `UnderlyingGeneralMapping(` map`) A`

  attribute for underlying relations of general mappings

  ## 31.3 Images under Mappings

- `ImagesSource(` map`) A`

  is the set of images of the source of the general mapping map.

  `ImagesSource` delegates to `ImagesSet`, it is introduced only to store the image of map as attribute value.

- `ImagesRepresentative(` map`,` elm`) O`

  If elm is an element of the source of the general mapping map then `ImagesRepresentative` returns either a representative of the set of images of elm under map or `fail`, the latter if and only if elm has no images under map.

  Anything may happen if elm is not an element of the source of map.

- `ImagesElm(` map`,` elm`) O`

  If elm is an element of the source of the general mapping map then `ImagesElm` returns the set of all images of elm under map.

  Anything may happen if elm is not an element of the source of map.

- `ImagesSet(` map`,` elms`) O`

  If elms is a subset of the source of the general mapping map then `ImagesSet` returns the set of all images of elms under map.

  Anything may happen if elms is not a subset of the source of map.

- `ImageElm(` map`,` elm`) O`

  If elm is an element of the source of the total and single-valued mapping map then `ImageElm` returns the unique image of elm under map.

  Anything may happen if elm is not an element of the source of map.

- `Image(` map`) F`
- `Image(` map`,` elm`) F`
- `Image(` map`,` coll`) F`

  `Image(` map`)` is the image of the general mapping map, i.e., the subset of elements of the range of map that are actually values of map. Note that in this case the argument may also be multi-valued.

  `Image(` map`,` elm`)` is the image of the element elm of the source of the mapping map under map, i.e., the unique element of the range to which map maps elm. This can also be expressed as ``elm`^` map``. Note that map must be total and single valued, a multi valued general mapping is not allowed (see Images).

  `Image(` map`,` coll`)` is the image of the subset coll of the source of the mapping map under map, i.e., the subset of the range to which map maps elements of coll. coll may be a proper set or a domain. The result will be either a proper set or a domain. Note that in this case map may also be multi-valued. (If coll and the result are lists then the positions of entries do in general not correspond.)

  `Image` delegates to `ImagesSource` when called with one argument, and to `ImageElm` resp. `ImagesSet` when called with two arguments.

  If the second argument is not an element or a subset of the source of the first argument, an error is signalled.

- `Images(` map`) F`
- `Images(` map`,` elm`) F`
- `Images(` map`,` coll`) F`

  `Images(` map`)` is the image of the general mapping map, i.e., the subset of elements of the range of map that are actually values of map.

  `Images(` map`,` elm`)` is the set of images of the element elm of the source of the general mapping map under map, i.e., the set of elements of the range to which map maps elm.

  `Images(` map`,` coll`)` is the set of images of the subset coll of the source of the general mapping map under map, i.e., the subset of the range to which map maps elements of coll. coll may be a proper set or a domain. The result will be either a proper set or a domain. (If coll and the result are lists then the positions of entries do in general not correspond.)

  `Images` delegates to `ImagesSource` when called with one argument, and to `ImagesElm` resp. `ImagesSet` when called with two arguments.

  If the second argument is not an element or a subset of the source of the first argument, an error is signalled.

  ## 31.4 Preimages under Mappings

- `PreImagesRange(` map`) A`

  is the set of preimages of the range of the general mapping map.

  `PreImagesRange` delegates to `PreImagesSet`, it is introduced only to store the preimage of map as attribute value.

- `PreImagesElm(` map`,` elm`) O`

  If elm is an element of the range of the general mapping map then `PreImagesElm` returns the set of all preimages of elm under map.

  Anything may happen if elm is not an element of the range of map.

- `PreImageElm(` map`,` elm`) O`

  If elm is an element of the range of the injective and surjective general mapping map then `PreImageElm` returns the unique preimage of elm under map.

  Anything may happen if elm is not an element of the range of map.

- `PreImagesRepresentative(` map`,` elm`) O`

  If elm is an element of the range of the general mapping map then `PreImagesRepresentative` returns either a representative of the set of preimages of elm under map or `fail`, the latter if and only if elm has no preimages under map.

  Anything may happen if elm is not an element of the range of map.

- `PreImagesSet(` map`,` elms`) O`

  If elms is a subset of the range of the general mapping map then `PreImagesSet` returns the set of all preimages of elms under map.

  Anything may happen if elms is not a subset of the range of map.

- `PreImage(` map`) F`
- `PreImage(` map`,` elm`) F`
- `PreImage(` map`,` coll`) F`

  `PreImage(` map`)` is the preimage of the general mapping map, i.e., the subset of elements of the source of map that actually have values under map. Note that in this case the argument may also be non-injective or non-surjective.

  `PreImage(` map`,` elm`)` is the preimage of the element elm of the range of the injective and surjective mapping map under map, i.e., the unique element of the source which is mapped under map to elm. Note that map must be injective and surjective (see PreImages).

  `PreImage(` map`,` coll`)` is the preimage of the subset coll of the range of the general mapping map under map, i.e., the subset of the source which is mapped under map to elements of coll. coll may be a proper set or a domain. The result will be either a proper set or a domain. Note that in this case map may also be non-injective or non-surjective. (If coll and the result are lists then the positions of entries do in general not correspond.)

  `PreImage` delegates to `PreImagesRange` when called with one argument, and to `PreImageElm` resp. `PreImagesSet` when called with two arguments.

  If the second argument is not an element or a subset of the range of the first argument, an error is signalled.

- `PreImages(` map`) F`
- `PreImages(` map`,` elm`) F`
- `PreImages(` map`,` coll`) F`

  `PreImages(` map`)` is the preimage of the general mapping map, i.e., the subset of elements of the source of map that have actually values under map.

  `PreImages(` map`,` elm`)` is the set of preimages of the element elm of the range of the general mapping map under map, i.e., the set of elements of the source which map maps to elm.

  `PreImages(` map`,` coll`)` is the set of images of the subset coll of the range of the general mapping map under map, i.e., the subset of the source which map maps to elements of coll. coll may be a proper set or a domain. The result will be either a proper set or a domain. (If coll and the result are lists then the positions of entries do in general not correspond.)

  `PreImages` delegates to `PreImagesRange` when called with one argument, and to `PreImagesElm` resp. `PreImagesSet` when called with two arguments.

  If the second argument is not an element or a subset of the range of the first argument, an error is signalled.

  ## 31.5 Arithmetic Operations for General Mappings

  General mappings are arithmetic objects. One can form groups and vector spaces of general mappings provided that they are invertible or can be added and admit scalar multiplication, respectively.

  For two general mappings with same source, range, preimage, and image, the sum is defined pointwise, i.e., the images of a point under the sum is the set of all sums with first summand in the images of the first general mapping and second summand in the images of the second general mapping.

  Scalar multiplication of general mappings is defined likewise.

  The product of two general mappings is defined as the composition. This multiplication is always associative. In addition to the composition via `*`, general mappings can be composed --in reversed order-- via `CompositionMapping`.

  General mappings are in the category of multiplicative elements with inverses. Similar to matrices, not every general mapping has an inverse or an identity, and we define the behaviour of `One` and `Inverse` for general mappings as follows. `One` returns `fail` when called for a general mapping whose source and range differ, otherwise `One` returns the identity mapping of the source. (Note that the source may differ from the preimage). `Inverse` returns `fail` when called for a non-bijective general mapping or for a general mapping whose source and range differ; otherwise `Inverse` returns the inverse mapping.

  Besides the usual inverse of multiplicative elements, which means that `Inverse(` g`) *` g`=` g`* Inverse(` g`) = One(` g`)`, for general mappings we have the attribute `InverseGeneralMapping`. If F is a general mapping with source S, range R, and underlying relation Rel then `InverseGeneralMapping(` F`)` has source R, range S, and underlying relation { (r,s) | (s,r) Î Rel }. For a general mapping that has an inverse in the usual sense, i.e., for a bijection of the source, of course both concepts coincide.

  `Inverse` may delegate to `InverseGeneralMapping`. `InverseGeneralMapping` must not delegate to `Inverse`, but a known value of `Inverse` may be fetched. So methods to compute the inverse of a general mapping should be installed for `InverseGeneralMapping`.

  (Note that in many respects, general mappings behave similar to matrices, for example one can define left and right identities and inverses, which do not fit into the current concepts of GAP.)

  ## 31.6 Mappings which are Compatible with Algebraic Structures

  From an algebraical point of view, the most important mappings are those which are compatible with a structure. For Magmas, Groups and Rings, GAP supports the following four types of such mappings:

  1. General mappings that respect multiplication

  2. General mappings that respect addition

  3. General mappings that respect scalar mult.

  4. General mappings that respect multiplicative and additive structure

  (Very technical note: GAP defines categories `IsSPGeneralMapping` and `IsNonSPGeneralMapping`. The distinction between these is orthogonal to the Structure Compatibility described here and should not be confused.)

  ## 31.7 Magma Homomorphisms

- `IsMagmaHomomorphism(` mapp`) P`

  A `MagmaHomomorphism` is a total single valued mapping which respects multiplication.

- `MagmaHomomorphismByFunctionNC(` G`,` H`,` fn`) F`

  Creates the homomorphism from G to H without checking that fn is a homomorphism.

- `NaturalHomomorphismByGenerators(` f`,` s`) O`

  returns a mapping from the magma f with n generators to the magma s with n generators, which maps the ith generator of f to the ith generator of s.

  ## 31.8 Mappings that Respect Multiplication

- `RespectsMultiplication(` mapp`) P`

  Let mapp be a general mapping with underlying relation F Í S ×R, where S and R are the source and the range of mapp, respectively. Then `RespectsMultiplication` returns `true` if S and R are magmas such that (s1,r1), (s2,r2) Î F implies (s1 * s2,r1 * r2) Î F, and `false` otherwise.

  If mapp is single-valued then `RespectsMultiplication` returns `true` if and only if the equation ``s1`^`mapp`*` s2`^`mapp`= (`s1`*`s2`)^`mapp`` holds for all s1, s2 in S.

- `RespectsOne(` mapp`) P`

  Let mapp be a general mapping with underlying relation F Í S ×R , where S and R are the source and the range of mapp, respectively. Then `RespectsOne` returns `true` if S and R are magmas-with-one such that ( `One``(`S `)`, `One``(`R `)` ) Î F, and `false` otherwise.

  If mapp is single-valued then `RespectsOne` returns `true` if and only if the equation `One(` S`)^`mapp`= One(` R`)` holds.

- `RespectsInverses(` mapp`) P`

  Let mapp be a general mapping with underlying relation F Í S ×R , where S and R are the source and the range of mapp, respectively. Then `RespectsInverses` returns `true` if S and R are magmas-with-inverses such that, for s Î S and r Î R , (s,r) Î F implies (s-1,r-1) Î F, and `false` otherwise.

  If mapp is single-valued then `RespectsInverses` returns `true` if and only if the equation `Inverse(` s`)^`mapp`= Inverse(` s`^`mapp`)` holds for all s in S.

  Mappings that are defined on a group and respect multiplication and inverses are group homomorphisms. Chapter Group Homomorphisms explains them in more detail.

- `IsGroupGeneralMapping(` mapp`) P`
- `IsGroupHomomorphism(` mapp`) P`

  A `GroupGeneralMapping` is a mapping which respects multiplication and inverses. If it is total and single valued it is called a group homomorphism.

- `KernelOfMultiplicativeGeneralMapping(` mapp`) A`

  Let mapp be a general mapping. Then `KernelOfMultiplicativeGeneralMapping` returns the set of all elements in the source of mapp that have the identity of the range in their set of images.

  (This is a monoid if mapp respects multiplication and one, and if the source of mapp is associative.)

- `CoKernelOfMultiplicativeGeneralMapping(` mapp`) A`

  Let mapp be a general mapping. Then `CoKernelOfMultiplicativeGeneralMapping` returns the set of all elements in the range of mapp that have the identity of the source in their set of preimages.

  (This is a monoid if mapp respects multiplication and one, and if the range of mapp is associative.)

  ## 31.9 Mappings that Respect Addition

- `RespectsAddition(` mapp`) P`

  Let mapp be a general mapping with underlying relation F Í S ×R, where S and R are the source and the range of mapp, respectively. Then `RespectsAddition` returns `true` if S and R are additive magmas such that (s1,r1), (s2,r2) Î F implies (s1 + s2,r1 + r2) Î F, and `false` otherwise.

  If mapp is single-valued then `RespectsAddition` returns `true` if and only if the equation ``s1`^`mapp`+` s2`^`mapp`= (`s1`+`s2`)^`mapp`` holds for all s1, s2 in S.

- `RespectsAdditiveInverses(` mapp`) P`

  Let mapp be a general mapping with underlying relation F Í S ×R, where S and R are the source and the range of mapp, respectively. Then `RespectsAdditiveInverses` returns `true` if S and R are additive-magmas-with-inverses such that (s,r) Î F implies (-s,-r) Î F, and `false` otherwise.

  If mapp is single-valued then `RespectsAdditiveInverses` returns `true` if and only if the equation `AdditiveInverse(` s`)^`mapp`= AdditiveInverse(` s`^`mapp`)` holds for all s in S.

- `RespectsZero(` mapp`) P`

  Let mapp be a general mapping with underlying relation F Í S ×R , where S and R are the source and the range of mapp, respectively. Then `RespectsZero` returns `true` if S and R are additive-magmas-with-zero such that ( `Zero``(`S `)`, `Zero``(`R `)` ) Î F, and `false` otherwise.

  If mapp is single-valued then `RespectsZero` returns `true` if and only if the equation `Zero(` S`)^`mapp`= Zero(` R`)` holds.

- `IsAdditiveGroupGeneralMapping(` mapp`) P`
- `IsAdditiveGroupHomomorphism(` mapp`) P`

- `KernelOfAdditiveGeneralMapping(` mapp`) A`

  Let mapp be a general mapping. Then `KernelOfAdditiveGeneralMapping` returns the set of all elements in the source of mapp that have the zero of the range in their set of images.

- `CoKernelOfAdditiveGeneralMapping(` mapp`) A`

  Let mapp be a general mapping. Then `CoKernelOfAdditiveGeneralMapping` returns the set of all elements in the range of mapp that have the zero of the source in their set of preimages.

  ## 31.10 Linear Mappings

  Also see Sections Mappings that Respect Multiplication and Mappings that Respect Addition.

- `RespectsScalarMultiplication(` mapp`) P`

  Let mapp be a general mapping, with underlying relation F Í S ×R, where S and R are the source and the range of mapp, respectively. Then `RespectsScalarMultiplication` returns `true` if S and R are left modules with the left acting domain D of S contained in the left acting domain of R and such that (s,r) Î F implies (c * s,c * r) Î F for all c Î D, and `false` otherwise.

  If mapp is single-valued then `RespectsScalarMultiplication` returns `true` if and only if the equation ``c`*` s`^`mapp`= (`c`*` s`)^`mapp`` holds for all c in D and s in S.

- `IsLeftModuleGeneralMapping(` mapp`) P`
- `IsLeftModuleHomomorphism(` mapp`) P`

- `IsLinearMapping(` F`,` mapp`) O`

  For a field F and a general mapping mapp, `IsLinearMapping` returns `true` if mapp is an F-linear mapping, and `false` otherwise.

  A mapping f is a linear mapping (or vector space homomorphism) if the source and range are vector spaces over the same division ring D, and if f( a + b ) = f(a) + f(b) and f( s * a ) = s * f(a) hold for all elements a, b in the source of f and s Î D.

  See also `KernelOfMultiplicativeGeneralMapping` (KernelOfMultiplicativeGeneralMapping) and `CoKernelOfMultiplicativeGeneralMapping` (CoKernelOfMultiplicativeGeneralMapping).

  ## 31.11 Ring Homomorphisms

- `IsRingGeneralMapping(` mapp`) P`
- `IsRingHomomorphism(` mapp`) P`

- `IsRingWithOneGeneralMapping(` mapp`) P`
- `IsRingWithOneHomomorphism(` mapp`) P`

- `IsAlgebraGeneralMapping(` mapp`) P`
- `IsAlgebraHomomorphism(` mapp`) P`

- `IsAlgebraWithOneGeneralMapping(` mapp`) P`
- `IsAlgebraWithOneHomomorphism(` mapp`) P`

- `IsFieldHomomorphism(` mapp`) P`

  A general mapping is a field homomorphism if and only if it is a ring homomorphism with source a field.

  ## 31.12 General Mappings

- `IsGeneralMapping(` map`) C`

  Each general mapping lies in the category `IsGeneralMapping`. It implies the categories `IsMultiplicativeElementWithInverse` (see IsMultiplicativeElementWithInverse) and `IsAssociativeElement` (see IsAssociativeElement); for a discussion of these implications, see Arithmetic Operations for General Mappings.

- `IsConstantTimeAccessGeneralMapping(` map`) P`

  is `true` if the underlying relation of the general mapping map knows its `AsList` value, and `false` otherwise.

  In the former case, map is allowed to use this list for calls to `ImagesElm` etc.

- `IsEndoGeneralMapping(` obj`) P`

  If a general mapping has this property then its source and range are equal.

  ## 31.13 Technical Matters Concerning General Mappings

  `Source` and `Range` are basic operations for general mappings. `UnderlyingRelation` is secondary, its default method sets up a domain that delegates tasks to the general mapping. (Note that this allows one to handle also infinite relations by generic methods if source or range of the general mapping is finite.)

  The distinction between basic operations and secondary operations for general mappings may be a little bit complicated. Namely, each general mapping must be in one of the two categories `IsNonSPGeneralMapping`, `IsSPGeneralMapping`. (The category `IsGeneralMapping` is defined as the disjoint union of these two.)

  For general mappings of the first category, `ImagesElm` and `PreImagesElm` are basic operations. (Note that in principle it is possible to delegate from `PreImagesElm` to `ImagesElm`.) Methods for the secondary operations `(Pre)ImageElm`, `(Pre)ImagesSet`, and `(Pre)ImagesRepresentative` may use `(Pre)ImagesElm`, and methods for `(Pre)ImagesElm` must not call the secondary operations. In particular, there are no generic methods for `(Pre)ImagesElm`.

  Methods for `(Pre)ImagesSet` must not use `PreImagesRange` and `ImagesSource`, e.g., compute the intersection of the set in question with the preimage of the range resp. the image of the source.

  For general mappings of the second category (which means structure preserving general mappings), the situation is different. The set of preimages under a group homomorphism, for example, is either empty or can be described as a coset of the (multiplicative) kernel. So it is reasonable to have `(Pre)ImagesRepresentative` and `Multplicative(Co)Kernel` as basic operations here, and to make `(Pre)ImagesElm` secondary operations that may delegate to these.

  In order to avoid infinite recursions, we must distinguish between the two different types of mappings.

  (Note that the basic domain operations such as `AsList` for the underlying relation of a general mapping may use either `ImagesElm` or `ImagesRepresentative` and the appropriate cokernel. Conversely, if `AsList` for the underlying relation is known then `ImagesElm` resp. `ImagesRepresentative` may delegate to it, the general mapping gets the property `IsConstantTimeAccessGeneralMapping` for this; note that this is not allowed if only an enumerator of the underlying relation is known.)

  Secondary operations are `IsInjective`, `IsSingleValued`, `IsSurjective`, `IsTotal`; they may use the basic operations, and must not be used by them.

- `IsSPGeneralMapping(` map`) C`
- `IsNonSPGeneralMapping(` map`) C`

- `IsGeneralMappingFamily(` obj`) C`

- `FamilyRange(` Fam`) A`

  is the elements family of the family of the range of each general mapping in the family Fam.

- `FamilySource(` Fam`) A`

  is the elements family of the family of the source of each general mapping in the family Fam.

- `FamiliesOfGeneralMappingsAndRanges(` Fam`) AM`

  is a list that stores at the odd positions the families of general mappings with source in the family Fam, at the even positions the families of ranges of the general mappings.

- `GeneralMappingsFamily(` sourcefam`,` rangefam`) F`

  All general mappings with same source family FS and same range family FR lie in the family `GeneralMappingsFamily(` FS`,` FR`)`.

- `TypeOfDefaultGeneralMapping(` source`,` range`,` filter`) F`

  is the type of mappings with `IsDefaultGeneralMappingRep` with source source and range range and additional categories filter.

  Methods for the operations `ImagesElm`, `ImagesRepresentative`, `ImagesSet`, `ImageElm`, `PreImagesElm`, `PreImagesRepresentative`, `PreImagesSet`, and `PreImageElm` take two arguments, a general mapping map and an element or collection of elements elm. These methods must not check whether elm lies in the source or the range of map. In the case that elm does not, `fail` may be returned as well as any other GAP object, and even an error message is allowed. Checks of the arguments are done only by the functions `Image`, `Images`, `PreImage`, and `PreImages`, which then delegate to the operations listed above.

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  GAP 4 manual
  May 2002