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32 Mappings

32.2 Creating Mappings

32.2-1 GeneralMappingByElements

32.2-2 MappingByFunction

32.2-3 InverseGeneralMapping

32.2-4 RestrictedInverseGeneralMapping

32.2-5 CompositionMapping

32.2-6 CompositionMapping2

32.2-7 IsCompositionMappingRep

32.2-8 ConstituentsCompositionMapping

32.2-9 ZeroMapping

32.2-10 IdentityMapping

32.2-11 Embedding

32.2-12 Projection

32.2-13 RestrictedMapping

32.2-1 GeneralMappingByElements

32.2-2 MappingByFunction

32.2-3 InverseGeneralMapping

32.2-4 RestrictedInverseGeneralMapping

32.2-5 CompositionMapping

32.2-6 CompositionMapping2

32.2-7 IsCompositionMappingRep

32.2-8 ConstituentsCompositionMapping

32.2-9 ZeroMapping

32.2-10 IdentityMapping

32.2-11 Embedding

32.2-12 Projection

32.2-13 RestrictedMapping

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 12.4). The corresponding attributes for general mappings are `Source`

(32.3-8), `Range`

(32.3-7), and `UnderlyingRelation`

(32.3-9).

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 32.6); both should not apply to domains.

Each element of an underlying relation of a general mapping lies in the category of direct product elements (see `IsDirectProductElement`

(32.1-1)).

For each s ∈ S, the set { r ∈ R | (s,r) ∈ Rel } is called the set of *images* of s. Analogously, for r ∈ R, the set { s ∈ S | (s,r) ∈ 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 32.7.

Some technical details of general mappings are described in section 32.13.

`‣ IsDirectProductElement` ( obj ) | ( category ) |

`IsDirectProductElement`

is a subcategory of the meet of `IsDenseList`

(21.1-2), `IsMultiplicativeElementWithInverse`

(31.14-13), `IsAdditiveElementWithInverse`

(31.14-7), and `IsCopyable`

(12.6-1), 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.

For an object in the filter `IsDirectProductElement`

, `ShallowCopy`

(12.7-1) returns a mutable plain list with the same entries. The sum and the product of a direct product element and a list in `IsListDefault`

(21.12-3) is the list of sums and products, respectively. The sum and the product of a direct product element and an object that is neither a list nor a collection is the direct product element of componentwise sums and products, respectively.

`‣ DirectProductFamily` ( args ) | ( function ) |

`args` must be a dense list of `CollectionsFamily`

(30.2-1) families, otherwise the function raises an error.

`DirectProductFamily`

returns `fam`

, a collections family of `IsDirectProductElement`

(32.1-1) objects.

`fam`

is the `CollectionsFamily`

(30.2-1) of `IsDirectProductElement`

(32.1-1) objects whose `i`

-th component is in `ElementsFamily(args[i])`

.

Note that a collection in `fam`

may not itself be a direct product; it just is a subcollection of a direct product.

gap> D8 := DihedralGroup(IsPermGroup, 8);; gap> FamilyObj(D8) = CollectionsFamily(PermutationsFamily); true gap> fam := DirectProductFamily([FamilyObj(D8), FamilyObj(D8)]);; gap> ComponentsOfDirectProductElementsFamily(ElementsFamily(fam)); [ <Family: "PermutationsFamily">, <Family: "PermutationsFamily"> ]

Also note that not all direct products in **GAP** are created via these families. For example if the arguments to `DirectProduct`

(49.1-1) are permutation groups, then it returns a permutation group as well, whose elements are not `IsDirectProductElement`

(32.1-1) objects.

gap> fam = FamilyObj(DirectProduct(D8, D8)); false gap> D4 := DihedralGroup(IsPcGroup, 4);; gap> fam2 := DirectProductFamily([FamilyObj(D8), FamilyObj(D4)]);; gap> fam2 = FamilyObj(DirectProduct(D8, D4)); true

`‣ GeneralMappingByElements` ( S, R, elms ) | ( function ) |

is the general mapping with source `S` and range `R`, and with underlying relation consisting of the collection `elms` of direct product elements.

`‣ MappingByFunction` ( S, R, fun[, invfun] ) | ( function ) |

`‣ MappingByFunction` ( S, R, fun, false, prefun ) | ( function ) |

`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.

If five arguments are given and the fourth argument is `false`

then the **GAP** function `prefun` can be used to compute a single preimage also if `map`

is not bijective.

The mapping returned by `MappingByFunction`

lies in the filter `IsNonSPGeneralMapping`

(32.14-1), see 32.14.

`‣ InverseGeneralMapping` ( map ) | ( attribute ) |

The *inverse general mapping* of a general mapping `map` is the general mapping whose underlying relation (see `UnderlyingRelation`

(32.3-9)) contains a pair (r,s) if and only if the underlying relation of `map` contains the pair (s,r).

See the introduction to Chapter 32 for the subtleties concerning the difference between `InverseGeneralMapping`

and `Inverse`

(31.10-8).

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( `

works.`map` )

`‣ RestrictedInverseGeneralMapping` ( map ) | ( attribute ) |

The *restricted inverse general mapping* of a general mapping `map` is the general mapping whose underlying relation (see `UnderlyingRelation`

(32.3-9)) contains a pair (r,s) if and only if the underlying relation of `map` contains the pair (s,r), and whose domain is restricted to the image of `map` and whose range is the domain of `map`.

`‣ CompositionMapping` ( map1, map2, ... ) | ( function ) |

`CompositionMapping`

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

(32.2-6). The result is a map that maps an element first under the last argument, then under the penultimate argument and so forth.

Additionally, the properties `IsInjective`

(32.3-4) and `IsSingleValued`

(32.3-2) are maintained. If the range of the i+1-th argument is identical to the range of the i-th argument, also `IsTotal`

(32.3-1) and `IsSurjective`

(32.3-5) are maintained. (So one should not call `CompositionMapping2`

(32.2-6) 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 algebraic structures and **GAP** can decompose elements of the source of `map2` into generators) or as an (iterated) composition (see `IsCompositionMappingRep`

(32.2-7)).

gap> f := GroupHomomorphismByImages(CyclicGroup(IsPermGroup, 2), > CyclicGroup(IsPermGroup, 1)); [ (1,2) ] -> [ () ] gap> g := GroupHomomorphismByImages(CyclicGroup(IsPermGroup, 6), > CyclicGroup(IsPermGroup, 2)); [ (1,2,3,4,5,6) ] -> [ (1,2) ] gap> CompositionMapping(f, g); [ (1,2,3,4,5,6) ] -> [ () ] gap> CompositionMapping(g, f); [ (1,2) ] -> [ () ]

`‣ CompositionMapping2` ( map2, map1 ) | ( operation ) |

`‣ CompositionMapping2General` ( map2, map1 ) | ( function ) |

`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 the multiplication `\*`

(31.12-1).

`CompositionMapping2General`

is the method that forms a composite mapping with two constituent mappings. (This is used in some algorithms.)

`‣ IsCompositionMappingRep` ( map ) | ( representation ) |

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`

(32.2-8).

`‣ ConstituentsCompositionMapping` ( map ) | ( function ) |

If `map` is stored in the representation `IsCompositionMappingRep`

(32.2-7) as composition of two mappings `map1` and `map2`, this function returns the two constituent mappings in a list `[ `

.`map1`, `map2` ]

`‣ ZeroMapping` ( S, R ) | ( operation ) |

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 ) | ( attribute ) |

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

`‣ Embedding` ( S, T ) | ( operation ) |

`‣ Embedding` ( S, i ) | ( operation ) |

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`

(64.1-3); for group products, see 49.6 for a general description, or for examples see 49.1 for direct products, 49.2 for semidirect products, or 49.4 for wreath products; or for magma rings see 65.3.

`‣ Projection` ( S, T ) | ( operation ) |

`‣ Projection` ( S, i ) | ( operation ) |

`‣ Projection` ( S ) | ( operation ) |

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 quotient domain of `S`. Various methods are defined for group products; see 49.6 for a general description, or for examples see 49.1 for direct products, 49.2 for semidirect products, 49.3 for subdirect products, or 49.4 for wreath products.

`‣ RestrictedMapping` ( map, subdom ) | ( operation ) |

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

`‣ IsTotal` ( map ) | ( property ) |

is `true`

if each element in the source S of the general mapping `map` has images, i.e., s^`map` ≠ ∅ for all s ∈ S, and `false`

otherwise.

`‣ IsSingleValued` ( map ) | ( property ) |

is `true`

if each element in the source S of the general mapping `map` has at most one image, i.e., |s^`map`| ≤ 1 for all s ∈ S, and `false`

otherwise.

Equivalently, `IsSingleValued( `

is `map` )`true`

if and only if the preimages of different elements in R are disjoint.

`‣ IsMapping` ( map ) | ( filter ) |

A *mapping* `map` is a general mapping that assigns to each element `elm`

of its source a unique element `Image( `

of its range.`map`, elm )

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

(32.3-1), `IsSingleValued`

(32.3-2)).

`‣ IsInjective` ( map ) | ( property ) |

is `true`

if the images of different elements in the source S of the general mapping `map` are disjoint, i.e., x^`map` ∩ y^`map` = ∅ for x ≠ y ∈ S, and `false`

otherwise.

Equivalently, `IsInjective( `

is `map` )`true`

if and only if each element in the range of `map` has at most one preimage in S.

`‣ IsSurjective` ( map ) | ( property ) |

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 ∈ S ∣ x ∈ s^`map` } ≠ ∅ for all x ∈ R, and `false`

otherwise.

`‣ IsBijective` ( map ) | ( property ) |

A general mapping `map` is *bijective* if and only if it is an injective and surjective mapping (see `IsMapping`

(32.3-3), `IsInjective`

(32.3-4), `IsSurjective`

(32.3-5)).

`‣ Range` ( map ) | ( attribute ) |

The range of a general mapping.

`‣ Source` ( map ) | ( attribute ) |

The source of a general mapping.

`‣ UnderlyingRelation` ( map ) | ( attribute ) |

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`

(32.3-8), `Range`

(32.3-7)), and r ∈ `ImagesElm( `

s`map`, ` )`

.

Each element of the underlying relation is represented by a direct product element (see `IsDirectProductElement`

(32.1-1)).

`‣ UnderlyingGeneralMapping` ( map ) | ( attribute ) |

attribute for underlying relations of general mappings

`‣ ImagesSource` ( map ) | ( attribute ) |

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

`ImagesSource`

delegates to `ImagesSet`

(32.4-4), it is introduced only to store the image of `map` as attribute value.

`‣ ImagesRepresentative` ( map, elm ) | ( operation ) |

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 ) | ( operation ) |

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 ) | ( operation ) |

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`.

The result will be either a proper set or a domain. Anything may happen if `elms` is not a subset of the source of `map`.

`‣ ImageElm` ( map, elm ) | ( operation ) |

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 ) | ( function ) |

`‣ Image` ( map, elm ) | ( function ) |

`‣ Image` ( map, coll ) | ( function ) |

`Image( `

is the `map` )*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( `

is the image of the element `map`, `elm` )`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` or as `map``( `

`elm`` )`

.

Note that `map` must be total and single valued, a multi valued general mapping is not allowed (see `Images`

(32.4-7)).

`Image( `

is the image of the subset `map`, `coll` )`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( `

can also be expressed as `map`, `coll` )

and `map`(`coll`)`Image( `

as `map`, `elm` )

. Those using this notation should remember that composition of mappings in `map`(`elm`)**GAP** still follows the conventions appropriate for mapping acting from the right, so that `(`

is equivalent to `map1`*`map2`)(`x`)`map2`(`map1`(`x`))

`Image`

delegates to `ImagesSource`

(32.4-1) when called with one argument, and to `ImageElm`

(32.4-5) resp. `ImagesSet`

(32.4-4) 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 ) | ( function ) |

`‣ Images` ( map, elm ) | ( function ) |

`‣ Images` ( map, coll ) | ( function ) |

`Images( `

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

`Images( `

is the set of images of the element `map`, `elm` )`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( `

is the set of images of the subset `map`, `coll` )`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`

(32.4-1) when called with one argument, and to `ImagesElm`

(32.4-3) resp. `ImagesSet`

(32.4-4) 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.

`‣ PreImagesRange` ( map ) | ( attribute ) |

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

`PreImagesRange`

delegates to `PreImagesSet`

(32.5-5), it is introduced only to store the preimage of `map` as attribute value.

`‣ PreImagesElm` ( map, elm ) | ( operation ) |

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 ) | ( operation ) |

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 ) | ( operation ) |

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 ) | ( operation ) |

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 ) | ( function ) |

`‣ PreImage` ( map, elm ) | ( function ) |

`‣ PreImage` ( map, coll ) | ( function ) |

`PreImage( `

is the preimage of the general mapping `map` )`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( `

is the preimage of the element `map`, `elm` )`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`

(32.5-7)).

`PreImage( `

is the preimage of the subset `map`, `coll` )`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`

(32.5-1) when called with one argument, and to `PreImageElm`

(32.5-3) resp. `PreImagesSet`

(32.5-5) 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 ) | ( function ) |

`‣ PreImages` ( map, elm ) | ( function ) |

`‣ PreImages` ( map, coll ) | ( function ) |

`PreImages( `

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

`PreImages( `

is the set of preimages of the element `map`, `elm` )`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( `

is the set of images of the subset `map`, `coll` )`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`

(32.5-1) when called with one argument, and to `PreImagesElm`

(32.5-2) resp. `PreImagesSet`

(32.5-5) 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.

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`

(32.2-5).

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`

(31.10-2) and `Inverse`

(31.10-8) for general mappings as follows. `One`

(31.10-2) returns `fail`

when called for a general mapping whose source and range differ, otherwise `One`

(31.10-2) returns the identity mapping of the source. (Note that the source may differ from the preimage). `Inverse`

(31.10-8) returns `fail`

when called for a non-bijective general mapping or for a general mapping whose source and range differ; otherwise `Inverse`

(31.10-8) returns the inverse mapping.

Besides the usual inverse of multiplicative elements, which means that `Inverse( `

, for general mappings we have the attribute `g` ) * `g` = `g` * Inverse( `g` ) = One( `g` )`InverseGeneralMapping`

(32.2-3). If `F` is a general mapping with source S, range R, and underlying relation Rel then `InverseGeneralMapping( `

has source R, range S, and underlying relation { (r,s) ∣ (s,r) ∈ 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.`F` )

`Inverse`

(31.10-8) may delegate to `InverseGeneralMapping`

(32.2-3). `InverseGeneralMapping`

(32.2-3) must not delegate to `Inverse`

(31.10-8), but a known value of `Inverse`

(31.10-8) may be fetched. So methods to compute the inverse of a general mapping should be installed for `InverseGeneralMapping`

(32.2-3).

(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**.)

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:

General mappings that respect multiplication

General mappings that respect addition

General mappings that respect scalar mult.

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.)

`‣ IsMagmaHomomorphism` ( mapp ) | ( filter ) |

A *magma homomorphism* is a total single valued mapping which respects multiplication.

`‣ MagmaHomomorphismByFunctionNC` ( G, H, fn ) | ( function ) |

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

`‣ NaturalHomomorphismByGenerators` ( f, s ) | ( operation ) |

returns a mapping from the magma `f` with n generators to the magma `s` with n generators, which maps the i-th generator of `f` to the i-th generator of `s`.

`‣ RespectsMultiplication` ( mapp ) | ( property ) |

Let `mapp` be a general mapping with underlying relation F ⊆ 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 (s_1,r_1), (s_2,r_2) ∈ F implies (s_1 * s_2,r_1 * r_2) ∈ F, and `false`

otherwise.

If `mapp` is single-valued then `RespectsMultiplication`

returns `true`

if and only if the equation

holds for all `s1`^`mapp` * `s2`^`mapp` = (`s1` * `s2`)^`mapp``s1`, `s2` in S.

`‣ RespectsOne` ( mapp ) | ( property ) |

Let `mapp` be a general mapping with underlying relation F ⊆ `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(`

) ∈ F, and `R`)`false`

otherwise.

If `mapp` is single-valued then `RespectsOne`

returns `true`

if and only if the equation `One( `

holds.`S` )^`mapp` = One( `R` )

`‣ RespectsInverses` ( mapp ) | ( property ) |

Let `mapp` be a general mapping with underlying relation F ⊆ `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 ∈ `S` and r ∈ `R`, (s,r) ∈ F implies (s^{-1},r^{-1}) ∈ F, and `false`

otherwise.

If `mapp` is single-valued then `RespectsInverses`

returns `true`

if and only if the equation `Inverse( `

holds for all `s` )^`mapp` = Inverse( `s`^`mapp` )`s` in S.

`‣ IsGroupGeneralMapping` ( mapp ) | ( filter ) |

`‣ IsGroupHomomorphism` ( mapp ) | ( filter ) |

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

Chapter 40 explains group homomorphisms in more detail.

`‣ KernelOfMultiplicativeGeneralMapping` ( mapp ) | ( attribute ) |

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 ) | ( attribute ) |

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.)

`‣ RespectsAddition` ( mapp ) | ( property ) |

Let `mapp` be a general mapping with underlying relation F ⊆ 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 (s_1,r_1), (s_2,r_2) ∈ F implies (s_1 + s_2,r_1 + r_2) ∈ F, and `false`

otherwise.

If `mapp` is single-valued then `RespectsAddition`

returns `true`

if and only if the equation

holds for all `s1`^`mapp` + `s2`^`mapp` = (`s1`+`s2`)^`mapp``s1`, `s2` in S.

`‣ RespectsAdditiveInverses` ( mapp ) | ( property ) |

Let `mapp` be a general mapping with underlying relation F ⊆ 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) ∈ F implies (-s,-r) ∈ F, and `false`

otherwise.

If `mapp` is single-valued then `RespectsAdditiveInverses`

returns `true`

if and only if the equation `AdditiveInverse( `

holds for all `s` )^`mapp` = AdditiveInverse( `s`^`mapp` )`s` in S.

`‣ RespectsZero` ( mapp ) | ( property ) |

Let `mapp` be a general mapping with underlying relation F ⊆ `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(`

) ∈ F, and `R`)`false`

otherwise.

If `mapp` is single-valued then `RespectsZero`

returns `true`

if and only if the equation `Zero( `

holds.`S` )^`mapp` = Zero( `R` )

`‣ IsAdditiveGroupGeneralMapping` ( mapp ) | ( filter ) |

`‣ IsAdditiveGroupHomomorphism` ( mapp ) | ( filter ) |

`IsAdditiveGroupGeneralMapping`

specifies whether a general mapping `mapp` respects addition (see `RespectsAddition`

(32.10-1)) and respects additive inverses (see `RespectsAdditiveInverses`

(32.10-2)).

`IsAdditiveGroupHomomorphism`

is a synonym for the meet of `IsAdditiveGroupGeneralMapping`

and `IsMapping`

(32.3-3).

`‣ KernelOfAdditiveGeneralMapping` ( mapp ) | ( attribute ) |

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 ) | ( attribute ) |

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.

Also see Sections 32.9, 32.10, and `KernelOfMultiplicativeGeneralMapping`

(32.9-5), `CoKernelOfMultiplicativeGeneralMapping`

(32.9-6).

`‣ RespectsScalarMultiplication` ( mapp ) | ( property ) |

Let `mapp` be a general mapping, with underlying relation F ⊆ 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) ∈ F implies (c * s,c * r) ∈ F for all c ∈ D, and `false`

otherwise.

If `mapp` is single-valued then `RespectsScalarMultiplication`

returns `true`

if and only if the equation

holds for all `c` * `s`^`mapp` = (`c` * `s`)^`mapp``c` in D and `s` in S.

`‣ IsLeftModuleGeneralMapping` ( mapp ) | ( filter ) |

`‣ IsLeftModuleHomomorphism` ( mapp ) | ( filter ) |

`IsLeftModuleGeneralMapping`

specifies whether a general mapping `mapp` satisfies the property `IsAdditiveGroupGeneralMapping`

(32.10-4) and respects scalar multiplication (see `RespectsScalarMultiplication`

(32.11-1)).

`IsLeftModuleHomomorphism`

is a synonym for the meet of `IsLeftModuleGeneralMapping`

and `IsMapping`

(32.3-3).

`‣ IsLinearMapping` ( F, mapp ) | ( operation ) |

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 ∈ D.

`‣ IsRingGeneralMapping` ( mapp ) | ( filter ) |

`‣ IsRingHomomorphism` ( mapp ) | ( filter ) |

`IsRingGeneralMapping`

specifies whether a general mapping `mapp` satisfies the property `IsAdditiveGroupGeneralMapping`

(32.10-4) and respects multiplication (see `RespectsMultiplication`

(32.9-1)).

`IsRingHomomorphism`

is a synonym for the meet of `IsRingGeneralMapping`

and `IsMapping`

(32.3-3).

`‣ IsRingWithOneGeneralMapping` ( mapp ) | ( filter ) |

`‣ IsRingWithOneHomomorphism` ( mapp ) | ( filter ) |

`‣ IsAlgebraGeneralMapping` ( mapp ) | ( filter ) |

`‣ IsAlgebraHomomorphism` ( mapp ) | ( filter ) |

`IsAlgebraGeneralMapping`

specifies whether a general mapping `mapp` satisfies both properties `IsRingGeneralMapping`

(32.12-1) and (see `IsLeftModuleGeneralMapping`

(32.11-2)).

`IsAlgebraHomomorphism`

is a synonym for the meet of `IsAlgebraGeneralMapping`

and `IsMapping`

(32.3-3).

`‣ IsAlgebraWithOneGeneralMapping` ( mapp ) | ( filter ) |

`‣ IsAlgebraWithOneHomomorphism` ( mapp ) | ( filter ) |

`IsAlgebraWithOneGeneralMapping`

specifies whether a general mapping `mapp` satisfies both properties `IsAlgebraGeneralMapping`

(32.12-3) and `RespectsOne`

(32.9-2).

`IsAlgebraWithOneHomomorphism`

is a synonym for the meet of `IsAlgebraWithOneGeneralMapping`

and `IsMapping`

(32.3-3).

`‣ IsFieldHomomorphism` ( mapp ) | ( property ) |

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

`‣ IsGeneralMapping` ( map ) | ( category ) |

Each general mapping lies in the category `IsGeneralMapping`

. It implies the categories `IsMultiplicativeElementWithInverse`

(31.14-13) and `IsAssociativeElement`

(31.15-1); for a discussion of these implications, see 32.6.

`‣ IsConstantTimeAccessGeneralMapping` ( map ) | ( property ) |

is `true`

if the underlying relation of the general mapping `map` knows its `AsList`

(30.3-8) value, and `false`

otherwise.

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

(32.4-3) etc.

`‣ IsEndoGeneralMapping` ( obj ) | ( property ) |

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

`Source`

(32.3-8) and `Range`

(32.3-7) are basic operations for general mappings. `UnderlyingRelation`

(32.3-9) 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`

(32.14-1), `IsSPGeneralMapping`

(32.14-1). (The category `IsGeneralMapping`

(32.13-1) is defined as the disjoint union of these two.)

For general mappings of the first category, `ImagesElm`

(32.4-3) and `PreImagesElm`

(32.5-2) are basic operations. (Note that in principle it is possible to delegate from `PreImagesElm`

(32.5-2) to `ImagesElm`

(32.4-3).) Methods for the secondary operations `ImageElm`

(32.4-5), `PreImageElm`

(32.5-3), `ImagesSet`

(32.4-4), `PreImagesSet`

(32.5-5), `ImagesRepresentative`

(32.4-2), and `PreImagesRepresentative`

(32.5-4) may use `ImagesElm`

(32.4-3) and `PreImagesElm`

(32.5-2), respectively, and methods for `ImagesElm`

(32.4-3), `PreImagesElm`

(32.5-2) must *not* call the secondary operations. In particular, there are no generic methods for `ImagesElm`

(32.4-3) and `PreImagesElm`

(32.5-2).

Methods for `ImagesSet`

(32.4-4) and `PreImagesSet`

(32.5-5) must *not* use `PreImagesRange`

(32.5-1) and `ImagesSource`

(32.4-1), 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 `ImagesRepresentative`

(32.4-2), `PreImagesRepresentative`

(32.5-4), `KernelOfMultiplicativeGeneralMapping`

(32.9-5), and `CoKernelOfMultiplicativeGeneralMapping`

(32.9-6) as basic operations here, and to make `ImagesElm`

(32.4-3) and `PreImagesElm`

(32.5-2) 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`

(30.3-8) for the underlying relation of a general mapping may use either `ImagesElm`

(32.4-3) or `ImagesRepresentative`

(32.4-2) and the appropriate cokernel. Conversely, if `AsList`

(30.3-8) for the underlying relation is known then `ImagesElm`

(32.4-3) resp. `ImagesRepresentative`

(32.4-2) may delegate to it, the general mapping gets the property `IsConstantTimeAccessGeneralMapping`

(32.13-2) for this; note that this is not allowed if only an enumerator of the underlying relation is known.)

Secondary operations are `IsInjective`

(32.3-4), `IsSingleValued`

(32.3-2), `IsSurjective`

(32.3-5), `IsTotal`

(32.3-1); they may use the basic operations, and must not be used by them.

Methods for the operations `ImagesElm`

(32.4-3), `ImagesRepresentative`

(32.4-2), `ImagesSet`

(32.4-4), `ImageElm`

(32.4-5), `PreImagesElm`

(32.5-2), `PreImagesRepresentative`

(32.5-4), `PreImagesSet`

(32.5-5), and `PreImageElm`

(32.5-3) 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`

(32.4-6), `Images`

(32.4-7), `PreImage`

(32.5-6), and `PreImages`

(32.5-7), which then delegate to the operations listed above.

`‣ IsSPGeneralMapping` ( map ) | ( category ) |

`‣ IsNonSPGeneralMapping` ( map ) | ( category ) |

`‣ IsGeneralMappingFamily` ( obj ) | ( category ) |

The family category of the category of general mappings.

`‣ FamilyRange` ( Fam ) | ( attribute ) |

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

`‣ FamilySource` ( Fam ) | ( attribute ) |

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

`‣ FamiliesOfGeneralMappingsAndRanges` ( Fam ) | ( attribute ) |

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 ) | ( function ) |

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

.`FS`, `FR` )

`‣ TypeOfDefaultGeneralMapping` ( source, range, filter ) | ( function ) |

is the type of mappings with `IsDefaultGeneralMappingRep`

with source `source` and range `range` and additional categories `filter`.

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