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32 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 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.

32.1 Direct Products and their Elements

32.1-1 IsDirectProductElement

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.

32.1-2 DirectProductFamily

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

32.2 Creating Mappings

32.2-1 GeneralMappingByElements

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

32.2-2 MappingByFunction

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.

32.2-3 InverseGeneralMapping

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( map ) works.

32.2-4 RestrictedInverseGeneralMapping

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.

32.2-5 CompositionMapping

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) ] -> [ () ]

32.2-6 CompositionMapping2

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.)

32.2-7 IsCompositionMappingRep

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).

32.2-8 ConstituentsCompositionMapping

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 ].

32.2-9 ZeroMapping

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.)

32.2-10 IdentityMapping

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

32.2-11 Embedding

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.

32.2-12 Projection

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.

32.2-13 RestrictedMapping

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

32.3 Properties and Attributes of (General) Mappings

32.3-1 IsTotal

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.

32.3-2 IsSingleValued

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

32.3-3 IsMapping

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 (32.3-1), IsSingleValued (32.3-2)).

32.3-4 IsInjective

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

32.3-5 IsSurjective

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.

32.3-6 IsBijective

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)).

32.3-7 Range

The range of a general mapping.

32.3-8 Source

The source of a general mapping.

32.3-9 UnderlyingRelation

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( map, s ).

Each element of the underlying relation is represented by a direct product element (see IsDirectProductElement (32.1-1)).

32.3-10 UnderlyingGeneralMapping

attribute for underlying relations of general mappings

32.4 Images under Mappings

32.4-1 ImagesSource

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.

32.4-2 ImagesRepresentative

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.

32.4-3 ImagesElm

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.

32.4-4 ImagesSet

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.

32.4-5 ImageElm

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.

32.4-6 Image

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 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( 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( map, coll ) can also be expressed as map(coll) and Image( map, elm ) as map(elm). Those using this notation should remember that composition of mappings in GAP still follows the conventions appropriate for mapping acting from the right, so that (map1*map2)(x) is equivalent to 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.

32.4-7 Images

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 (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.

32.5 Preimages under Mappings

32.5-1 PreImagesRange

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.

32.5-2 PreImagesElm

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.

32.5-3 PreImageElm

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.

32.5-4 PreImagesRepresentative

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.

32.5-5 PreImagesSet

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.

32.5-6 PreImage

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 (32.5-7)).

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 (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.

32.5-7 PreImages

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 (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.

32.6 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 (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( g ) * g = g * Inverse( g ) = One( g ), for general mappings we have the attribute InverseGeneralMapping (32.2-3). 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) ∈ 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 (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.)

32.7 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.)

32.8 Magma Homomorphisms

32.8-1 IsMagmaHomomorphism

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

32.8-2 MagmaHomomorphismByFunctionNC

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

32.8-3 NaturalHomomorphismByGenerators

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.

32.9 Mappings that Respect Multiplication

32.9-1 RespectsMultiplication

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

32.9-2 RespectsOne

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(R)) ∈ 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.

32.9-3 RespectsInverses

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

32.9-4 IsGroupGeneralMapping

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.

32.9-5 KernelOfMultiplicativeGeneralMapping

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.)

32.9-6 CoKernelOfMultiplicativeGeneralMapping

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.)

32.10 Mappings that Respect Addition

32.10-1 RespectsAddition

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

32.10-2 RespectsAdditiveInverses

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

32.10-3 RespectsZero

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(R)) ∈ 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.

32.10-4 IsAdditiveGroupGeneralMapping

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).

32.10-5 KernelOfAdditiveGeneralMapping

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.

32.10-6 CoKernelOfAdditiveGeneralMapping

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.

32.11 Linear Mappings

Also see Sections 32.9, 32.10, and KernelOfMultiplicativeGeneralMapping (32.9-5), CoKernelOfMultiplicativeGeneralMapping (32.9-6).

32.11-1 RespectsScalarMultiplication

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

32.11-2 IsLeftModuleGeneralMapping

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).

32.11-3 IsLinearMapping

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.

32.12 Ring Homomorphisms

32.12-1 IsRingGeneralMapping

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).

32.12-2 IsRingWithOneGeneralMapping

32.12-3 IsAlgebraGeneralMapping

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).

32.12-4 IsAlgebraWithOneGeneralMapping

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).

32.12-5 IsFieldHomomorphism

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

32.13 General Mappings

32.13-1 IsGeneralMapping

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.

32.13-2 IsConstantTimeAccessGeneralMapping

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.

32.13-3 IsEndoGeneralMapping

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

32.14 Technical Matters Concerning General Mappings

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.

32.14-1 IsSPGeneralMapping

32.14-2 IsGeneralMappingFamily

The family category of the category of general mappings.

32.14-3 FamilyRange

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

32.14-4 FamilySource

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

32.14-5 FamiliesOfGeneralMappingsAndRanges

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.

32.14-6 GeneralMappingsFamily

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

32.14-7 TypeOfDefaultGeneralMapping

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

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