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.
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.
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
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.
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.)
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.
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.)
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.
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).
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.
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