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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 IsDirectProductElement (Filter)
  
  32.1-1 IsDirectProductElement
  
  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   a  non-list  is  the  direct  product  element  of
  componentwise sums and products, respectively.
  
  
  32.2 Creating Mappings
  
  32.2-1 GeneralMappingByElements
  
  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.
  
  
  32.2-2 MappingByFunction
  
  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.
  
  32.2-3 InverseGeneralMapping
  
  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( map ) works.
  
  32.2-4 RestrictedInverseGeneralMapping
  
  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.
  
  32.2-5 CompositionMapping
  
  CompositionMapping( map1, map2, ... )  function
  
  CompositionMapping  allows one to compose arbitrarily many general mappings,
  and delegates each step to CompositionMapping2 (32.2-6).
  
  Additionally,   the   properties  IsInjective  (32.3-4)  and  IsSingleValued
  (32.3-2)  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 (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)).
  
  32.2-6 CompositionMapping2
  
  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.)
  
  32.2-7 IsCompositionMappingRep
  
  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).
  
  32.2-8 ConstituentsCompositionMapping
  
  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 ].
  
  32.2-9 ZeroMapping
  
  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.)
  
  32.2-10 IdentityMapping
  
  IdentityMapping( D )  attribute
  
  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
  
  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.
  
  
  32.2-12 Projection
  
  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.
  
  32.2-13 RestrictedMapping
  
  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.
  
  
  32.3 Properties and Attributes of (General) Mappings
  
  32.3-1 IsTotal
  
  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.
  
  32.3-2 IsSingleValued
  
  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( map ) is true if and only if the preimages of
  different elements in R are disjoint.
  
  32.3-3 IsMapping
  
  IsMapping( map )  filter
  
  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
  
  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( 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
  
  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.
  
  32.3-6 IsBijective
  
  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)).
  
  32.3-7 Range
  
  Range( map )  attribute
  
  The range of a general mapping.
  
  32.3-8 Source
  
  Source( map )  attribute
  
  The source of a general mapping.
  
  32.3-9 UnderlyingRelation
  
  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( map, s ).
  
  Each  element  of the underlying relation is represented by a direct product
  element (see IsDirectProductElement (32.1-1)).
  
  32.3-10 UnderlyingGeneralMapping
  
  UnderlyingGeneralMapping( map )  attribute
  
  attribute for underlying relations of general mappings
  
  
  32.4 Images under Mappings
  
  32.4-1 ImagesSource
  
  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.
  
  32.4-2 ImagesRepresentative
  
  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.
  
  32.4-3 ImagesElm
  
  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.
  
  32.4-4 ImagesSet
  
  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.
  
  32.4-5 ImageElm
  
  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.
  
  
  32.4-6 Image
  
  Image( map )  function
  Image( map, elm )  function
  Image( map, coll )  function
  
  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
  (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  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 )  function
  Images( map, elm )  function
  Images( map, coll )  function
  
  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
  
  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.
  
  32.5-2 PreImagesElm
  
  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.
  
  32.5-3 PreImageElm
  
  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.
  
  32.5-4 PreImagesRepresentative
  
  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.
  
  32.5-5 PreImagesSet
  
  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.
  
  
  32.5-6 PreImage
  
  PreImage( map )  function
  PreImage( map, elm )  function
  PreImage( map, coll )  function
  
  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 )  function
  PreImages( map, elm )  function
  PreImages( map, coll )  function
  
  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
  
  IsMagmaHomomorphism( mapp )  filter
  
  A  magma  homomorphism  is  a  total  single  valued  mapping which respects
  multiplication.
  
  32.8-2 MagmaHomomorphismByFunctionNC
  
  MagmaHomomorphismByFunctionNC( G, H, fn )  function
  
  Creates  the  homomorphism  from  G  to  H  without  checking  that  fn is a
  homomorphism.
  
  32.8-3 NaturalHomomorphismByGenerators
  
  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.
  
  
  32.9 Mappings that Respect Multiplication
  
  32.9-1 RespectsMultiplication
  
  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 s1^mapp * s2^mapp = (s1 * s2)^mapp holds for all s1, s2
  in S.
  
  32.9-2 RespectsOne
  
  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(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
  
  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( s )^mapp = Inverse( s^mapp ) holds for all s in S.
  
  32.9-4 IsGroupGeneralMapping
  
  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.
  
  32.9-5 KernelOfMultiplicativeGeneralMapping
  
  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.)
  
  32.9-6 CoKernelOfMultiplicativeGeneralMapping
  
  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.)
  
  
  32.10 Mappings that Respect Addition
  
  32.10-1 RespectsAddition
  
  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 s1^mapp + s2^mapp = (s1+s2)^mapp holds for all s1, s2 in S.
  
  32.10-2 RespectsAdditiveInverses
  
  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( s )^mapp = AdditiveInverse( s^mapp )
  holds for all s in S.
  
  32.10-3 RespectsZero
  
  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(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( 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).
  
  32.10-5 KernelOfAdditiveGeneralMapping
  
  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.
  
  32.10-6 CoKernelOfAdditiveGeneralMapping
  
  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.
  
  
  32.11 Linear Mappings
  
  Also  see  Sections 32.9,  32.10,  and  KernelOfMultiplicativeGeneralMapping
  (32.9-5), CoKernelOfMultiplicativeGeneralMapping (32.9-6).
  
  32.11-1 RespectsScalarMultiplication
  
  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 c * s^mapp = (c * s)^mapp holds for all c in D and
  s in S.
  
  32.11-2 IsLeftModuleGeneralMapping
  
  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).
  
  32.11-3 IsLinearMapping
  
  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.
  
  
  32.12 Ring Homomorphisms
  
  32.12-1 IsRingGeneralMapping
  
  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).
  
  32.12-2 IsRingWithOneGeneralMapping
  
  IsRingWithOneGeneralMapping( mapp )  filter
  IsRingWithOneHomomorphism( mapp )  filter
  
  32.12-3 IsAlgebraGeneralMapping
  
  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).
  
  32.12-4 IsAlgebraWithOneGeneralMapping
  
  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).
  
  32.12-5 IsFieldHomomorphism
  
  IsFieldHomomorphism( mapp )  property
  
  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
  
  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.
  
  32.13-2 IsConstantTimeAccessGeneralMapping
  
  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.
  
  32.13-3 IsEndoGeneralMapping
  
  IsEndoGeneralMapping( obj )  property
  
  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
  
  IsSPGeneralMapping( map )  Category
  IsNonSPGeneralMapping( map )  Category
  
  32.14-2 IsGeneralMappingFamily
  
  IsGeneralMappingFamily( obj )  Category
  
  The family category of the category of general mappings.
  
  32.14-3 FamilyRange
  
  FamilyRange( Fam )  attribute
  
  is the elements family of the family of the range of each general mapping in
  the family Fam.
  
  32.14-4 FamilySource
  
  FamilySource( Fam )  attribute
  
  is  the  elements family of the family of the source of each general mapping
  in the family Fam.
  
  32.14-5 FamiliesOfGeneralMappingsAndRanges
  
  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.
  
  32.14-6 GeneralMappingsFamily
  
  GeneralMappingsFamily( sourcefam, rangefam )  function
  
  All general mappings with same source family FS and same range family FR lie
  in the family GeneralMappingsFamily( FS, FR ).
  
  32.14-7 TypeOfDefaultGeneralMapping
  
  TypeOfDefaultGeneralMapping( source, range, filter )  function
  
  is  the  type of mappings with IsDefaultGeneralMappingRep with source source
  and range range and additional categories filter.