  
  [1X4 Groupoids[0X
  
  [13XMany  of  the names of the functions described in this chapter have changed,
  due  to  the  introduction of magmas with objects, so the chapter is full of
  errors. A new version will be released as soon as possible.[0m
  
  A   [13Xgroupoid[0m  is  a  (mathematical)  category  in  which  every  element  is
  invertible.  It  consists  of  a set of [13Xpieces[0m, each of which is a connected
  groupoid.  (The  usual  terminology  is  `connected  component',  but in [5XGAP[0m
  `component' is used for `record component'.)
  
  A  [13Xsingle piece groupoid[0m is determined by a set of [13Xobjects[0m [10Xobs[0m and an [13Xobject
  group[0m [10Xgrp[0m. The objects of a single piece groupoid are generally chosen to be
  consecutive  negative  integers, but any suitable ordered set is acceptable,
  and  `consecutive'  is  not a requirement. The object groups will usually be
  taken  to  be  permutation  groups,  but  pc-groups  and  fp-groups are also
  supported.
  
  A [13Xgroup[0m is a single piece groupoid with one object.
  
  A  [13Xgroupoid[0m is a set of one or more single piece groupoids, its single piece
  [13Xpieces[0m,    and    is    represented   as   [10XIsGroupoidRep[0m,   with   attribute
  [10XPiecesOfGroupoid[0m.
  
  For  the  definitions of the standard properties of groupoids we refer to R.
  Brown's   book  ``Topology''  [Bro88],  recently  revised  and  reissued  as
  ``Topology and Groupoids'' [Bro06].
  
  
  [1X4.1 Groupoids: their elements and attributes[0X
  
  [1X4.1-1 SinglePieceGroupoid[0m
  
  [2X> SinglePieceGroupoid( [0X[3Xgrp, obs[0X[2X ) _________________________________[0Xoperation
  [2X> Groupoid( [0X[3Xargs[0X[2X ) _________________________________________________[0Xfunction
  
  There  are  a  variety  of  constructors  for  groupoids,  with  one  or two
  parameters.  The global function [10XGroupoid[0m will normally find the appropriate
  one to call, the options being:
  
  --    the object group, a list of objects;
  
  --    a group being converted to a groupoid, a single object;
  
  --    a list of groupoids which have already been constructed.
  
  Methods for [10XViewObj[0m, [10XPrintObj[0m and [10XDisplay[0m are provided for groupoids and the
  other types of object in this package. Users are advised to supply names for
  all the groups and groupoids they construct.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> d8 := Group( (1,2,3,4), (1,3) );;[0X
    [4Xgap> SetName( d8, "d8" );[0X
    [4Xgap> Gd8 := SinglePieceGroupoid( d8, [-9,-8,-7] );[0X
    [4XPerm single piece groupoid:[0X
    [4X< d8, [ -9, -8, -7 ] >[0X
    [4Xgap> c6 := Group( (5,6,7)(8,9) );;[0X
    [4Xgap> SetName( c6, "c6" );[0X
    [4Xgap> Gc6 := DomainWithSingleObject( c6, -6 );[0X
    [4XPerm SinglePiece Groupoid:[0X
    [4X< c6, [ -6 ] >[0X
    [4Xgap> Gd8c6 := UnionOfPieces( [ Gd8, Gc6 ] );;[0X
    [4Xgap> Display( Gd8c6 );[0X
    [4XPerm Groupoid with 2 pieces:[0X
    [4X< objects: [ -9, -8, -7 ][0X
    [4X    group: d8 = <[ (1,2,3,4), (1,3) ]> >[0X
    [4X< objects: [ -6 ][0X
    [4X    group: c6 = <[ (5,6,7)(8,9) ]> >[0X
    [4Xgap> SetName( Gd8, "Gd8" );  SetName( Gc6, "Gc6" );  SetName( Gd8c6, "Gd8+Gc6" );[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.1-2 Pieces[0m
  
  [2X> Pieces( [0X[3Xgpd[0X[2X ) ___________________________________________________[0Xattribute
  [2X> ObjectList( [0X[3Xgpd[0X[2X ) _______________________________________________[0Xattribute
  
  When  a  groupoid  consists  of  two or more pieces, we require their object
  lists  to  be  disjoint.  The pieces are sorted by the first object in their
  object  lists,  which  must  be  disjoint.  The  list [10XObjectsOfGroupoid[0m of a
  groupoid is the sorted concatenation of the objects in the pieces.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> Pieces( Gd8c6 );[0X
    [4X[ Gd8, Gc6 ][0X
    [4Xgap> ObjectList( Gd8c6 );[0X
    [4X[ -9, -8, -7, -6 ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.1-3 IsPermGroupoid[0m
  
  [2X> IsPermGroupoid( [0X[3Xgpd[0X[2X ) ____________________________________________[0Xproperty
  [2X> IsPcGroupoid( [0X[3Xgpd[0X[2X ) ______________________________________________[0Xproperty
  [2X> IsFpGroupoid( [0X[3Xgpd[0X[2X ) ______________________________________________[0Xproperty
  
  A  groupoid  is  a  permutation  groupoid if all its pieces have permutation
  groups.  Most of the examples in this chapter are permutation groupoids, but
  in principle any type of group known to [5XGAP[0m may be used.
  
  In the following example [10XGf2[0m is an fp-groupoid, while [10XGq8[0m is a pc-groupoid.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> f2 := FreeGroup( 2 );;[0X
    [4Xgap> SetName( f2, "f2" );[0X
    [4Xgap> Gf2 := Groupoid( f2, -22 );;[0X
    [4Xgap> q8 := SmallGroup( 8, 4 );;[0X
    [4Xgap> Gq8 := Groupoid( q8, [ -28, -27 ] );;[0X
    [4Xgap> SetName( q8, "q8" );  SetName( Gq8, "Gq8" );[0X
    [4Xgap> Gf2q8 := Groupoid( [ Gf2, Gq8 ] );;[0X
    [4Xgap> [ IsFpGroupoid( Gf2 ), IsPcGroupoid( Gq8 ), IsPcGroupoid( Gf2q8 ) ];[0X
    [4X[ true, true, false ][0X
    [4Xgap> G4 := Groupoid( [ Gd8c6, Gf2, Gq8 ] );;[0X
    [4Xgap> Display( G4 );[0X
    [4XGroupoid with 4 pieces:[0X
    [4X< objects: [ -28, -27 ][0X
    [4X    group: q8 = <[ f1, f2, f3 ]> >[0X
    [4X< objects: [ -22 ][0X
    [4X    group: f2 = <[ f1, f2 ]> >[0X
    [4X< objects: [ -9, -8, -7 ][0X
    [4X    group: d8 = <[ (1,2,3,4), (1,3) ]> >[0X
    [4X< objects: [ -6 ][0X
    [4X    group: c6 = <[ (5,6,7)(8,9) ]> >[0X
    [4Xgap> G4 = Groupoid( [ Gq8, Gf2, Gd8c6 ] );[0X
    [4Xtrue[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.1-4 GroupoidElement[0m
  
  [2X> GroupoidElement( [0X[3Xgpd, elt, tail, head[0X[2X ) _________________________[0Xoperation
  [2X> IsElementOfGroupoid( [0X[3Xelt[0X[2X ) _______________________________________[0Xproperty
  [2X> Arrow( [0X[3Xelt[0X[2X ) ____________________________________________________[0Xattribute
  [2X> Arrowtail( [0X[3Xelt[0X[2X ) ________________________________________________[0Xattribute
  [2X> Arrowhead( [0X[3Xelt[0X[2X ) ________________________________________________[0Xattribute
  [2X> Size( [0X[3Xgpd[0X[2X ) _____________________________________________________[0Xattribute
  
  A  [13Xgroupoid element[0m [10Xe[0m is a triple consisting of a group element, [10XArrow(e)[0m or
  [10Xe![1][0m;  the  tail  (source)  object,  [10XArrowtail(e)[0m  or  [10Xe![2][0m;  and the head
  (target) object, [10XArrowhead(e)[0m or [10Xe![3][0m.
  
  The  [10XSize[0m  of  a  groupoid is the number of its elements which, for a single
  piece  groupoid,  is the product of the size of the group with the square of
  the number of objects.
  
  Groupoid elements have a [13Xpartial composition[0m: two elements may be multiplied
  when the head of the first coincides with the tail of the second.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> e1 := GroupoidElement( Gd8, (1,2)(3,4), -9, -8 );[0X
    [4X[(1,2)(3,4) : -9 -> -8][0X
    [4Xgap> e2 := GroupoidElement( Gd8, (1,3), -8, -7 );;[0X
    [4Xgap> Print( [ Arrow( e2 ), Arrowtail( e2 ), Arrowhead( e2 ) ], "\n" );[0X
    [4X[ (1,3), -8, -7 ][0X
    [4Xgap> prod := e1*e2;[0X
    [4X[(1,2,3,4) : -9 -> -7][0X
    [4Xgap> e3 := GroupoidElement( Gd8, (1,3)(2,4), -7, -9 );;[0X
    [4Xgap> cycle := prod*e3;[0X
    [4X[(1,4,3,2) : -9 -> -9][0X
    [4Xgap> cycle^2;[0X
    [4X[(1,3)(2,4) : -9 -> -9][0X
    [4Xgap> Order( cycle );[0X
    [4X4[0X
    [4Xgap> cycle^e1;[0X
    [4X[(1,2,3,4) : -8 -> -8][0X
    [4Xgap> [ Size( Gd8 ), Size( Gc6 ), Size( Gd8c6 ), Size( Gf2q8 ) ];[0X
    [4X[ 72, 6, 78, infinity ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.1-5 IsSinglePiece[0m
  
  [2X> IsSinglePiece( [0X[3Xgpd[0X[2X ) _____________________________________________[0Xproperty
  [2X> IsDiscrete( [0X[3Xgpd[0X[2X ) ________________________________________________[0Xproperty
  
  The  forgetful  functor, which forgets the composition of elements, maps the
  category  of  groupoids  and their morphisms to the category of digraphs and
  their  morphisms.  Applying  this functor to a particular groupoid gives the
  [13Xunderlying  digraph[0m  of  the  groupoid.  A  groupoid  is  [13Xconnected[0m  if  its
  underlying digraph is connected (and so complete). A groupoid is [13Xdiscrete[0m if
  it  is a union of groups, so that all the arcs in its underlying digraph are
  loops. It is sometimes convenient to call a groupoid element a [13Xloop[0m when its
  tail and head are the same object.
  
  
  [1X4.2 Subgroupoids[0X
  
  [1X4.2-1 SubgroupoidByPieces[0m
  
  [2X> SubgroupoidByPieces( [0X[3Xgpd, obhoms[0X[2X ) ______________________________[0Xoperation
  [2X> Subgroupoid( [0X[3Xargs[0X[2X ) ______________________________________________[0Xfunction
  [2X> FullSubgroupoid( [0X[3Xgpd, obs[0X[2X ) _____________________________________[0Xoperation
  [2X> MaximalDiscreteSubgroupoid( [0X[3Xgpd[0X[2X ) _______________________________[0Xattribute
  [2X> DiscreteSubgroupoid( [0X[3Xgpd, obs, sgps[0X[2X ) ___________________________[0Xoperation
  [2X> FullIdentitySubgroupoid( [0X[3Xgpd[0X[2X ) __________________________________[0Xattribute
  [2X> DiscreteIdentitySubgroupoid( [0X[3Xgpd[0X[2X ) ______________________________[0Xattribute
  
  A  [13Xsubgroupoid[0m [10Xsgpd[0m of [10Xgpd[0m has as objects a subset of the objects of [10Xgpd[0m. It
  is  [13Xwide[0m if all the objects are included. It is [13Xfull[0m if, for any two objects
  in [10Xsgpd[0m, the [10XHomset[0m is the same as in [10Xgpd[0m. The elements of [10Xsgpd[0m are a subset
  of  those  of [10Xgpd[0m, closed under multiplication and with tail and head in the
  chosen object set.
  
  There  are  a  variety  of constructors for a subgroupoid of a groupoid. The
  operation  [10XSubgroupoidByPieces[0m is the most general. Its two parameters are a
  groupoid  and  a  list  of  pieces,  where each piece is specified as a list
  [10X[obs,sgp][0m,  [10Xobs[0m  is a subset of the objects in one of the pieces of [10Xgpd[0m, and
  [10Xsgp[0m is a subgroup of the group in that piece.
  
  The  [10XFullSubgroupoid[0m  of  a  groupoid  [10Xgpd[0m  on  a  subset [10Xobs[0m of its objects
  contains all the elements of [10Xgpd[0m with tail and head in [10Xobs[0m.
  
  A   subgroupoid   is   [13Xdiscrete[0m   if   it   is   a   union  of  groups.  The
  [10XMaximalDiscreteSubgroupoid[0m of [10Xgpd[0m is the union of all the single-object full
  subgroupoids of [10Xgpd[0m.
  
  An [13Xidentity subgroupoid[0m has trivial object groups, but need not be discrete.
  A single piece identity groupoid is sometimes called a [13Xtree groupoid[0m.
  
  The global function [10XSubgroupoid[0m should call the appropriate operation.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> c4d := Subgroup( d8, [ (1,2,3,4) ] );;[0X
    [4Xgap> k4d := Subgroup( d8, [ (1,2)(3,4), (1,3)(2,4) ] );;[0X
    [4Xgap> SetName( c4d, "c4d" );  SetName( k4d, "k4d" );[0X
    [4Xgap> Ud8 := Subgroupoid( Gd8, [ [ k4d,[-9] ], [ c4d, [-8,-7] ] ] );;[0X
    [4Xgap> SetName( Ud8, "Ud8" );[0X
    [4Xgap> Display( Ud8 );[0X
    [4XPerm Groupoid with 2 pieces:[0X
    [4X< objects: [ -9 ][0X
    [4X    group: k4d = <[ (1,2)(3,4), (1,3)(2,4) ]> >[0X
    [4X< objects: [ -8, -7 ][0X
    [4X    group: c4d = <[ (1,2,3,4) ]> >[0X
    [4Xgap> FullSubgroupoid( Gd8c6, [-7,-6] );[0X
    [4XPerm Groupoid with pieces:[0X
    [4X< [ -7 ], d8 >[0X
    [4X< [ -6 ], c6 >[0X
    [4Xgap> DiscreteSubgroupoid( Gd8c6, [-9,-8], [ c4d, k4d ] );[0X
    [4XPerm Groupoid with pieces:[0X
    [4X< [ -9 ], c4d >[0X
    [4X< [ -8 ], k4d >[0X
    [4Xgap> FullIdentitySubgroupoid( Ud8 );[0X
    [4XPerm Groupoid with pieces:[0X
    [4X< [ -9 ], id(k4d) >[0X
    [4X< [ -8, -7 ], id(c4d) >[0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.3 Stars, Costars and Homsets[0X
  
  [1X4.3-1 ObjectStar[0m
  
  [2X> ObjectStar( [0X[3Xgpd, obj[0X[2X ) __________________________________________[0Xoperation
  [2X> ObjectCostar( [0X[3Xgpd, obj[0X[2X ) ________________________________________[0Xoperation
  [2X> Homset( [0X[3Xgpd, tail, head[0X[2X ) _______________________________________[0Xoperation
  
  The  [13Xstar[0m  at  [10Xobj[0m  is  the set of groupoid elements which have [10Xobj[0m as tail,
  while  the  [13Xcostar[0m is the set of elements which have [10Xobj[0m as head. The [13Xhomset[0m
  from  [10Xobj1[0m  to [10Xobj2[0m is the set of elements with the specified tail and head,
  and  so  is  bijective  with  the elements of the group. Thus every star and
  every costar is a union of homsets.
  
  In  order  not  to  create  unneccessary long lists, these operations return
  objects  of  type  [10XIsHomsetCosetsRep[0m  for which an [10XIterator[0m is provided. (An
  [10XEnumerator[0m is not yet implemented.)
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> star9 := ObjectStar( Gd8, -9 );[0X
    [4X<star at [ -9 ] with group d8>[0X
    [4Xgap> for e in star9 do[0X
    [4X>      if ( Order( e![1] ) = 4 ) then Print( e, "\n" ); fi;[0X
    [4X>    od;[0X
    [4X[(1,4,3,2) : -9 -> -9][0X
    [4X[(1,4,3,2) : -9 -> -8][0X
    [4X[(1,4,3,2) : -9 -> -7][0X
    [4X[(1,2,3,4) : -9 -> -9][0X
    [4X[(1,2,3,4) : -9 -> -8][0X
    [4X[(1,2,3,4) : -9 -> -7][0X
    [4Xgap> costar6 := ObjectCostar( Gc6, -6 );[0X
    [4X<costar at [ -6 ] with group c6>[0X
    [4Xgap> hset78 := Homset( Ud8, -7, -8 );[0X
    [4X<homset -7 -> -8 with group c4d>[0X
    [4Xgap> for e in hset78 do  Print( e, "\n" );  od;[0X
    [4X[() : -7 -> -8][0X
    [4X[(1,3)(2,4) : -7 -> -8][0X
    [4X[(1,4,3,2) : -7 -> -8][0X
    [4X[(1,2,3,4) : -7 -> -8][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.3-2 IdentityElement[0m
  
  [2X> IdentityElement( [0X[3Xgpd, obj[0X[2X ) _____________________________________[0Xoperation
  
  The  identity  groupoid element [10X1\_{o}[0m of [10Xgpd[0m at object [10Xo[0m is [10X[e,o,o][0m where [10Xe[0m
  is the identity group element in the object group. It is a left identity for
  the star and a right identity for the costar at that object.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> i7 := IdentityElement( Gd8, -7 );;[0X
    [4Xgap> i8 := IdentityElement( Gd8, -8 );;[0X
    [4Xgap> L := [ i7, i8 ];;[0X
    [4Xgap> for e in hset78 do  Add( L, i7*e*i8 = e );  od;[0X
    [4Xgap> L;[0X
    [4X[ [() : -7 -> -7], [() : -8 -> -8], true, true, true, true ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.4 Left, right and double cosets[0X
  
  [1X4.4-1 RightCoset[0m
  
  [2X> RightCoset( [0X[3XG, U, elt[0X[2X ) _________________________________________[0Xoperation
  [2X> RightCosetRepresentatives( [0X[3XG, U[0X[2X ) _______________________________[0Xoperation
  [2X> RightCosetsNC( [0X[3XG, U[0X[2X ) ___________________________________________[0Xoperation
  [2X> LeftCoset( [0X[3XG, U, elt[0X[2X ) __________________________________________[0Xoperation
  [2X> LeftCosetRepresentatives( [0X[3XG, U[0X[2X ) ________________________________[0Xoperation
  [2X> LeftCosetRepresentativesFromObject( [0X[3XG, U, obj[0X[2X ) _________________[0Xoperation
  [2X> LeftCosetsNC( [0X[3XG, U[0X[2X ) ____________________________________________[0Xoperation
  [2X> DoubleCoset( [0X[3XG, U, elt, V[0X[2X ) _____________________________________[0Xoperation
  [2X> DoubleCosetRepresentatives( [0X[3XG, U, V[0X[2X ) ___________________________[0Xoperation
  [2X> DoubleCosetsNC( [0X[3XG, U, V[0X[2X ) _______________________________________[0Xoperation
  
  If  [10XU[0m  is  a  wide  subgroupoid  of  G,  the  [13Xright cosets[0m of U in G are the
  equivalence classes of the relation on the elements of G where g1 is related
  to  g2  if  and  only  if g2 = u*g1 for some element u of U. The right coset
  containing  g is written Ug. These right cosets refine the costars of G and,
  in  particular,  U1_o  is the costar of U at o, so that (unlike groups) U is
  itself a coset only when G has a single object.
  
  The  [13Xright  coset  representatives[0m  for  U  in  G form a list containing one
  groupoid element for each coset where, in a particular piece of U, the group
  element  chosen  is  the right coset representative of the group of U in the
  group of G.
  
  Similarly, the [13Xleft cosets[0m gU refine the stars of G, while [13Xdouble cosets[0m are
  unions    of    left   cosets   and   of   right   cosets.   The   operation
  [10XLeftCosetRepresentativesFromObject(  G,  U,  obj )[0m is used in Chapter 4, and
  returns only those representatives which have tail at [10Xobj[0m.
  
  As  with stars and homsets, these cosets are implemented with representation
  [10XIsHomsetCosetsRep[0m  and provided with an iterator. Note that, when U has more
  than one piece, cosets may have differing lengths.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> re2 := RightCoset( Gd8, Ud8, e2 );[0X
    [4XRightCoset(c4d,[(1,3) : -8 -> -7])[0X
    [4Xgap> for x in re2 do Print( x, "\n" ); od;[0X
    [4X[(1,3) : -8 -> -8][0X
    [4X[(1,3) : -7 -> -8][0X
    [4X[(2,4) : -8 -> -8][0X
    [4X[(2,4) : -7 -> -8][0X
    [4X[(1,4)(2,3) : -8 -> -8][0X
    [4X[(1,4)(2,3) : -7 -> -8][0X
    [4X[(1,2)(3,4) : -8 -> -8][0X
    [4X[(1,2)(3,4) : -7 -> -8][0X
    [4Xgap> rcrd8 := RightCosetRepresentatives( Gd8, Ud8 );[0X
    [4X[ [() : -9 -> -9], [() : -9 -> -8], [() : -9 -> -7], [(2,4) : -9 -> -9],[0X
    [4X  [(2,4) : -9 -> -8], [(2,4) : -9 -> -7], [() : -8 -> -9], [() : -8 -> -8],[0X
    [4X  [() : -8 -> -7], [(2,4) : -8 -> -9], [(2,4) : -8 -> -8], [(2,4) : -8 -> -7][0X
    [4X ][0X
    [4Xgap> lcr7 := LeftCosetRepresentativesFromObject( Gd8, Ud8, -7 );[0X
    [4X[ [() : -7 -> -9], [(2,4) : -7 -> -9], [() : -7 -> -8], [(2,4) : -7 -> -8] ][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.5 Conjugation[0X
  
  [1X4.5-1 \^[0m
  
  [2X> \^( [0X[3Xe1, e2[0X[2X ) ____________________________________________________[0Xoperation
  
  When  e2  = c : p -> q and e1 has group element b, the conjugate e1^e2 has a
  complicated  definition,  but  may be remembered as follows. All objects are
  fixed  except  p,q, which are interchanged. For p,q as source, multiply b on
  the  left  by  c^-1,c respectively; and for p,q as target, multiply b on the
  right by c,c^-1. This product gives the group element of the conjugate.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4X[0X
    [4Xgap> x := GroupoidElement( Gd8, (2,4), -9, -9 );; [0X
    [4Xgap> y := GroupoidElement( Gd8, (1,2,3,4), -8, -9 );; [0X
    [4Xgap> z := GroupoidElement( Gd8, (1,3)(2,4), -7, -8 );; [0X
    [4Xgap> Print( "\nConjugation with elements x, y, and z in Gd8:\n" );[0X
    [4Xgap> Print( "x = ", x, ",  y = ", y, ",  z = ", z, "\n" );[0X
    [4Xx = [(2,4) : -9 -> -9],  y = [(1,2,3,4) : -8 -> -9],  z = [(1,3) : -8 -> -8][0X
    [4Xgap> Print( "x^x = ", x^x, ",  x^y = ", x^y, ",  x^z = ", x^z, "\n" );[0X
    [4Xx^x = [(2,4) : -9 -> -9],  x^y = [(1,3) : -8 -> -8],  x^z = [(2,4) : -9 -> -9][0X
    [4Xgap> Print( "y^x = ", y^x, ",  y^y = ", y^y, ",  y^z = ", y^z, "\n" );[0X
    [4Xy^x = [() : -8 -> -9],  y^y = [(1,4,3,2) : -9 -> -8],  y^z = [(1,4)(2,3) : -8 -> -9][0X
    [4Xgap> Print( "z^x = ", z^x, ",  z^y = ", z^y, ",  z^z = ", z^z, "\n" );[0X
    [4Xz^x = [(1,3) : -8 -> -8],  z^y = [(2,4) : -9 -> -9],  z^z = [(1,3) : -8 -> -8][0X
    [4X[0X
  [4X------------------------------------------------------------------[0X
  
  More  examples  of  all  these  operations  may be found in the example file
  [11Xgpd/examples/gpd.g[0m.
  
