  
  [1X4 [33X[0;0YProjective Spaces[133X[101X
  
  [33X[0;0YIn this chapter we describe how to use [5XFinInG[105X to work with finite projective
  spaces.[133X
  
  
  [1X4.1 [33X[0;0YProjective Spaces and basic operations[133X[101X
  
  [33X[0;0YA [13Xprojective space[113X is a point-line incidence geometry, satisfying a few well
  known axioms. An axiomatic treatment can be found in [VY65a] and [VY65b]. In
  [5XFinInG[105X, we deal with [13Xfinite Desarguesian projective spaces[113X. It is well known
  that  these  geometries can be described completely using vector spaces over
  finite  fields.  The  elements  of  the  projective space are all nontrivial
  subspaces   of   the   vector  space.  So  the  projective  points  are  the
  one-dimensional  subspaces,  the  projective  lines  are the two-dimensional
  subspaces,  and  so on. From the axiomatic point of view, a projective space
  is  a point-line geometry, and has rank at least 2. But a projective line is
  obtained  if  we  start with a two dimensional vector space. Starting with a
  one dimensional vector space yields a single projective point. Both examples
  are  not  a  projective  space in the axiomatic point of view, but in [5XFinInG[105X
  they are considered as projective spaces.[133X
  
  [1X4.1-1 IsProjectiveSpace[101X
  
  [33X[1;0Y[29X[2XIsProjectiveSpace[102X [32X Category[133X
  
  [33X[0;0YThis  category  is  a  subcategory of [10XIsLieGeometry[110X, and contains all finite
  Desarguesian projective spaces.[133X
  
  [33X[0;0YWe refer the reader to [HT91] for the necessary background theory in case it
  is not provided in the manual.[133X
  
  [1X4.1-2 ProjectiveSpace[101X
  
  [33X[1;0Y[29X[2XProjectiveSpace[102X( [3Xd[103X, [3XF[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XProjectiveSpace[102X( [3Xd[103X, [3Xq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPG[102X( [3Xd[103X, [3Xq[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya projective space[133X
  
  [33X[0;0Y[3Xd[103X  must  be  a  positive  integer.  In  the first form, [3XF[103X is a field and the
  function  returns  the projective space of dimension [3Xd[103X over [3XF[103X. In the second
  form, [3Xq[103X is a prime power specifying the size of the field. The user may also
  use an alias, namely, the common abbreviation [10XPG(d, q)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XProjectiveSpace(3,GF(3));[127X[104X
    [4X[28XProjectiveSpace(3, 3)[128X[104X
    [4X[25Xgap>[125X [27XProjectiveSpace(3,3);[127X[104X
    [4X[28XProjectiveSpace(3, 3)[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.1-3 ProjectiveDimension[101X
  
  [33X[1;0Y[29X[2XProjectiveDimension[102X( [3Xps[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDimension[102X( [3Xps[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRank[102X( [3Xps[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe projective dimension of the projective space [3Xps[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(5,8);[127X[104X
    [4X[28XProjectiveSpace(5, 8)[128X[104X
    [4X[25Xgap>[125X [27XProjectiveDimension(ps);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XDimension(ps);[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XRank(ps);[127X[104X
    [4X[28X5[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.1-4 BaseField[101X
  
  [33X[1;0Y[29X[2XBaseField[102X( [3Xps[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yreturns the base field for the projective space [3Xps[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBaseField(ProjectiveSpace(3,81));[127X[104X
    [4X[28XGF(3^4)[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.1-5 UnderlyingVectorSpace[101X
  
  [33X[1;0Y[29X[2XUnderlyingVectorSpace[102X( [3Xps[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya vector space[133X
  
  [33X[0;0YIf  [3Xps[103X  is a projective space of dimension [22Xn[122X over the field of order [22Xq[122X, then
  this  operation  simply  returns  the  underlying vector space, i.e. the [22Xn+1[122X
  dimensional vector space over the field of order [22Xq[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := ProjectiveSpace(4,7);[127X[104X
    [4X[28XProjectiveSpace(4, 7)[128X[104X
    [4X[25Xgap>[125X [27Xvs := UnderlyingVectorSpace(ps);[127X[104X
    [4X[28X( GF(7)^5 )[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.1-6 AmbientSpace[101X
  
  [33X[1;0Y[29X[2XAmbientSpace[102X( [3Xps[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ya projective space[133X
  
  [33X[0;0YThe  ambient  space of a projective space [3Xps[103X is the projective space itself.
  Hence, simply [3Xps[103X will be returned.[133X
  
  
  [1X4.2 [33X[0;0YSubspaces of projective spaces[133X[101X
  
  [33X[0;0YThe  elements  of  a  projective  space [22XmathrmPG(n,q)[122X are the subspaces of a
  suitable  dimension.  The  empty subspace, also called the trivial subspace,
  has  dimension [22X-1[122X and corresponds to the zero dimensional vector subspace of
  the  underlying  vector space of [22XmathrmPG(n,q)[122X , and is hence represented by
  the  zero  vector  of length [22Xn+1[122X over the underlying field [22XmathrmGF(q)[122X . The
  trivial   subspace   and  the  whole  projective  space  are  mathematically
  considered  as a subspace of the projective geometry, but not as elements of
  the  incidence  geometry,  and hence do in [5XFinInG[105X [12Xnot[112X belong to the category
  [10XIsSubspaceOfProjectiveSpace[110X.[133X
  
  [1X4.2-1 VectorSpaceToElement[101X
  
  [33X[1;0Y[29X[2XVectorSpaceToElement[102X( [3Xgeo[103X, [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan element[133X
  
  [33X[0;0Y[3Xgeo[103X is a projective space, and [3Xv[103X is either a row vector (for points) or an [22Xm
  ×  n[122X matrix (for an [22X(m-1)[122X-subspace of projective space of dimension [22Xn-1[122X). In
  the case that [3Xv[103X is a matrix, the rows represent generators for the subspace.
  An  exceptional  case  is when [3Xv[103X is a zero-vector, in which case the trivial
  subspace is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := ProjectiveSpace(6,7);[127X[104X
    [4X[28XProjectiveSpace(6, 7)[128X[104X
    [4X[25Xgap>[125X [27Xv := [3,5,6,0,3,2,3]*Z(7)^0;[127X[104X
    [4X[28X[ Z(7), Z(7)^5, Z(7)^3, 0*Z(7), Z(7), Z(7)^2, Z(7) ][128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(ps,v);[127X[104X
    [4X[28X<a point in ProjectiveSpace(6, 7)>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(p);[127X[104X
    [4X[28X[142.131][128X[104X
    [4X[25Xgap>[125X [27Xps := ProjectiveSpace(3,4);[127X[104X
    [4X[28XProjectiveSpace(3, 4)[128X[104X
    [4X[25Xgap>[125X [27Xv := [1,1,0,1]*Z(4)^0;[127X[104X
    [4X[28X[ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(ps,v);[127X[104X
    [4X[28X<a point in ProjectiveSpace(3, 4)>[128X[104X
    [4X[25Xgap>[125X [27Xmat := [[1,0,0,1],[0,1,1,0]]*Z(4)^0;[127X[104X
    [4X[28X[ [ Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ] ][128X[104X
    [4X[25Xgap>[125X [27Xline := VectorSpaceToElement(ps,mat);[127X[104X
    [4X[28X<a line in ProjectiveSpace(3, 4)>[128X[104X
    [4X[25Xgap>[125X [27Xe := VectorSpaceToElement(ps,[]);[127X[104X
    [4X[28XError, <v> does not represent any element called from[128X[104X
    [4X[28X<function "unknown">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 17 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-2 EmptySubspace[101X
  
  [33X[1;0Y[29X[2XEmptySubspace[102X( [3Xps[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe trivial subspace in the projective [3Xps[103X[133X
  
  [33X[0;0YThe  object  returned  by  this  operation  is contained in every projective
  subspace  of  the  projective  space [3Xps[103X, but is not an element of [3Xps[103X. Hence,
  testing incidence results in an error message.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xe := EmptySubspace(PG(5,9));[127X[104X
    [4X[28X< empty subspace >[128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(PG(5,9),[1,0,0,0,0,0]*Z(9)^0);[127X[104X
    [4X[28X<a point in ProjectiveSpace(5, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xe*p;[127X[104X
    [4X[28XError, no method found! For debugging hints type ?Recovery from NoMethodFound[128X[104X
    [4X[28XError, no 1st choice method found for `*' on 2 arguments called from[128X[104X
    [4X[28X<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 10 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
    [4X[25Xgap>[125X [27Xe in p;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-3 ProjectiveDimension[101X
  
  [33X[1;0Y[29X[2XProjectiveDimension[102X( [3Xsub[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  projective dimension of a subspace of a projective space. The
            operation   [11XProjectiveDimension[111X   is   also   applicable   on  the
            EmptySubspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(2,5);[127X[104X
    [4X[28XProjectiveSpace(2, 5)[128X[104X
    [4X[25Xgap>[125X [27Xv := [[1,1,0],[0,3,2]]*Z(5)^0;[127X[104X
    [4X[28X[ [ Z(5)^0, Z(5)^0, 0*Z(5) ], [ 0*Z(5), Z(5)^3, Z(5) ] ][128X[104X
    [4X[25Xgap>[125X [27Xline := VectorSpaceToElement(ps,v);[127X[104X
    [4X[28X<a line in ProjectiveSpace(2, 5)>[128X[104X
    [4X[25Xgap>[125X [27XProjectiveDimension(line);[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XDimension(line);[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(ps,[1,2,3]*Z(5)^0);[127X[104X
    [4X[28X<a point in ProjectiveSpace(2, 5)>[128X[104X
    [4X[25Xgap>[125X [27XProjectiveDimension(p);[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XDimension(p);[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XProjectiveDimension(EmptySubspace(ps));[127X[104X
    [4X[28X-1[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-4 ElementsOfIncidenceStructure[101X
  
  [33X[1;0Y[29X[2XElementsOfIncidenceStructure[102X( [3Xps[103X, [3Xj[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe collection of elements of the projective space [3Xps[103X of type [3Xj[103X[133X
  
  [33X[0;0YFor the projective space [3Xps[103X of dimension [22Xd[122X and the type [3Xj[103X (where [22X1 ≤ j ≤ d[122X),
  this operation returns the collection of [22Xj-1[122X dimensional subspaces. An error
  message  is  produced  when  the  projective space [3Xps[103X has no elements of the
  required type.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := ProjectiveSpace(6,7);[127X[104X
    [4X[28XProjectiveSpace(6, 7)[128X[104X
    [4X[25Xgap>[125X [27XElementsOfIncidenceStructure(ps,1);[127X[104X
    [4X[28X<points of ProjectiveSpace(6, 7)>[128X[104X
    [4X[25Xgap>[125X [27XElementsOfIncidenceStructure(ps,2);[127X[104X
    [4X[28X<lines of ProjectiveSpace(6, 7)>[128X[104X
    [4X[25Xgap>[125X [27XElementsOfIncidenceStructure(ps,3);[127X[104X
    [4X[28X<planes of ProjectiveSpace(6, 7)>[128X[104X
    [4X[25Xgap>[125X [27XElementsOfIncidenceStructure(ps,4);[127X[104X
    [4X[28X<solids of ProjectiveSpace(6, 7)>[128X[104X
    [4X[25Xgap>[125X [27XElementsOfIncidenceStructure(ps,5);[127X[104X
    [4X[28X<proj. 4-subspaces of ProjectiveSpace(6, 7)>[128X[104X
    [4X[25Xgap>[125X [27XElementsOfIncidenceStructure(ps,6);[127X[104X
    [4X[28X<proj. 5-subspaces of ProjectiveSpace(6, 7)>[128X[104X
    [4X[25Xgap>[125X [27XElementsOfIncidenceStructure(ps,7);[127X[104X
    [4X[28XError, <ps> has no elements of type <j> called from[128X[104X
    [4X[28X<function "unknown">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 15 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X4.2-5 [33X[0;0YShort names for ElementsOfIncidenceStructure[133X[101X
  
  [33X[1;0Y[29X[2XPoints[102X( [3Xps[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XLines[102X( [3Xps[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPlanes[102X( [3Xps[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSolids[102X( [3Xps[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XHyperplanes[102X( [3Xps[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe  elements  of  [3Xps[103X  of  respective  type  1,  2,  3, 4, and the
            hyperplanes[133X
  
  [33X[0;0YAn error message is produced when the projective space [3Xps[103X has no elements of
  a required type.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(6,13);[127X[104X
    [4X[28XProjectiveSpace(6, 13)[128X[104X
    [4X[25Xgap>[125X [27XPoints(ps);[127X[104X
    [4X[28X<points of ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XLines(ps);[127X[104X
    [4X[28X<lines of ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XPlanes(ps);[127X[104X
    [4X[28X<planes of ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XSolids(ps);[127X[104X
    [4X[28X<solids of ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XHyperplanes(ps);[127X[104X
    [4X[28X<proj. 5-subspaces of ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27Xps := PG(2,2);[127X[104X
    [4X[28XProjectiveSpace(2, 2)[128X[104X
    [4X[25Xgap>[125X [27XHyperplanes(ps);[127X[104X
    [4X[28X<lines of ProjectiveSpace(2, 2)>[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X4.2-6 [33X[0;0YIncidence and containment[133X[101X
  
  [33X[1;0Y[29X[2XIsIncident[102X( [3Xel1[103X, [3Xel2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2X\*[102X( [3Xel1[103X, [3Xel2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2X\in[102X( [3Xel1[103X, [3Xel2[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YRecall  that  for  projective  spaces, incidence is symmetrized containment,
  where  the  empty  subspace  and  the whole projective space are excluded as
  arguments  for  this operation, since they are not considered as elements of
  the geometry, but both the empty subspace and the whole projective space are
  allowed as arguments for [11X\in[111X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := ProjectiveSpace(5,9);[127X[104X
    [4X[28XProjectiveSpace(5, 9)[128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(ps,[1,1,1,1,0,0]*Z(9)^0);[127X[104X
    [4X[28X<a point in ProjectiveSpace(5, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xl := VectorSpaceToElement(ps,[[1,1,1,1,0,0],[0,0,0,0,1,0]]*Z(9)^0);[127X[104X
    [4X[28X<a line in ProjectiveSpace(5, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xplane := VectorSpaceToElement(ps,[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]*Z(9)^0);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(5, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xp * l;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xl * p;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsIncident(p,l);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xp in l;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xl in p;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xp * plane;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xl * plane;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xl in plane;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xe := EmptySubspace(ps);[127X[104X
    [4X[28X< empty subspace >[128X[104X
    [4X[25Xgap>[125X [27Xe * l;[127X[104X
    [4X[28XError, no method found! For debugging hints type ?Recovery from NoMethodFound[128X[104X
    [4X[28XError, no 1st choice method found for `*' on 2 arguments called from[128X[104X
    [4X[28X<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 21 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
    [4X[25Xgap>[125X [27Xe in l;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xl in ps;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-7 StandardFrame[101X
  
  [33X[1;0Y[29X[2XStandardFrame[102X( [3Xps[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe standard frame of the projective space [3Xps[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XStandardFrame(PG(5,4));[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(5, 4)>, <a point in ProjectiveSpace(5, 4)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(5, 4)>, <a point in ProjectiveSpace(5, 4)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(5, 4)>, <a point in ProjectiveSpace(5, 4)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(5, 4)> ][128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X[ NewRowVector(IsCVecRep,GF(2,2),[Z(2)^0,0*Z(2),0*Z(2),0*Z(2),0*Z(2),0*Z(2),])[128X[104X
    [4X[28X    , NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),Z(2)^0,0*Z(2),0*Z(2),0*Z(2),[128X[104X
    [4X[28X    0*Z(2),]), NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),0*Z(2),Z(2)^0,0*Z(2),[128X[104X
    [4X[28X    0*Z(2),0*Z(2),]), NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),0*Z(2),0*Z(2),[128X[104X
    [4X[28X    Z(2)^0,0*Z(2),0*Z(2),]), NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),0*Z(2),[128X[104X
    [4X[28X    0*Z(2),0*Z(2),Z(2)^0,0*Z(2),]), NewRowVector(IsCVecRep,GF(2,2),[0*Z(2),[128X[104X
    [4X[28X    0*Z(2),0*Z(2),0*Z(2),0*Z(2),Z(2)^0,]), NewRowVector(IsCVecRep,GF(2,2),[[128X[104X
    [4X[28X    Z(2)^0,Z(2)^0,Z(2)^0,Z(2)^0,Z(2)^0,Z(2)^0,]) ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-8 Coordinates[101X
  
  [33X[1;0Y[29X[2XCoordinates[102X( [3Xp[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe homogeneous coordinates of the projective point [3Xp[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp := Random(Points(PG(5,16)));[127X[104X
    [4X[28X<a point in ProjectiveSpace(5, 16)>[128X[104X
    [4X[25Xgap>[125X [27XCoordinates(p);[127X[104X
    [4X[28X[ Z(2)^0, Z(2^4)^13, Z(2)^0, Z(2^4)^8, Z(2^4)^3, Z(2^4)^7 ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-9 DualCoordinatesOfHyperplane[101X
  
  [33X[1;0Y[29X[2XDualCoordinatesOfHyperplane[102X( [3Xhyp[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YThe  argument  [3Xhyp[103X  is  a  hyperplane  of a projective space. This operation
  returns  the  dual coordinates of the hyperplane [3Xhyp[103X, i.e. the list with the
  coefficients  of  the  equation  defining the hyperplane [3Xhyp[103X as an algebraic
  variety.[133X
  
  [1X4.2-10 HyperplaneByDualCoordinates[101X
  
  [33X[1;0Y[29X[2XHyperplaneByDualCoordinates[102X( [3Xpg[103X, [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya hyperplane of a projective space[133X
  
  [33X[0;0YThe  argument [3Xpg[103X is a projective space, and [3Xlist[103X is the coordinate vector of
  a  point  of  [3Xpg[103X. This operation returns the hyperplane that has [3Xlist[103X as the
  list of coefficients of the equation defining the hyperplane as an algebraic
  variety.[133X
  
  [1X4.2-11 EquationOfHyperplane[101X
  
  [33X[1;0Y[29X[2XEquationOfHyperplane[102X( [3Xh[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe equation of the hyperplane [3Xh[103X of a projective space[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xhyperplane := VectorSpaceToElement(PG(3,2),[[1,1,0,0],[0,0,1,0],[0,0,0,1]]*Z(2)^0);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(3, 2)>[128X[104X
    [4X[25Xgap>[125X [27XEquationOfHyperplane(hyperplane);[127X[104X
    [4X[28Xx_1+x_2[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-12 AmbientSpace[101X
  
  [33X[1;0Y[29X[2XAmbientSpace[102X( [3Xel[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yreturns the ambient space of an element [3Xel[103X of a projective space[133X
  
  [33X[0;0YThis operation is also applicable on the empty subspace and the whole space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(3,27);[127X[104X
    [4X[28XProjectiveSpace(3, 27)[128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(ps,[1,2,1,0]*Z(3)^3);[127X[104X
    [4X[28X<a point in ProjectiveSpace(3, 27)>[128X[104X
    [4X[25Xgap>[125X [27XAmbientSpace(p);[127X[104X
    [4X[28XProjectiveSpace(3, 27)[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-13 BaseField[101X
  
  [33X[1;0Y[29X[2XBaseField[102X( [3Xel[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yreturns the base field of an element [3Xel[103X of a projective space[133X
  
  [33X[0;0YThis  operation  is  also  applicable  on the trivial subspace and the whole
  space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(5,8);[127X[104X
    [4X[28XProjectiveSpace(5, 8)[128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(ps,[1,1,1,0,0,1]*Z(2));[127X[104X
    [4X[28X<a point in ProjectiveSpace(5, 8)>[128X[104X
    [4X[25Xgap>[125X [27XBaseField(p);[127X[104X
    [4X[28XGF(2^3)[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-14 Random[101X
  
  [33X[1;0Y[29X[2XRandom[102X( [3Xelements[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya random element from the collection [3Xelements[103X[133X
  
  [33X[0;0YThe    collection    [3Xelements[103X    is    an    object    in    the    category
  [10XIsElementsOfIncidenceStructure[110X,  i.e.  an  object  representing  the  set of
  elements  of  a  certain  incidence  structure  of  a given type. The latter
  information can be derived e.g. using [10XAmbientSpace[110X and [10XType[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(9,49);[127X[104X
    [4X[28XProjectiveSpace(9, 49)[128X[104X
    [4X[25Xgap>[125X [27XRandom(Lines(ps));[127X[104X
    [4X[28X<a line in ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27XRandom(Points(ps));[127X[104X
    [4X[28X<a point in ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27XRandom(Solids(ps));[127X[104X
    [4X[28X<a solid in ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27XRandom(Hyperplanes(ps));[127X[104X
    [4X[28X<a proj. 8-space in ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27Xelts := ElementsOfIncidenceStructure(ps,6);[127X[104X
    [4X[28X<proj. 5-subspaces of ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27XRandom(elts);[127X[104X
    [4X[28X<a proj. 5-space in ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X[[Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^14,Z(7^2)^44,Z(7^2)^14,[128X[104X
    [4X[28XZ(7)^5,][128X[104X
    [4X[28X [0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^29,Z(7^2)^13,Z(7^2)^19,[128X[104X
    [4X[28XZ(7^2)^27,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),Z(7^2)^20,Z(7^2)^10,Z(7^2)^18,[128X[104X
    [4X[28XZ(7^2)^27,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),Z(7),Z(7^2)^30,Z(7^2)^18,Z(7^2)^[128X[104X
    [4X[28X14,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),Z(7^2)^10,Z(7^2)^28,Z(7^2)^47,[128X[104X
    [4X[28XZ(7^2)^29,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,Z(7^2)^9,Z(7^2)^42,Z(7^2)^34,[128X[104X
    [4X[28XZ(7^2)^25,][128X[104X
    [4X[28X][128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps,3);[127X[104X
    [4X[28X<a solid in ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X[[Z(7)^0,0*Z(7),0*Z(7),0*Z(7),Z(7^2)^17,Z(7^2)^33,Z(7^2)^4,0*Z(7),Z(7^2),[128X[104X
    [4X[28XZ(7^2)^33,][128X[104X
    [4X[28X [0*Z(7),Z(7)^0,0*Z(7),0*Z(7),Z(7^2)^30,Z(7)^2,Z(7)^3,Z(7)^0,Z(7^2)^20,Z(7^2)^[128X[104X
    [4X[28X42,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),Z(7)^0,0*Z(7),Z(7^2)^20,Z(7^2)^30,Z(7^2)^11,Z(7^2)^39,Z(7)^3,[128X[104X
    [4X[28XZ(7),][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),Z(7)^0,Z(7^2)^21,Z(7)^0,Z(7^2)^11,Z(7^2)^45,Z(7^2),[128X[104X
    [4X[28XZ(7^2)^9,][128X[104X
    [4X[28X][128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps,7);[127X[104X
    [4X[28X<a proj. 7-space in ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X[[Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^42,Z(7^2)^[128X[104X
    [4X[28X35,][128X[104X
    [4X[28X [0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^43,Z(7^2),][128X[104X
    [4X[28X [0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^44,Z(7)^4,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7^2)^41,Z(7^2)^[128X[104X
    [4X[28X10,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),0*Z(7),Z(7^2)^37,Z(7^2)^[128X[104X
    [4X[28X12,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),0*Z(7),Z(7^2)^11,Z(7^2)^[128X[104X
    [4X[28X39,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,0*Z(7),Z(7^2)^22,Z(7^2)^[128X[104X
    [4X[28X10,][128X[104X
    [4X[28X [0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),0*Z(7),Z(7)^0,Z(7^2)^43,Z(7^2)^[128X[104X
    [4X[28X22,][128X[104X
    [4X[28X][128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(9, 49)>[128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a proj. 6-space in ProjectiveSpace(9, 49)>[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-15 RandomSubspace[101X
  
  [33X[1;0Y[29X[2XRandomSubspace[102X( [3Xps[103X, [3Xi[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRandomSubspace[102X( [3Xps[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  first  variant  returns  a  random  element  of type [3Xi[103X of the
            projective  space  [3Xps[103X. The second variant returns a random element
            of a random type of the projective space [3Xps[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(8,16);[127X[104X
    [4X[28XProjectiveSpace(8, 16)[128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a point in ProjectiveSpace(8, 16)>[128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a proj. 5-space in ProjectiveSpace(8, 16)>[128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a proj. 7-space in ProjectiveSpace(8, 16)>[128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a proj. 4-space in ProjectiveSpace(8, 16)>[128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(8, 16)>[128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(8, 16)>[128X[104X
    [4X[25Xgap>[125X [27XRandomSubspace(ps);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(8, 16)>[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-16 Span[101X
  
  [33X[1;0Y[29X[2XSpan[102X( [3Xu[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSpan[102X( [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan element or the empty subspace or the whole space[133X
  
  [33X[0;0YWhen  [3Xu[103X  and [3Xv[103X are elements of a projective space. This function returns the
  span  of  the  two  elements.  When  [3Xlist[103X  is a list of elements of the same
  projective  space,  then  this  function returns the span of all elements in
  [3Xlist[103X.  It  is  checked whether the elements [3Xu[103X and [3Xv[103X are elements of the same
  projective  space.  Although  the  trivial subspace and the whole projective
  space  are not objects in the category [10XIsSubspaceOfProjectiveSpace[110X, they are
  allowed  as arguments for this operation, and also for the second variant of
  this operation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := ProjectiveSpace(3,3);[127X[104X
    [4X[28XProjectiveSpace(3, 3)[128X[104X
    [4X[25Xgap>[125X [27Xp := Random(Planes(ps));[127X[104X
    [4X[28X<a plane in ProjectiveSpace(3, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xq := Random(Planes(ps));[127X[104X
    [4X[28X<a plane in ProjectiveSpace(3, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xs := Span(p,q);[127X[104X
    [4X[28XProjectiveSpace(3, 3)[128X[104X
    [4X[25Xgap>[125X [27Xs = Span([p,q]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xt := Span(EmptySubspace(ps),p);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(3, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xt = p;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSpan(ps,p);[127X[104X
    [4X[28XProjectiveSpace(3, 3)[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-17 Meet[101X
  
  [33X[1;0Y[29X[2XMeet[102X( [3Xu[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMeet[102X( [3Xlist[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan element or the empty subspace or the whole space[133X
  
  [33X[0;0YWhen  [3Xu[103X  and [3Xv[103X are elements of a projective space. This function returns the
  intersection  of  the  two  elements. When [3Xlist[103X is a list of elements of the
  same  projective  space,  then this function returns the intersection of all
  elements in [3Xlist[103X. It is checked whether the elements [3Xu[103X and [3Xv[103X are elements of
  the  same  projective  space.  Although  the  trivial subspace and the whole
  projective     space     are     not     objects     in     the     category
  [10XIsSubspaceOfProjectiveSpace[110X,   they   are  allowed  as  arguments  for  this
  operation, and also for the second variant of this operation. We remark that
  the result of a [11XMeet[111X operation can be the empty subspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XProjectiveSpace(7, 8)[128X[104X
    [4X[25Xgap>[125X [27Xp := Random(Solids(ps));[127X[104X
    [4X[28X<a solid in ProjectiveSpace(7, 8)>[128X[104X
    [4X[25Xgap>[125X [27Xq := Random(Solids(ps));[127X[104X
    [4X[28X<a solid in ProjectiveSpace(7, 8)>[128X[104X
    [4X[25Xgap>[125X [27Xs := Meet(p,q);[127X[104X
    [4X[28X< empty subspace >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(s);[127X[104X
    [4X[28X< empty subspace >[128X[104X
    [4X[25Xgap>[125X [27Xr := Random(Hyperplanes(ps));[127X[104X
    [4X[28X<a proj. 6-space in ProjectiveSpace(7, 8)>[128X[104X
    [4X[25Xgap>[125X [27XMeet(p,r);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(7, 8)>[128X[104X
    [4X[25Xgap>[125X [27XMeet(q,r);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(7, 8)>[128X[104X
    [4X[25Xgap>[125X [27XMeet([p,q,r]);[127X[104X
    [4X[28X< empty subspace >[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-18 FlagOfIncidenceStructure[101X
  
  [33X[1;0Y[29X[2XFlagOfIncidenceStructure[102X( [3Xps[103X, [3Xels[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  flag  of the projetcive space [3Xps[103X, determined by the subspaces
            of  [3Xps[103X  in  the  list  [3Xels[103X.  When  [3Xels[103X is empty, the empty flag is
            returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := ProjectiveSpace(12,11);[127X[104X
    [4X[28XProjectiveSpace(12, 11)[128X[104X
    [4X[25Xgap>[125X [27Xs1 := RandomSubspace(ps,8);[127X[104X
    [4X[28X<a proj. 8-space in ProjectiveSpace(12, 11)>[128X[104X
    [4X[25Xgap>[125X [27Xs2 := RandomSubspace(s1,6);[127X[104X
    [4X[28X<a proj. 6-space in ProjectiveSpace(12, 11)>[128X[104X
    [4X[25Xgap>[125X [27Xs3 := RandomSubspace(s2,4);[127X[104X
    [4X[28X<a proj. 4-space in ProjectiveSpace(12, 11)>[128X[104X
    [4X[25Xgap>[125X [27Xs4 := Random(Solids(s3));[127X[104X
    [4X[28X<a solid in ProjectiveSpace(12, 11)>[128X[104X
    [4X[25Xgap>[125X [27Xs5 := Random(Points(s4));[127X[104X
    [4X[28X<a point in ProjectiveSpace(12, 11)>[128X[104X
    [4X[25Xgap>[125X [27Xflag := FlagOfIncidenceStructure(ps,[s1,s3,s2,s5,s4]);[127X[104X
    [4X[28X<a flag of ProjectiveSpace(12, 11)>[128X[104X
    [4X[25Xgap>[125X [27Xps := PG(4,5);[127X[104X
    [4X[28XProjectiveSpace(4, 5)[128X[104X
    [4X[25Xgap>[125X [27Xp := Random(Points(ps));[127X[104X
    [4X[28X<a point in ProjectiveSpace(4, 5)>[128X[104X
    [4X[25Xgap>[125X [27Xl := Random(Lines(ps));[127X[104X
    [4X[28X<a line in ProjectiveSpace(4, 5)>[128X[104X
    [4X[25Xgap>[125X [27Xv := Random(Solids(ps));[127X[104X
    [4X[28X<a solid in ProjectiveSpace(4, 5)>[128X[104X
    [4X[25Xgap>[125X [27Xflag := FlagOfIncidenceStructure(ps,[v,l,p]);[127X[104X
    [4X[28XError, <els> does not determine a flag> called from[128X[104X
    [4X[28X<function "unknown">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 19 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
    [4X[25Xgap>[125X [27Xflag := FlagOfIncidenceStructure(ps,[]);[127X[104X
    [4X[28X<a flag of ProjectiveSpace(4, 5)>[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.2-19 IsEmptyFlag[101X
  
  [33X[1;0Y[29X[2XIsEmptyFlag[102X( [3Xflag[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yreturn true if [3Xflag[103X is the empty flag[133X
  
  [1X4.2-20 IsChamberOfIncidenceStructure[101X
  
  [33X[1;0Y[29X[2XIsChamberOfIncidenceStructure[102X( [3Xflag[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ytrue if [3Xflag[103X is a chamber[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(3,13);[127X[104X
    [4X[28XProjectiveSpace(3, 13)[128X[104X
    [4X[25Xgap>[125X [27Xplane := Random(Planes(ps));[127X[104X
    [4X[28X<a plane in ProjectiveSpace(3, 13)>[128X[104X
    [4X[25Xgap>[125X [27Xline := Random(Lines(plane));[127X[104X
    [4X[28X<a line in ProjectiveSpace(3, 13)>[128X[104X
    [4X[25Xgap>[125X [27Xpoint := Random(Points(line));[127X[104X
    [4X[28X<a point in ProjectiveSpace(3, 13)>[128X[104X
    [4X[25Xgap>[125X [27Xflag := FlagOfIncidenceStructure(ps,[point,line,plane]);[127X[104X
    [4X[28X<a flag of ProjectiveSpace(3, 13)>[128X[104X
    [4X[25Xgap>[125X [27XIsChamberOfIncidenceStructure(flag);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X4.3 [33X[0;0YShadows of Projective Subspaces[133X[101X
  
  [1X4.3-1 ShadowOfElement[101X
  
  [33X[1;0Y[29X[2XShadowOfElement[102X( [3Xps[103X, [3Xel[103X, [3Xi[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XShadowOfElement[102X( [3Xps[103X, [3Xel[103X, [3Xstr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  elements  of  type  [3Xi[103X  incident  with  [3Xel[103X. The second variant
            determines  the  type  [3Xi[103X  from  the  position  of  [3Xstr[103X in the list
            returned by [11XTypesOfElementsOfIncidenceStructurePlural[111X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(4,3);[127X[104X
    [4X[28XProjectiveSpace(4, 3)[128X[104X
    [4X[25Xgap>[125X [27Xplane := Random(Planes(ps));[127X[104X
    [4X[28X<a plane in ProjectiveSpace(4, 3)>[128X[104X
    [4X[25Xgap>[125X [27Xshadowpoints := ShadowOfElement(ps,plane,1);[127X[104X
    [4X[28X<shadow points in ProjectiveSpace(4, 3)>[128X[104X
    [4X[25Xgap>[125X [27XList(shadowpoints);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(4, 3)>, <a point in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(4, 3)> ][128X[104X
    [4X[25Xgap>[125X [27Xshadowlines := ShadowOfElement(ps,plane,2);[127X[104X
    [4X[28X<shadow lines in ProjectiveSpace(4, 3)>[128X[104X
    [4X[25Xgap>[125X [27XList(shadowlines);[127X[104X
    [4X[28X[ <a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(4, 3)>, <a line in ProjectiveSpace(4, 3)>, [128X[104X
    [4X[28X  <a line in ProjectiveSpace(4, 3)> ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.3-2 ShadowOfFlag[101X
  
  [33X[1;0Y[29X[2XShadowOfFlag[102X( [3Xps[103X, [3Xflag[103X, [3Xi[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XShadowOfFlag[102X( [3Xps[103X, [3Xflag[103X, [3Xstr[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  type [3Xi[103X shadow elements of the flag [3Xflag[103X, i.e. the elements of
            type  [3Xi[103X  incident  with  all  elements of [3Xflag[103X. The second variant
            determines  the  type  [3Xi[103X  from  the  position  of  [3Xstr[103X in the list
            returned by [11XTypesOfElementsOfIncidenceStructurePlural[111X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(5,7);[127X[104X
    [4X[28XProjectiveSpace(5, 7)[128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(ps,[1,0,0,0,0,0]*Z(7)^0);[127X[104X
    [4X[28X<a point in ProjectiveSpace(5, 7)>[128X[104X
    [4X[25Xgap>[125X [27Xl := VectorSpaceToElement(ps,[[1,0,0,0,0,0],[0,1,0,0,0,0]]*Z(7)^0);[127X[104X
    [4X[28X<a line in ProjectiveSpace(5, 7)>[128X[104X
    [4X[25Xgap>[125X [27Xv := VectorSpaceToElement(ps,[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]*Z(7)^0);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(5, 7)>[128X[104X
    [4X[25Xgap>[125X [27Xflag := FlagOfIncidenceStructure(ps,[v,l,p]);[127X[104X
    [4X[28X<a flag of ProjectiveSpace(5, 7)>[128X[104X
    [4X[25Xgap>[125X [27Xs := ShadowOfFlag(ps,flag,4);[127X[104X
    [4X[28X<shadow solids in ProjectiveSpace(5, 7)>[128X[104X
    [4X[25Xgap>[125X [27Xs := ShadowOfFlag(ps,flag,"solids");[127X[104X
    [4X[28X<shadow solids in ProjectiveSpace(5, 7)>[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.3-3 ElementsIncidentWithElementOfIncidenceStructure[101X
  
  [33X[1;0Y[29X[2XElementsIncidentWithElementOfIncidenceStructure[102X( [3Xel[103X, [3Xi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe  elements of type [3Xi[103X incident with [3Xel[103X, in other words, the type
            [3Xi[103X shadow of the element [3Xel[103X[133X
  
  [33X[0;0YInternally,  the  function [11XFlagOfIncidenceStructure[111X is used to create a flag
  from [3Xlist[103X. This function also performs the checking.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(6,9);[127X[104X
    [4X[28XProjectiveSpace(6, 9)[128X[104X
    [4X[25Xgap>[125X [27Xp := VectorSpaceToElement(ps,[1,0,1,0,0,0,0]*Z(9)^0);[127X[104X
    [4X[28X<a point in ProjectiveSpace(6, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xels := ElementsIncidentWithElementOfIncidenceStructure(p,3);[127X[104X
    [4X[28X<shadow planes in ProjectiveSpace(6, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xline := VectorSpaceToElement(ps,[[1,1,1,1,0,0,0],[0,0,0,0,1,1,1]]*Z(9)^0);[127X[104X
    [4X[28X<a line in ProjectiveSpace(6, 9)>[128X[104X
    [4X[25Xgap>[125X [27Xels := ElementsIncidentWithElementOfIncidenceStructure(line,1);[127X[104X
    [4X[28X<shadow points in ProjectiveSpace(6, 9)>[128X[104X
    [4X[25Xgap>[125X [27XList(els);[127X[104X
    [4X[28X[ <a point in ProjectiveSpace(6, 9)>, <a point in ProjectiveSpace(6, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(6, 9)>, <a point in ProjectiveSpace(6, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(6, 9)>, <a point in ProjectiveSpace(6, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(6, 9)>, <a point in ProjectiveSpace(6, 9)>, [128X[104X
    [4X[28X  <a point in ProjectiveSpace(6, 9)>, <a point in ProjectiveSpace(6, 9)> ][128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X4.3-4 [33X[0;0YShort names for [11XElementsIncidentWithElementOfIncidenceStructure[111X[101X[1X[133X[101X
  
  [33X[1;0Y[29X[2XPoints[102X( [3Xps[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XLines[102X( [3Xps[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPlanes[102X( [3Xps[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSolids[102X( [3Xps[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XHyperplanes[102X( [3Xinc[103X, [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPoints[102X( [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XLines[102X( [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPlanes[102X( [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSolids[102X( [3Xv[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XHyperplanes[102X( [3Xv[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe  elements  of the incidence geometry of the according type. If
            [3Xps[103X  is  not  given  as  an  argument,  it is deduced from [3Xv[103X as its
            ambient geometry.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xps := PG(6,13);[127X[104X
    [4X[28XProjectiveSpace(6, 13)[128X[104X
    [4X[25Xgap>[125X [27Xplane := Random(Planes(ps));[127X[104X
    [4X[28X<a plane in ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XPoints(plane);[127X[104X
    [4X[28X<shadow points in ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XLines(plane);[127X[104X
    [4X[28X<shadow lines in ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XSolids(plane);[127X[104X
    [4X[28X<shadow solids in ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XHyperplanes(plane);[127X[104X
    [4X[28X<shadow lines in ProjectiveSpace(6, 13)>[128X[104X
    [4X[25Xgap>[125X [27XElementsIncidentWithElementOfIncidenceStructure(plane,6);[127X[104X
    [4X[28X<shadow proj. 5-subspaces in ProjectiveSpace(6, 13)>[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  
  [1X4.4 [33X[0;0YEnumerating subspaces of a projective space[133X[101X
  
  [1X4.4-1 Iterator[101X
  
  [33X[1;0Y[29X[2XIterator[102X( [3Xsubspaces[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan iterator for the collection [3Xsubspaces[103X[133X
  
  [33X[0;0YWe refer to the GAP operation [2XIterator[102X for the definition of an iterator.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(5,7);[127X[104X
    [4X[28XProjectiveSpace(5, 7)[128X[104X
    [4X[25Xgap>[125X [27Xplanes := Planes(pg);[127X[104X
    [4X[28X<planes of ProjectiveSpace(5, 7)>[128X[104X
    [4X[25Xgap>[125X [27Xiter := Iterator(planes);[127X[104X
    [4X[28X<iterator>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator(iter);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(5, 7)>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator(iter);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(5, 7)>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator(iter);[127X[104X
    [4X[28X<a plane in ProjectiveSpace(5, 7)>[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.4-2 Enumerator[101X
  
  [33X[1;0Y[29X[2XEnumerator[102X( [3Xsubspaces[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan enumerator for the collection [3Xsubspaces[103X[133X
  
  [33X[0;0YFor complete collections of subspaces of a given type of a projective space,
  currently,  no non-trivial enumerator is installed, i.e. this operation just
  returns  a  list containing all elements of the collection [3Xsubspaces[103X. Such a
  list  can,  of  course,  be  used  as  an enumerator, but this might be time
  consuming.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(3,4);[127X[104X
    [4X[28XProjectiveSpace(3, 4)[128X[104X
    [4X[25Xgap>[125X [27Xlines := Lines(pg);[127X[104X
    [4X[28X<lines of ProjectiveSpace(3, 4)>[128X[104X
    [4X[25Xgap>[125X [27Xenum := Enumerator(lines);;[127X[104X
    [4X[25Xgap>[125X [27XLength(enum);[127X[104X
    [4X[28X357[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X4.4-3 List[101X
  
  [33X[1;0Y[29X[2XList[102X( [3Xsubspaces[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XAsList[102X( [3Xsubspaces[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ythe complete list of elements in the collection [3Xsubspaces[103X[133X
  
  [33X[0;0YThe  operation  [11XList[111X  will return a complete list, the operation [11XAsList[111X will
  return  an  [5Xorb[105X  object, representing a complete orbit, i.e. representing in
  this  case  a  complete  list. To obtain the elements explicitly, one has to
  issue  the  [11XList[111X  operation  with as argument the [5Xorb[105X object again. Applying
  [11XList[111X directly to a collection of subspaces, refers to the enumerator for the
  collection,  while  using [11XAsList[111X uses the [5Xorb[105X to compute all subspaces as an
  orbit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpg := PG(3,4);[127X[104X
    [4X[28XProjectiveSpace(3, 4)[128X[104X
    [4X[25Xgap>[125X [27Xlines := Lines(pg);[127X[104X
    [4X[28X<lines of ProjectiveSpace(3, 4)>[128X[104X
    [4X[25Xgap>[125X [27Xlist := List(lines);;[127X[104X
    [4X[25Xgap>[125X [27XLength(list);[127X[104X
    [4X[28X357[128X[104X
    [4X[25Xgap>[125X [27Xaslist := AsList(lines);[127X[104X
    [4X[28X<closed orbit, 357 points>[128X[104X
    [4X[25Xgap>[125X [27Xlist2 := List(aslist);;[127X[104X
    [4X[25Xgap>[125X [27XLength(list2);[127X[104X
    [4X[28X357[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
