  
  [1X7 [33X[0;0YHomomorphisms of Right Modules over Path Algebras[133X[101X
  
  [33X[0;0YThis  chapter  describes  the  categories,  representations, attributes, and
  operations on homomorphisms between representations of quivers.[133X
  
  [33X[0;0YGiven  two  homorphisms  [23Xf\colon  L\to  M[123X  and  [23Xg\colon  M\to  N[123X,  then  the
  composition   is   written   [23Xf*g[123X.   The  elements  in  the  modules  or  the
  representations  of  a  quiver  are row vectors. Therefore the homomorphisms
  between  two  modules are acting on these row vectors, that is, if [23Xm_i[123X is in
  [23XM[i][123X and [23Xg_i\colon M[i]\to N[i][123X represents the linear map, then the value of
  [23Xg[123X applied to [23Xm_i[123X is the matrix product [23Xm_i*g_i[123X.[133X
  
  [33X[0;0YThe example used throughout this chapter is the following.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQ := Quiver(3,[[1,2,"a"],[1,2,"b"],[2,2,"c"],[2,3,"d"],[3,1,"e"]]);;[127X[104X
    [4X[25Xgap>[125X [27XKQ := PathAlgebra(Rationals, Q);;[127X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables(KQ);;[127X[104X
    [4X[25Xgap>[125X [27Xrels := [d*e,c^2,a*c*d-b*d,e*a];;[127X[104X
    [4X[25Xgap>[125X [27XA := KQ/rels;;[127X[104X
    [4X[25Xgap>[125X [27Xmat :=[["a",[[1,2],[0,3],[1,5]]],["b",[[2,0],[3,0],[5,0]]],[127X[104X
    [4X[25X>[125X [27X["c",[[0,0],[1,0]]],["d",[[1,2],[0,1]]],["e",[[0,0,0],[0,0,0]]]];;[127X[104X
    [4X[25Xgap>[125X [27XN := RightModuleOverPathAlgebra(A,mat);; [127X[104X
  [4X[32X[104X
  
  
  [1X7.1 [33X[0;0YCategories and representation of homomorphisms[133X[101X
  
  [1X7.1-1 IsPathAlgebraModuleHomomorphism[101X
  
  [29X[2XIsPathAlgebraModuleHomomorphism[102X( [3Xf[103X ) [32X filter
  
  [33X[0;0YArguments: [3Xf[103X - any object in GAP.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue  or  false  depending  on  if  [3Xf[103X  belongs  to  the categories
            [10XIsPathAlgebraModuleHomomorphism[110X.[133X
  
  [33X[0;0YThis defines the category [2XIsPathAlgebraModuleHomomorphism[102X.[133X
  
  [1X7.1-2 RightModuleHomOverAlgebra[101X
  
  [29X[2XRightModuleHomOverAlgebra[102X( [3XM[103X, [3XN[103X, [3Xmats[103X ) [32X operation
  
  [33X[0;0YArguments:  [3XM[103X,  [3XN[103X  - two modules over the same (quotient of a) path algebra,
  [3Xmats[103X  -  a  list  of matrices, one for each vertex in the quiver of the path
  algebra.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  homomorphism  in  the  category [10XIsPathAlgebraModuleHomomorphism[110X
            from the module [3XM[103X to the module [3XN[103X given by the matrices [3Xmats[103X.[133X
  
  [33X[0;0YThe  arguments  [3XM[103X  and  [3XN[103X  are  two  modules  over the same algebra (this is
  checked),  and  [3Xmats[103X is a list of matrices [10Xmats[i][110X, where [10Xmats[i][110X represents
  the  linear map from [10XM[i][110X to [10XN[i][110X with [10Xi[110X running through all the vertices in
  the  same  order  as  when  the  underlying  quiver  was  created.  If  both
  [10XDimensionVector(M)[i][110X  and  [10XDimensionVector(N)[i][110X are non-zero, then [10Xmats[i][110X
  is    a    [10XDimensionVector(M)[i][110X   by   [10XDimensionVector(N)[i][110X   matrix.   If
  [10XDimensionVector(M)[i][110X  is  zero  and [10XDimensionVector(N)[i][110X is non-zero, then
  [10Xmats[i][110X  must  be  the zero [10X1[110X by [10XDimensionVector(N)[i][110X matrix. Similarly for
  the    other    way    around.    If    both    [10XDimensionVector(M)[i][110X    and
  [10XDimensionVector(N)[i][110X are zero, then [10Xmats[i][110X must be the [10X1[110X by [10X1[110X zero matrix.
  The  function  checks  if  [3Xmats[103X  is  a homomorphism from the module [3XM[103X to the
  module  [3XN[103X  by checking that the matrices given in [3Xmats[103X have the correct size
  and  satisfy  the  appropriate commutativity conditions with the matrices in
  the  modules  given  by  [3XM[103X  and  [3XN[103X. The source (or domain) and the range (or
  codomain)  of  the  homomorphism  constructed can by obtained again by [2XRange[102X
  ([14X7.2-22[114X) and by [2XSource[102X ([14X7.2-24[114X), respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := RightModuleOverPathAlgebra(A,[["a",[0,1]],["b",[0,1]],[127X[104X
    [4X[25X>[125X [27X["c",[[0]]],["d",[[1]]],["e",[1,0]]]);[127X[104X
    [4X[28X<[ 0, 1, 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XDimensionVector(L);[127X[104X
    [4X[28X[ 0, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xf := RightModuleHomOverAlgebra(L,N,[[[0,0,0]], [[1,0]], [127X[104X
    [4X[25X>[125X [27X[[1,2]]]);[127X[104X
    [4X[28X<<[ 0, 1, 1 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIsPathAlgebraMatModuleHomomorphism(f);[127X[104X
    [4X[28Xtrue [128X[104X
  [4X[32X[104X
  
  
  [1X7.2 [33X[0;0YGeneralities of homomorphisms[133X[101X
  
  [1X7.2-1 \= (maps)[101X
  
  [29X[2X\= (maps)[102X( [3Xf[103X, [3Xg[103X ) [32X operation
  
  [33X[0;0YArguments: [3Xf[103X, [3Xg[103X - two homomorphisms between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue,  if  [10XSource(f)  =  Source(g)[110X,  [10XRange(f)  = Range(g)[110X, and the
            matrices defining the maps [3Xf[103X and [3Xg[103X coincide.[133X
  
  [1X7.2-2 \+ (maps)[101X
  
  [29X[2X\+ (maps)[102X( [3Xf[103X, [3Xg[103X ) [32X operation
  
  [33X[0;0YArguments: [3Xf[103X, [3Xg[103X - two homomorphisms between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe sum [3Xf+g[103X of the maps [3Xf[103X and [3Xg[103X.[133X
  
  [33X[0;0YThe function checks if the maps have the same source and the same range, and
  returns an error message otherwise.[133X
  
  [1X7.2-3 \* (maps)[101X
  
  [29X[2X\* (maps)[102X( [3Xf[103X, [3Xg[103X ) [32X operation
  
  [33X[0;0YArguments:  [3Xf[103X,  [3Xg[103X - two homomorphisms between two modules, or one scalar and
  one homomorphism between modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  composition  [3Xfg[103X  of  the  maps [3Xf[103X and [3Xg[103X, if the input are maps
            between  representations  of  the  same  quivers.  If  [3Xf[103X or [3Xg[103X is a
            scalar,  it  returns  the  natural  action  of scalars on the maps
            between representations.[133X
  
  [33X[0;0YThe function checks if the maps are composable, in the first case and in the
  second  case it checks if the scalar is in the correct field, and returns an
  error message otherwise.[133X
  
  [1X7.2-4 CoKernelOfWhat[101X
  
  [29X[2XCoKernelOfWhat[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  homomorphism  [3Xg[103X,  if [3Xf[103X has been computed as the cokernel of the
            homomorphism [3Xg[103X.[133X
  
  [1X7.2-5 IdentityMapping[101X
  
  [29X[2XIdentityMapping[102X( [3XM[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X - a module.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe identity map between [3XM[103X and [3XM[103X.[133X
  
  [1X7.2-6 ImageElm[101X
  
  [29X[2XImageElm[102X( [3Xf[103X, [3Xelem[103X ) [32X operation
  
  [33X[0;0YArguments:  [3Xf[103X - a homomorphism between two modules, [3Xelem[103X - an element in the
  source of [3Xf[103X.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  image  of  the  element [3Xelem[103X in the source (or domain) of the
            homomorphism [3Xf[103X.[133X
  
  [33X[0;0YThe function checks if [3Xelem[103X is an element in the source of [3Xf[103X, and it returns
  an error message otherwise.[133X
  
  [1X7.2-7 ImagesSet[101X
  
  [29X[2XImagesSet[102X( [3Xf[103X, [3Xelts[103X ) [32X operation
  
  [33X[0;0YArguments:  [3Xf[103X - a homomorphism between two modules, [3Xelts[103X - an element in the
  source of [3Xf[103X, or the source of [3Xf[103X.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe non-zero images of a set of elements [3Xelts[103X in the source of the
            homomorphism  [3Xf[103X, or if [3Xelts[103X is the source of [3Xf[103X, it returns a basis
            of the image.[133X
  
  [33X[0;0YThe  function checks if the set of elements [3Xelts[103X consists of elements in the
  source of [3Xf[103X, and it returns an error message otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XB := BasisVectors(Basis(N)); [127X[104X
    [4X[28X[ [ [ 1, 0, 0 ], [ 0, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 1, 0 ], [ 0, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 1 ], [ 0, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 1, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 1 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 0 ], [ 1, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 0 ], [ 0, 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative(f,B[4]);     [127X[104X
    [4X[28X[ [ 0 ], [ 1 ], [ 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative(f,B[5]);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XBL := BasisVectors(Basis(L));[127X[104X
    [4X[28X[ [ [ 0 ], [ 1 ], [ 0 ] ], [ [ 0 ], [ 0 ], [ 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XImageElm(f,BL[1]);[127X[104X
    [4X[28X[ [ 0, 0, 0 ], [ 1, 0 ], [ 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XImagesSet(f,L);[127X[104X
    [4X[28X[ [ [ 0, 0, 0 ], [ 1, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 0 ], [ 1, 2 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XImagesSet(f,BL);[127X[104X
    [4X[28X[ [ [ 0, 0, 0 ], [ 1, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 0 ], [ 1, 2 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xz := Zero(f);;[127X[104X
    [4X[25Xgap>[125X [27Xf = z;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XRange(f) = Range(z);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xy := ZeroMapping(L,N);;[127X[104X
    [4X[25Xgap>[125X [27Xy = z;            [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xid := IdentityMapping(N);;[127X[104X
    [4X[25Xgap>[125X [27Xf*id;;[127X[104X
    [4X[25Xgap>[125X [27X#This causes an error![127X[104X
    [4X[25Xgap>[125X [27Xid*f;[127X[104X
    [4X[28XError, codomain of the first argument is not equal to the domain of th\[128X[104X
    [4X[28Xe second argument,  called from[128X[104X
    [4X[28X<function>( <arguments> ) called from read-eval-loop[128X[104X
    [4X[28XEntering break read-eval-print loop ...[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 [27X2*f + z;[127X[104X
    [4X[28X<<[ 0, 1, 1 ]> ---> <[ 3, 2, 2 ]>> [128X[104X
  [4X[32X[104X
  
  [1X7.2-8 ImageOfWhat[101X
  
  [29X[2XImageOfWhat[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  homomorphism  [3Xg[103X, if [3Xf[103X has been computed as the image projection
            or the image inclusion of the homomorphism [3Xg[103X.[133X
  
  [1X7.2-9 IsInjective[101X
  
  [29X[2XIsInjective[102X( [3Xf[103X ) [32X property
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if the homomorphism [3Xf[103X is one-to-one.[133X
  
  [1X7.2-10 IsIsomorphism[101X
  
  [29X[2XIsIsomorphism[102X( [3Xf[103X ) [32X operation
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if the homomorphism [3Xf[103X is an isomorphism.[133X
  
  [1X7.2-11 IsLeftMinimal[101X
  
  [29X[2XIsLeftMinimal[102X( [3Xf[103X ) [32X property
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if the homomorphism [3Xf[103X is left minimal.[133X
  
  [1X7.2-12 IsRightMinimal[101X
  
  [29X[2XIsRightMinimal[102X( [3Xf[103X ) [32X property
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if the homomorphism [3Xf[103X is right minimal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := RightModuleOverPathAlgebra(A,[["a",[0,1]],["b",[0,1]],[127X[104X
    [4X[25X>[125X [27X["c",[[0]]],["d",[[1]]],["e",[1,0]]]);;[127X[104X
    [4X[25Xgap>[125X [27Xf := RightModuleHomOverAlgebra(L,N,[[[0,0,0]], [[1,0]], [127X[104X
    [4X[25X>[125X [27X[[1,2]]]);[127X[104X
    [4X[28X<<[ 0, 1, 1 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xg := CoKernelProjection(f);[127X[104X
    [4X[28X<<[ 3, 2, 2 ]> ---> <[ 3, 1, 1 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XCoKernelOfWhat(g) = f;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xh := ImageProjection(f);[127X[104X
    [4X[28X<<[ 0, 1, 1 ]> ---> <[ 0, 1, 1 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XImageOfWhat(h) = f;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsInjective(f); IsSurjective(f); IsIsomorphism(f); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphism(h);[127X[104X
    [4X[28Xtrue [128X[104X
  [4X[32X[104X
  
  [1X7.2-13 IsSplitEpimorphism[101X
  
  [29X[2XIsSplitEpimorphism[102X( [3Xf[103X ) [32X property
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X  if the homomorphism [3Xf[103X is a splittable epimorphism, otherwise
            [10Xfalse[110X.[133X
  
  [1X7.2-14 IsSplitMonomorphism[101X
  
  [29X[2XIsSplitMonomorphism[102X( [3Xf[103X ) [32X property
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if the homomorphism [3Xf[103X is a splittable monomorphism, otherwise
            [10Xfalse[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := SimpleModules(A)[1];;[127X[104X
    [4X[25Xgap>[125X [27XH := HomOverAlgebra(N,S);; [127X[104X
    [4X[25Xgap>[125X [27XIsSplitMonomorphism(H[1]);  [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsSplitEpimorphism(H[1]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X7.2-15 IsSurjective[101X
  
  [29X[2XIsSurjective[102X( [3Xf[103X ) [32X property
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if the homomorphism [3Xf[103X is onto.[133X
  
  [1X7.2-16 IsZero[101X
  
  [29X[2XIsZero[102X( [3Xf[103X ) [32X property
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if the homomorphism [3Xf[103X is a zero homomorphism.[133X
  
  [1X7.2-17 KernelOfWhat[101X
  
  [29X[2XKernelOfWhat[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  homomorphism  [3Xg[103X,  if  [3Xf[103X  has been computed as the kernel of the
            homomorphism [3Xg[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := RightModuleOverPathAlgebra(A,[["a",[0,1]],["b",[0,1]],[127X[104X
    [4X[25X>[125X [27X["c",[[0]]],["d",[[1]]],["e",[1,0]]]);[127X[104X
    [4X[28X<[ 0, 1, 1 ]>[128X[104X
    [4X[25Xgap>[125X [27Xf := RightModuleHomOverAlgebra(L,N,[[[0,0,0]], [[1,0]], [127X[104X
    [4X[25X>[125X [27X[[1,2]]]);;[127X[104X
    [4X[25Xgap>[125X [27XIsZero(0*f);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xg := KernelInclusion(f);[127X[104X
    [4X[28X<<[ 0, 0, 0 ]> ---> <[ 0, 1, 1 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XKnownAttributesOfObject(g);[127X[104X
    [4X[28X[ "Range", "Source", "PathAlgebraOfMatModuleMap", "KernelOfWhat" ][128X[104X
    [4X[25Xgap>[125X [27XKernelOfWhat(g) = f;[127X[104X
    [4X[28Xtrue [128X[104X
  [4X[32X[104X
  
  [1X7.2-18 LeftInverseOfHomomorphism[101X
  
  [29X[2XLeftInverseOfHomomorphism[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xfalse[110X  if  the  homomorphism  [3Xf[103X  is  not a splittable epimorphism,
            otherwise it returns a splitting of the split epimorphism [3Xf[103X.[133X
  
  [1X7.2-19 MatricesOfPathAlgebraMatModuleHomomorphism[101X
  
  [29X[2XMatricesOfPathAlgebraMatModuleHomomorphism[102X( [3Xf[103X ) [32X operation
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe matrices defining the homomorphism [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMatricesOfPathAlgebraMatModuleHomomorphism(f);[127X[104X
    [4X[28X[ [ [ 0, 0, 0 ] ], [ [ 1, 0 ] ], [ [ 1, 2 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XRange(f);[127X[104X
    [4X[28X<[ 3, 2, 2 ]>[128X[104X
    [4X[25Xgap>[125X [27XSource(f);[127X[104X
    [4X[28X<[ 0, 1, 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XSource(f) = L;[127X[104X
    [4X[28Xtrue [128X[104X
  [4X[32X[104X
  
  [1X7.2-20 PathAlgebraOfMatModuleMap[101X
  
  [29X[2XPathAlgebraOfMatModuleMap[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments:   [3Xf[103X   --   a   homomorphism  between  two  path  algebra  modules
  ([10XPathAlgebraMatModule[110X).[133X
  
  [6XReturns:[106X  [33X[0;10Ythe   algebra   over  which  the  range  and  the  source  of  the
            homomorphism [3Xf[103X is defined.[133X
  
  [1X7.2-21 PreImagesRepresentative[101X
  
  [29X[2XPreImagesRepresentative[102X( [3Xf[103X, [3Xelem[103X ) [32X operation
  
  [33X[0;0YArguments:  [3Xf[103X - a homomorphism between two modules, [3Xelem[103X - an element in the
  range of [3Xf[103X.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  preimage  of  the  element  [3Xelem[103X in the range (or codomain) the
            homomorphism [3Xf[103X if a preimage exists, otherwise it returns [10Xfail[110X.[133X
  
  [33X[0;0YThe  function  checks if [3Xelem[103X is an element in the range of [3Xf[103X and returns an
  error message if not.[133X
  
  [1X7.2-22 Range[101X
  
  [29X[2XRange[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe range (or codomain) the homomorphism [3Xf[103X.[133X
  
  [1X7.2-23 RightInverseOfHomomorphism[101X
  
  [29X[2XRightInverseOfHomomorphism[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Y[10Xfalse[110X  if  the  homomorphism  [3Xf[103X  is not a splittable monomorphism,
            otherwise it returns a splitting of the split monomorphism [3Xf[103X.[133X
  
  [1X7.2-24 Source[101X
  
  [29X[2XSource[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe source (or domain) the homomorphism [3Xf[103X.[133X
  
  [1X7.2-25 Zero[101X
  
  [29X[2XZero[102X( [3Xf[103X ) [32X operation
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe zero map between [10XSource(f)[110X and [10XRange(f)[110X.[133X
  
  [1X7.2-26 ZeroMapping[101X
  
  [29X[2XZeroMapping[102X( [3XM[103X, [3XN[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X, [3XN[103X - two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe zero map between [3XM[103X and [3XN[103X.[133X
  
  [1X7.2-27 HomomorphismFromImages[101X
  
  [29X[2XHomomorphismFromImages[102X( [3XM[103X, [3XN[103X, [3XgenImages[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X, [3XN[103X -- two modules, [3XgenImages[103X -- a list.[133X
  
  [6XReturns:[106X  [33X[0;10YA map [22Xf[122X between [3XM[103X and [3XN[103X, given by [3XgenImages[103X.[133X
  
  [33X[0;0YLet  [10XB[110X  be  the  basis  [10XBasisVectors(  Basis( M ) )[110X of [3XM[103X. Then the number of
  elements  of  [10XgenImages[110X  should be equal to the number of elements of [10XB[110X, and
  [10XgenImages[i][110X  is  an  element of [10XN[110X and the image of [10XB[i][110X under [10Xf[110X. The method
  fails  if [10Xf[110X is not a homomorphism, or if [10XB[i][110X and [10XgenImages[i][110X are supported
  in different vertices.[133X
  
  
  [1X7.3 [33X[0;0YHomomorphisms and modules constructed from homomorphisms and modules[133X[101X
  
  [1X7.3-1 CoKernel[101X
  
  [29X[2XCoKernel[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe cokernel of a homomorphism [3Xf[103X between two modules.[133X
  
  [33X[0;0YThis function returns the cokernel of the homomorphism [3Xf[103X as a module.[133X
  
  [1X7.3-2 CoKernelProjection[101X
  
  [29X[2XCoKernelProjection[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe cokernel of a homomorphism [3Xf[103X between two modules.[133X
  
  [33X[0;0YThis  function  returns the cokernel of the homomorphism [3Xf[103X as the projection
  homomorphism  from  the  range  of the homomorphism [3Xf[103X to the cokernel of the
  homomorphism [3Xf[103X.[133X
  
  [1X7.3-3 EndModuloProjOverAlgebra[101X
  
  [29X[2XEndModuloProjOverAlgebra[102X( [3XM[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X - a module.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  natural  homomorphism  from the endomorphism ring of [3XM[103X to the
            endomorphism  ring  of  [3XM[103X  modulo  the  ideal  generated  by those
            endomorphisms of [3XM[103X which factor through a projective module.[133X
  
  [33X[0;0YThe  operation  returns an error message if the zero module is entered as an
  argument.[133X
  
  [1X7.3-4 EndOfModuleAsQuiverAlgebra[101X
  
  [29X[2XEndOfModuleAsQuiverAlgebra[102X( [3XM[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X - a PathAlgebraMatModule.[133X
  
  [6XReturns:[106X  [33X[0;10Ya list of three elements, (i) the endomorphism ring of [3XM[103X, (ii) the
            adjacency  matrix of the quiver of the endomorphism ring and (iii)
            the endomorphism ring as a quiver algebra.[133X
  
  [33X[0;0YSuppose  [3XM[103X  is  a  module over a quiver algebra over a field [22XK[122X. The function
  checks  if  the endomorphism ring of [3XM[103X is K-elementary (not necessary for it
  to  be  a quiver algebra, but this is a TODO improvement), and returns error
  message otherwise.[133X
  
  [1X7.3-5 EndOverAlgebra[101X
  
  [29X[2XEndOverAlgebra[102X( [3XM[103X ) [32X attribute
  
  [33X[0;0YArguments: [3XM[103X - a module.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  endomorphism  ring  of [3XM[103X as a subalgebra of the direct sum of
            the    full    matrix    rings    of    [10XDimensionVector(M)[i]    x
            DimensionVector(M)[i][110X,  where  [3Xi[103X  runs  over  all  vertices  where
            [10XDimensionVector(M)[i][110X is non-zero.[133X
  
  [33X[0;0YThe  endomorphism  is  an  algebra  with  one, and one can apply for example
  [10XRadicalOfAlgebra[110X to find the radical of the endomorphism ring.[133X
  
  [1X7.3-6 FromEndMToHomMM[101X
  
  [29X[2XFromEndMToHomMM[102X( [3Xf[103X ) [32X operation
  
  [33X[0;0YArguments: [3Xf[103X -- an element in [10XEndOverAlgebra(M)[110X.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe homomorphism from [3XM[103X to [3XM[103X corresponding to the element [3Xf[103X in the
            endomorphism ring [10XEndOverAlgebra(M)[110X of [3XM[103X.[133X
  
  [1X7.3-7 FromHomMMToEndM[101X
  
  [29X[2XFromHomMMToEndM[102X( [3Xf[103X ) [32X operation
  
  [33X[0;0YArguments: [3Xf[103X -- an element in [10XHomOverAlgebra(M,M)[110X.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  element  [3Xf[103X  in  the  endomorphism ring [10XEndOverAlgebra(M)[110X of [3XM[103X
            corresponding to the the homomorphism from [3XM[103X to [3XM[103X given by [3Xf[103X.[133X
  
  [1X7.3-8 HomFactoringThroughProjOverAlgebra[101X
  
  [29X[2XHomFactoringThroughProjOverAlgebra[102X( [3XM[103X, [3XN[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X, [3XN[103X - two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  basis  for  the vector space of homomorphisms from [3XM[103X to [3XN[103X which
            factors through a projective module.[133X
  
  [33X[0;0YThe  function  checks  if  [3XM[103X  and  [3XN[103X  are modules over the same algebra, and
  returns an error message otherwise.[133X
  
  [1X7.3-9 HomFromProjective[101X
  
  [29X[2XHomFromProjective[102X( [3Xm[103X, [3XM[103X ) [32X operation
  
  [33X[0;0YArguments: [3Xm[103X, [3XM[103X - an element and a module.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe homomorphism from the indecomposable projective module defined
            by the support of the element [3Xm[103X to the module [3XM[103X.[133X
  
  [33X[0;0YThe  function  checks  if  [3Xm[103X  is  an  elememt  in  [3XM[103X and if the element [3Xm[103X is
  supported in only one vertex. Otherwise it returns fail.[133X
  
  [1X7.3-10 HomOverAlgebra[101X
  
  [29X[2XHomOverAlgebra[102X( [3XM[103X, [3XN[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X, [3XN[103X - two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ya basis for the vector space of homomorphisms from [3XM[103X to [3XN[103X.[133X
  
  [33X[0;0YThe  function  checks  if  [3XM[103X  and  [3XN[103X  are modules over the same algebra, and
  returns an error message and fail otherwise.[133X
  
  [1X7.3-11 Image[101X
  
  [29X[2XImage[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe image of a homomorphism [3Xf[103X as a module.[133X
  
  [1X7.3-12 ImageInclusion[101X
  
  [29X[2XImageInclusion[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  inclusion  of the image of a homomorphism [3Xf[103X into the range of
            [3Xf[103X.[133X
  
  [1X7.3-13 ImageProjection[101X
  
  [29X[2XImageProjection[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  projection  from  the  source  of  [3Xf[103X  to  the  image  of  the
            homomorphism [3Xf[103X.[133X
  
  [1X7.3-14 ImageProjectionInclusion[101X
  
  [29X[2XImageProjectionInclusion[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Yboth  the  projection  from  the  source  of [3Xf[103X to the image of the
            homomorphism  [3Xf[103X and the inclusion of the image of a homomorphism [3Xf[103X
            into  the  range  of  [3Xf[103X  as  a  list  of  two  elements (first the
            projection and then the inclusion).[133X
  
  [1X7.3-15 IsomorphismOfModules[101X
  
  [29X[2XIsomorphismOfModules[102X( [3XM[103X, [3XN[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM, N[103X - two PathAlgebraMatModules.[133X
  
  [6XReturns:[106X  [33X[0;10Yfalse  if  [3XM[103X  and  [3XN[103X  are two non-isomorphic modules, otherwise it
            returns an isomorphism from [3XM[103X to [3XN[103X.[133X
  
  [33X[0;0YThe  function  checks  if  [3XM[103X  and  [3XN[103X  are modules over the same algebra, and
  returns an error message otherwise.[133X
  
  [1X7.3-16 Kernel[101X
  
  [29X[2XKernel[102X( [3Xf[103X ) [32X attribute
  [29X[2XKernelInclusion[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe kernel of a homomorphism [3Xf[103X between two modules.[133X
  
  [33X[0;0YThe  first  variant  [2XKernel[102X  returns  the  kernel of the homomorphism [3Xf[103X as a
  module,  while  the  latter  one  returns  the inclusion homomorphism of the
  kernel into the source of the homomorphism [3Xf[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xhom := HomOverAlgebra(N,N);[127X[104X
    [4X[28X[ <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X    , <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X    , <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X    , <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X    , <<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X     ][128X[104X
    [4X[25Xgap>[125X [27Xg := hom[1];[127X[104X
    [4X[28X<<[ 3, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XM := CoKernel(g);[127X[104X
    [4X[28X<[ 2, 2, 2 ]>[128X[104X
    [4X[25Xgap>[125X [27Xf := CoKernelProjection(g);[127X[104X
    [4X[28X<<[ 3, 2, 2 ]> ---> <[ 2, 2, 2 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XRange(f) = M;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xendo := EndOverAlgebra(N);[127X[104X
    [4X[28X<algebra-with-one of dimension 5 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XRadicalOfAlgebra(endo);[127X[104X
    [4X[28X<algebra of dimension 3 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XB := BasisVectors(Basis(N));[127X[104X
    [4X[28X[ [ [ 1, 0, 0 ], [ 0, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 1, 0 ], [ 0, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 1 ], [ 0, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 1, 0 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 1 ], [ 0, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 0 ], [ 1, 0 ] ], [128X[104X
    [4X[28X  [ [ 0, 0, 0 ], [ 0, 0 ], [ 0, 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xp := HomFromProjective(B[1],N);[127X[104X
    [4X[28X<<[ 1, 4, 3 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XU := Image(p);[127X[104X
    [4X[28X<[ 1, 2, 2 ]>[128X[104X
    [4X[25Xgap>[125X [27Xprojinc := ImageProjectionInclusion(p);[127X[104X
    [4X[28X[ <<[ 1, 4, 3 ]> ---> <[ 1, 2, 2 ]>>[128X[104X
    [4X[28X    , <<[ 1, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X     ][128X[104X
    [4X[25Xgap>[125X [27XU = Range(projinc[1]);                                      [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XKernel(p);[127X[104X
    [4X[28X<[ 0, 2, 1 ]> [128X[104X
  [4X[32X[104X
  
  [1X7.3-17 LeftMinimalVersion[101X
  
  [29X[2XLeftMinimalVersion[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  left  minimal  version [3Xf'[103X of the homomorphism [3Xf[103X together with
            the  a  list [10XB[110X of modules such that the direct sum of the modules,
            [10XRange(f')[110X and the modules in the list [10XB[110X is isomorphic to [10XRange(f)[110X.[133X
  
  [1X7.3-18 RightMinimalVersion[101X
  
  [29X[2XRightMinimalVersion[102X( [3Xf[103X ) [32X attribute
  
  [33X[0;0YArguments: [3Xf[103X - a homomorphism between two modules.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  right  minimal version [3Xf'[103X of the homomorphism [3Xf[103X together with
            the  a  list [10XB[110X of modules such that the direct sum of the modules,
            [10XSource(f')[110X  and  the  modules  on  the  list  [10XB[110X  is  isomorphic to
            [10XSource(f)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XH:= HomOverAlgebra(N,N);;[127X[104X
    [4X[25Xgap>[125X [27XRightMinimalVersion(H[1]);   [127X[104X
    [4X[28X[ <<[ 1, 0, 0 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X    , [ <[ 2, 2, 2 ]> ] ][128X[104X
    [4X[25Xgap>[125X [27XLeftMinimalVersion(H[1]);             [127X[104X
    [4X[28X[ <<[ 3, 2, 2 ]> ---> <[ 1, 0, 0 ]>>[128X[104X
    [4X[28X    , [ <[ 2, 2, 2 ]> ] ][128X[104X
    [4X[25Xgap>[125X [27XS := SimpleModules(A)[1];;[127X[104X
    [4X[25Xgap>[125X [27XMinimalRightApproximation(N,S);[127X[104X
    [4X[28X<<[ 1, 0, 0 ]> ---> <[ 1, 0, 0 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XS := SimpleModules(A)[3];;[127X[104X
    [4X[25Xgap>[125X [27XMinimalLeftApproximation(S,N);   [127X[104X
    [4X[28X<<[ 0, 0, 1 ]> ---> <[ 2, 2, 2 ]>> [128X[104X
  [4X[32X[104X
  
  [1X7.3-19 RadicalOfModuleInclusion[101X
  
  [29X[2XRadicalOfModuleInclusion[102X( [3XM[103X ) [32X attribute
  
  [33X[0;0YArguments: [3XM[103X - a module.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe inclusion of the radical of the module [3XM[103X into [3XM[103X.[133X
  
  [33X[0;0YThe  radical  of  [3XM[103X  can  be  accessed  using  [10XSource[110X, or it can be computed
  directly via the command [2XRadicalOfModule[102X ([14X6.4-33[114X). If the algebra over which
  [3XM[103X  is  a  module  is  not a finite dimensional path algebra or an admissible
  quotient of a path algebra, then it will search for other methods.[133X
  
  [1X7.3-20 RejectOfModule[101X
  
  [29X[2XRejectOfModule[102X( [3XM[103X, [3XN[103X ) [32X operation
  
  [33X[0;0YArguments: [3XN[103X, [3XM[103X -- two path algebra modules ([10XPathAlgebraMatModule[110X).[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  reject  of  the  module  [3XM[103X  in  the  module [3XN[103X as an inclusion
            homomorhpism from the reject of [3XM[103X into [3XN[103X.[133X
  
  [1X7.3-21 SocleOfModuleInclusion[101X
  
  [29X[2XSocleOfModuleInclusion[102X( [3XM[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X - a module.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe inclusion of the socle of the module [3XM[103X into [3XM[103X.[133X
  
  [33X[0;0YThe  socle of [3XM[103X can be accessed using [10XSource[110X, or it can be computed directly
  via the command [2XSocleOfModule[102X ([14X6.4-36[114X).[133X
  
  [1X7.3-22 SubRepresentationInclusion[101X
  
  [29X[2XSubRepresentationInclusion[102X( [3XM[103X, [3Xgens[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X - a module, [3Xgens[103X - a list of elements in [3XM[103X.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  inclusion  of  the submodule generated by the generators [3Xgens[103X
            into the module [3XM[103X.[133X
  
  [33X[0;0YThe  function checks if [3Xgens[103X consists of elements in [3XM[103X, and returns an error
  message  otherwise.  The  module  given  by  the  submodule generated by the
  generators [3Xgens[103X can be accessed using [10XSource[110X.[133X
  
  [1X7.3-23 TopOfModuleProjection[101X
  
  [29X[2XTopOfModuleProjection[102X( [3XM[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X - a module.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe projection from the module [3XM[103X to the top of the module [3XM[103X.[133X
  
  [33X[0;0YThe  module  given by the top of the module [3XM[103X can be accessed using [10XRange[110X of
  the homomorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := RadicalOfModuleInclusion(N);[127X[104X
    [4X[28X<<[ 0, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XradN := Source(f);[127X[104X
    [4X[28X<[ 0, 2, 2 ]>[128X[104X
    [4X[25Xgap>[125X [27Xg := SocleOfModuleInclusion(N);[127X[104X
    [4X[28X<<[ 1, 0, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XU := SubRepresentationInclusion(N,[B[5]+B[6],B[7]]);[127X[104X
    [4X[28X<<[ 0, 2, 2 ]> ---> <[ 3, 2, 2 ]>>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xh := TopOfModuleProjection(N);[127X[104X
    [4X[28X<<[ 3, 2, 2 ]> ---> <[ 3, 0, 0 ]>> [128X[104X
  [4X[32X[104X
  
  [1X7.3-24 TraceOfModule[101X
  
  [29X[2XTraceOfModule[102X( [3XM[103X, [3XN[103X ) [32X operation
  
  [33X[0;0YArguments: [3XM[103X, [3XC[103X -- two path algebra modules ([10XPathAlgebraMatModule[110X).[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  trace  of  the  module  [3XM[103X  in  the  module  [3XN[103X as an inclusion
            homomorhpism from the trace of [3XM[103X to [3XN[103X.[133X
  
