  
  [1X2 [33X[0;0YAlgebras and their Actions[133X[101X
  
  [33X[0;0YAll  the  algebras  considered  in  this  package  will  be  associative and
  commutative. Scalars belong to a commutative ring [12Xk[112X with [22X1 ≠ 0[122X.[133X
  
  [33X[0;0Y[13XWhy not a field? A group ring over the integers is not an algebra.[113X[133X
  
  
  [1X2.1 [33X[0;0YMultipliers[133X[101X
  
  [33X[0;0YA [13Xmultiplier[113X in a commutative algebra [22XA[122X is a function [22Xμ : A -> A[122X such that[133X
  
  
  [24X[33X[0;6Y\mu(ab) ~=~ (\mu a)b ~=~ a(\mu b) \quad \forall~ a,b \in A.[133X
  
  [124X
  
  [33X[0;0YThe [13Xregular multipliers[113X of [22XA[122X are the functions[133X
  
  
  [24X[33X[0;6Y\mu_a : A \to A ~:~ \mu_ab = ab \quad \forall~ b \in A.[133X
  
  [124X
  
  [33X[0;0YWhen [22XA[122X has a one, it follows from the defining condition that [22Xμ(b1) = (μ 1)b[122X
  and  so  [22Xμ  =  μ_a[122X  where  [22Xa  =  μ  1[122X. Since an ideal [22XI[122X of [22XA[122X is closed under
  multiplication, a multiplier [22Xμ[122X may be restricted to [22XI[122X.[133X
  
  [33X[0;0Y[12XQuestion:[112X  Is  there  an  example  of  an  algebra [22XA[122X [13Xwithout[113X a one which has
  multipliers [13Xnot[113X of the form [22Xμ_a[122X?[133X
  
  [1X2.1-1 RegularAlgebraMultiplier[101X
  
  [33X[1;0Y[29X[2XRegularAlgebraMultiplier[102X( [3XA[103X, [3XI[103X, [3Xa[103X ) [32X operation[133X
  
  [33X[0;0YThis operation defines the multiplier [22Xμ_a : I -> I[122X on an ideal [22XI[122X of [22XA[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XA5c6 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );;[127X[104X
    [4X[25Xgap>[125X [27XvecA := BasisVectors( Basis( A5c6 ) );; [127X[104X
    [4X[25Xgap>[125X [27Xv := vecA[1] + vecA[3] + vecA[5];[127X[104X
    [4X[28X(Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4)[128X[104X
    [4X[25Xgap>[125X [27XI5c6 := Ideal( A5c6, [v] );; [127X[104X
    [4X[25Xgap>[125X [27Xv2 := vecA[2];[127X[104X
    [4X[28X(Z(5)^0)*(1,2,3,4,5,6)[128X[104X
    [4X[25Xgap>[125X [27Xm2 := RegularAlgebraMultiplier( A5c6, I5c6, v2 ); [127X[104X
    [4X[28X[ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), [128X[104X
    [4X[28X  (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) ] -> [128X[104X
    [4X[28X[ (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2), [128X[104X
    [4X[28X  (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 IsAlgebraMultiplier[101X
  
  [33X[1;0Y[29X[2XIsAlgebraMultiplier[102X( [3Xm[103X ) [32X operation[133X
  
  [33X[0;0YThis function tests the condition [22Xμ(ab) = (μ a)b = a(μ b)[122X for all [22Xa,b[122X in the
  basis for [22XA[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIsAlgebraMultiplier( m2 ); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xone := One( A5c6 );; [127X[104X
    [4X[25Xgap>[125X [27XL := List( vecA, v -> one );; [127X[104X
    [4X[25Xgap>[125X [27Xm1 := LeftModuleHomomorphismByImages( A5c6, A5c6, vecA, L ); [127X[104X
    [4X[28X[ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), [128X[104X
    [4X[28X  (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2) [128X[104X
    [4X[28X ] -> [ (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), [128X[104X
    [4X[28X  (Z(5)^0)*() ][128X[104X
    [4X[25Xgap>[125X [27XIsAlgebraMultiplier( m1 );                                                [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.1-3 MultiplierAlgebraOfIdealBySubalgebra [101X
  
  [33X[1;0Y[29X[2XMultiplierAlgebraOfIdealBySubalgebra [102X( [3XA[103X, [3XI[103X, [3XB[103X ) [32X operation[133X
  
  [33X[0;0YThe regular multipliers [22Xμ_b : I -> I[122X for all [22Xb ∈ B[122X, where [22XI[122X is an ideal in [22XA[122X
  and [22XB[122X is a subalgebra of [22XA[122X, form an algebra with product [22Xμ_b ∘ μ_b' = μ_bb'[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xv3 := vecA[3];[127X[104X
    [4X[28X(Z(5)^0)*(1,3,5)(2,4,6)[128X[104X
    [4X[25Xgap>[125X [27XB5c3 := Subalgebra( A5c6, [ v3 ] );; [127X[104X
    [4X[25Xgap>[125X [27XM := MultiplierAlgebraOfIdealBySubalgebra( A5c6, I5c6, B5c3 );[127X[104X
    [4X[28X<algebra of dimension 1 over GF(5)>[128X[104X
    [4X[25Xgap>[125X [27XvecM := BasisVectors( Basis( M ) );; [127X[104X
    [4X[25Xgap>[125X [27XvecM[1];[127X[104X
    [4X[28X<linear mapping by matrix, [128X[104X
    [4X[28X<two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, (dimension 2[128X[104X
    [4X[28X )> -> <two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, [128X[104X
    [4X[28X  (dimension 2)>>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.1-4 MultiplierAlgebra[101X
  
  [33X[1;0Y[29X[2XMultiplierAlgebra[102X( [3XA[103X ) [32X attribute[133X
  
  [33X[0;0YThe  regular  multipliers  [22Xμ_a  :  A  ->  A[122X  for  all  [22Xa ∈ A[122X form an algebra
  isomorphic   to   [22XA[122X   by   the   map   [22Xa   ↦  μ_a[122X.  This  operation  returns
  [10XMultiplierAlgebraOfIdealBySubalgebra(A,A,A);[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XMA5c6 := RegularMultiplierAlgebra( A5c6 );[127X[104X
    [4X[28X<algebra of dimension 6 over GF(5)>[128X[104X
    [4X[25Xgap>[125X [27XvecM := BasisVectors( Basis( MA5c6 ) );; [127X[104X
    [4X[25Xgap>[125X [27XvecM[3];[127X[104X
    [4X[28X<linear mapping by matrix, <algebra-with-one of dimension [128X[104X
    [4X[28X6 over GF(5)> -> <algebra-with-one of dimension 6 over GF(5)>>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.1-5 MultiplierHomomorphism[101X
  
  [33X[1;0Y[29X[2XMultiplierHomomorphism[102X( [3XM[103X ) [32X attribute[133X
  
  [33X[0;0YIf [22XM[122X is a multiplier algebra with elements of algebra [22XA[122X multiplying an ideal
  [22XI[122X then this operation returns the homomorphism from [22XA[122X to [22XM[122X mapping [22Xa[122X to [22Xμ_a[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xhom := MultiplierHomomorphism( MA5c6 );; [127X[104X
    [4X[25Xgap>[125X [27XImageElm( hom, vecA[2] ); [127X[104X
    [4X[28XBasis( <two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, [128X[104X
    [4X[28X  (dimension 2)>, [128X[104X
    [4X[28X[ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), [128X[104X
    [4X[28X  (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) [128X[104X
    [4X[28X ] ) -> [128X[104X
    [4X[28X[ (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2), [128X[104X
    [4X[28X  (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YCommutative actions[133X[101X
  
  [33X[0;0YIf [22XS[122X and [22XR[122X are commutative [12Xk[112X-algebras, a map[133X
  
  
  [24X[33X[0;6YR \times S ~\to~ S, \qquad (r,s) ~\mapsto~ r \cdot s[133X
  
  [124X
  
  [33X[0;0Yis a commutative action if and only if the following five axioms hold:[133X
  
  [30X    [33X[0;6Y[22Xk(r ⋅ s) ~=~ (kr) ⋅ s ~=~ r ⋅ (ks)[122X,[133X
  
  [30X    [33X[0;6Y[22Xr ⋅ (s + s') ~=~ r ⋅ s + r ⋅ s', qquad[122X (so [22Xr ⋅ 0_S = 0_S ~∀~ r ∈ R[122X),[133X
  
  [30X    [33X[0;6Y[22X(r + r') ⋅ s ~=~ r ⋅ s + r' ⋅ s, qquad[122X (so [22X0_R ⋅ s = 0_S ~∀~ s ∈ S[122X),[133X
  
  [30X    [33X[0;6Y[22Xr ⋅ (ss') ~=~ (r ⋅ s)s' = s(r ⋅ s')[122X,[133X
  
  [30X    [33X[0;6Y[22X(rr') ⋅ s ~=~ r ⋅ (r' ⋅ s), qquad[122X (so [22X1_R ⋅ s = s ~∀~ s ∈ S[122X when [22XR[122X has
        a one),[133X
  
  [33X[0;0Yfor all [22Xk ∈[122X[12Xk[112X, [22Xr,r' ∈ R[122X, and [22Xs,s' ∈ S[122X.[133X
  
  [1X2.2-1 AlgebraActionByMultipliers[101X
  
  [33X[1;0Y[29X[2XAlgebraActionByMultipliers[102X( [3XA[103X, [3XI[103X ) [32X operation[133X
  
  [33X[0;0YWhen [22XI[122X is an ideal in [22XA[122X we have seen that the multiplier homomorphism from [22XA[122X
  to [10XMultiplierAlgebraOf(Ideal(A,I)[110X is an action.[133X
  
  [33X[0;0YIn  the  example  the algebra is the group ring of the cyclic group [22XC_6[122X over
  the  field  [22XGF(5)[122X.  The  ideal  is  generated  by  [22Xv = () + (1,3,5)(2,4,6) +
  (1,5,3)(2,6,4)[122X.  The generator [22Xr = (1,2,3,4,5,6)[122X acts on [22Xv[122X by multiplication
  to give the vector [22Xr ⋅ v = (1,2,3,4,5,6) + (1,4)(2,5)(3,6) + (1,6,5,4,3,2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XA5c6 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );;[127X[104X
    [4X[25Xgap>[125X [27XvecA := BasisVectors( Basis( A5c6 ) );; [127X[104X
    [4X[25Xgap>[125X [27Xv := vecA[1] + vecA[3] + vecA[5];[127X[104X
    [4X[28X(Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4)[128X[104X
    [4X[25Xgap>[125X [27XI5c6 := Ideal( A5c6, [v] );; [127X[104X
    [4X[25Xgap>[125X [27Xactm := AlgebraActionByMultipliers( A5c6, I5c6 );; [127X[104X
    [4X[25Xgap>[125X [27Xactm2 := Image( actm, vecA[2] );; [127X[104X
    [4X[25Xgap>[125X [27XImage( actm2, v );[127X[104X
    [4X[28X(Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.2-2 AlgebraActionBySurjection[101X
  
  [33X[1;0Y[29X[2XAlgebraActionBySurjection[102X( [3Xhom[103X ) [32X operation[133X
  
  [33X[0;0YLet [22Xθ : S -> R[122X be a surjective algebra homomorphism such that [22Xks = 0_S ~∀~ k
  ∈ K = kerθ[122X. Then [22XR[122X acts on [22XS[122X with [22Xr ⋅ s = (θ^-1r)s[122X. Note that thus action is
  well defined since if [22Xθ p = r[122X then [22Xθ^-1r = { p+k ~|~ k ∈ kerθ }[122X and [22X(p+k)s =
  ps+ks = ps+0[122X.[133X
  
  [33X[0;0YContinuing with the previous example, we construct the quotient algebra [22XQ5c6
  =  A5c6/I5c6[122X, and the natural homomorphism [22Xθ : A5c6 -> Q5c6[122X. The kernel of [22Xθ[122X
  is  not  contained  in  the  annihilator  of [22XA5c6[122X, so an attempt to form the
  action fails.[133X
  
  [33X[0;0YAn alternative example involves a single-generator matrix algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xtheta := NaturalHomomorphismByIdeal( A5c6, I5c6 );[127X[104X
    [4X[28X<linear mapping by matrix, <algebra-with-one of dimension [128X[104X
    [4X[28X6 over GF(5)> -> <algebra of dimension 4 over GF(5)>>[128X[104X
    [4X[25Xgap>[125X [27XList( vecA, v -> ImageElm( theta, v ) ); [127X[104X
    [4X[28X[ v.1, v.2, v.3, v.4, (Z(5)^2)*v.1+(Z(5)^2)*v.3, (Z(5)^2)*v.2+(Z(5)^2)*v.4 ][128X[104X
    [4X[25Xgap>[125X [27Xactp := AlgebraActionBySurjection( theta );[127X[104X
    [4X[28Xkernel of hom is not in the annihilator of A[128X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27X## an example which does not fail: [127X[104X
    [4X[25Xgap>[125X [27Xm := [ [0,1,2,3], [0,0,1,2], [0,0,0,1], [0,0,0,0] ];; [127X[104X
    [4X[25Xgap>[125X [27Xm^2;[127X[104X
    [4X[28X[ [ 0, 0, 1, 4 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xm^3;[127X[104X
    [4X[28X[ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XA1 := Algebra( Rationals, [m] );;[127X[104X
    [4X[25Xgap>[125X [27XA3 := Subalgebra( A1, [m^3] );; [127X[104X
    [4X[25Xgap>[125X [27Xnat3 := NaturalHomomorphismByIdeal( A1, A3 ); [127X[104X
    [4X[28X<linear mapping by matrix, <algebra of dimension [128X[104X
    [4X[28X3 over Rationals> -> <algebra of dimension 2 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27Xact3 := AlgebraActionBySurjection( nat3 );; [127X[104X
    [4X[25Xgap>[125X [27Xa3 := Image( act3, BasisVectors( Basis( Image( nat3 ) ) )[1] );;  [127X[104X
    [4X[25Xgap>[125X [27X[ Image( a3, m ) = m^2, Image( a3, m^2 ) = m^3 ];[127X[104X
    [4X[28X[ true, true ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.2-3 SemidirectProductOfAlgebras[101X
  
  [33X[1;0Y[29X[2XSemidirectProductOfAlgebras[102X( [3XR[103X, [3Xact[103X, [3XS[103X ) [32X operation[133X
  
  [33X[0;0YWhen  [22XR,S[122X  are  commutative  algebras  and  [22XR[122X acts on [22XS[122X then we can form the
  semidirect product [22XR ⋉ S[122X, where the product is given by:[133X
  
  
  [24X[33X[0;6Y(r_1,s_1)(r_2,s_2) ~=~ (r_1r_2,~ r_1 \cdot s_2 + r_2 \cdot s_1 + s_1s_2).[133X
  
  [124X
  
  [33X[0;0YThis   product,   as   wekll   as   being   commutative,   is   associative:
  [22X(r_1,s_1)(r_2,s_2)(r_3,s_3)[122X expands as:[133X
  
  
  [24X[33X[0;6Y(r_1r_2r_3,~  \left (r_1r_2)\cdot s3 + (r_1r_3)\cdot s_2 + (r_2r_3)\cdot s_1
  +  r_1  \cdot (s_2s_3) + r_2 \cdot (s_1s_3) + r_3 \cdot (s_1s_2) + s_1s_2s_3
  \right).[133X
  
  [124X
  
  [33X[0;0YIf [22XB_R, B_S[122X are the sets of basis vectors for [22XR[122X and [22XS[122X then [22XR ⋉ S[122X has basis[133X
  
  
  [24X[33X[0;6Y\{(r,0_S) ~|~ r \in B_R\} ~\cup~ \{(0_R,s) ~|~ s \in B_S\}[133X
  
  [124X
  
  [33X[0;0Ywith defining products[133X
  
  
  [24X[33X[0;6Y(r_1,0_S)(r_2,0_S)  = (r_1r_2,0_S), \qquad (r,0_S)(0_R,s) = (0_R,r \cdot s),
  \qquad (0_R,s_1)(0_R,s_2) = (0_R,s_1s_2).[133X
  
  [124X
  
  [33X[0;0YContinuing the example above,[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XP := SemidirectProductOfAlgebras( A5c6, actm, I5c6 ); [127X[104X
    [4X[25Xgap>[125X [27XEmbedding( P, 1 );[127X[104X
    [4X[28X[ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), [128X[104X
    [4X[28X  (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2) [128X[104X
    [4X[28X ] -> [ v.1, v.2, v.3, v.4, v.5, v.6 ][128X[104X
    [4X[25Xgap>[125X [27XEmbedding( P, 2 );[127X[104X
    [4X[28X[ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), [128X[104X
    [4X[28X  (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) ] -> [128X[104X
    [4X[28X[ v.7, v.8 ][128X[104X
    [4X[25Xgap>[125X [27XProjection( P, 1 );[127X[104X
    [4X[28X[ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8 ] -> [128X[104X
    [4X[28X[ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), [128X[104X
    [4X[28X  (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2), [128X[104X
    [4X[28X  <zero> of ..., <zero> of ... ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.2-4 SemidirectProductOfAlgebrasInfo[101X
  
  [33X[1;0Y[29X[2XSemidirectProductOfAlgebrasInfo[102X( [3XP[103X ) [32X attribute[133X
  
  [33X[0;0YThe [10XSemidirectProductOfAlgebrasInfo(P)[110X for [22XP = R ⋉ S[122X is a record with fields
  [10XP.action[110X; [10XP.algebras[110X; [10XP.embeddings[110X; and [10XP.projections[110X.[133X
  
  
  [1X2.3 [33X[0;0YLists of algebra homomorphisms[133X[101X
  
  [1X2.3-1 AllAlgebraHomomorphisms[101X
  
  [33X[1;0Y[29X[2XAllAlgebraHomomorphisms[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XAllBijectiveAlgebraHomomorphisms[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XAllIdempotentAlgebraHomomorphisms[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  
  [33X[0;0YThese  three  operations  list  all  the  homomorphisms  from  [22XA[122X to [22XB[122X of the
  specified type. These lists can get very long, so the operations should only
  be used with small algebras.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XA2c6 := GroupRing( GF(2), Group( (1,2,3,4,5,6) ) );;[127X[104X
    [4X[25Xgap>[125X [27XR2c3 := GroupRing( GF(2), Group( (7,8,9) ) );;[127X[104X
    [4X[25Xgap>[125X [27XhomAR := AllAlgebraHomomorphisms( A2c6, R2c3 );;[127X[104X
    [4X[25Xgap>[125X [27XList( homAR, h -> MappingGeneratorsImages(h) );[127X[104X
    [4X[28X[ [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ <zero> of ... ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*() ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*()+(Z(2)^0)*(7,8,9) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [128X[104X
    [4X[28X      [ (Z(2)^0)*()+(Z(2)^0)*(7,8,9)+(Z(2)^0)*(7,9,8) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*()+(Z(2)^0)*(7,9,8) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(7,8,9) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(7,8,9)+(Z(2)^0)*(7,9,8) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(7,9,8) ] ] ][128X[104X
    [4X[25Xgap>[125X [27XhomRA := AllAlgebraHomomorphisms( R2c3, A2c6 );;[127X[104X
    [4X[25Xgap>[125X [27XList( homRA, h -> MappingGeneratorsImages(h) );[127X[104X
    [4X[28X[ [ [ (Z(2)^0)*(7,8,9) ], [ <zero> of ... ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*() ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(7,8,9) ], [128X[104X
    [4X[28X      [ (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,5,3)(2,6,4) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*()+(Z(2)^0)*(1,5,3)(2,6,4) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*(1,3,5)(2,4,6) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,5,3)(2,6,4) ] [128X[104X
    [4X[28X     ], [ [ (Z(2)^0)*(7,8,9) ], [ (Z(2)^0)*(1,5,3)(2,6,4) ] ] ][128X[104X
    [4X[25Xgap>[125X [27XbijAA := AllBijectiveAlgebraHomomorphisms( A2c6, A2c6 );;[127X[104X
    [4X[25Xgap>[125X [27XList( bijAA, h -> MappingGeneratorsImages(h) );[127X[104X
    [4X[28X[ [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [128X[104X
    [4X[28X      [ (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)(3,6) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [128X[104X
    [4X[28X      [ (Z(2)^0)*()+(Z(2)^0)*(1,4)(2,5)(3,6)+(Z(2)^0)*(1,5,3)(2,6,4) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(1,2,3,4,5,6) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [128X[104X
    [4X[28X      [ (Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,5,3)[128X[104X
    [4X[28X            (2,6,4) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [128X[104X
    [4X[28X      [ (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*[128X[104X
    [4X[28X            (1,6,5,4,3,2) ] ], [128X[104X
    [4X[28X  [ [ (Z(2)^0)*(1,6,5,4,3,2) ], [ (Z(2)^0)*(1,6,5,4,3,2) ] ] ][128X[104X
    [4X[25Xgap>[125X [27XideAA := AllIdempotentAlgebraHomomorphisms( A2c6, A2c6 );; [127X[104X
    [4X[25Xgap>[125X [27XLength( ideAA );[127X[104X
    [4X[28X14[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
