  
  [1X65 [33X[0;0YMagma Rings[133X[101X
  
  [33X[0;0YGiven  a  magma [22XM[122X then the [13Xfree magma ring[113X (or [13Xmagma ring[113X for short) [22XRM[122X of [22XM[122X
  over  a ring-with-one [22XR[122X is the set of finite sums [22X∑_{i ∈ I} r_i m_i[122X with [22Xr_i
  ∈  R[122X,  and [22Xm_i ∈ M[122X. With the obvious addition and [22XR[122X-action from the left, [22XRM[122X
  is  a  free [22XR[122X-module with [22XR[122X-basis [22XM[122X, and with the usual convolution product,
  [22XRM[122X is a ring.[133X
  
  [33X[0;0YTypical examples of free magma rings are[133X
  
  [30X    [33X[0;6Y(multivariate) polynomial rings (see [14X66.15[114X), where the magma is a free
        abelian monoid generated by the indeterminates,[133X
  
  [30X    [33X[0;6Ygroup rings (see [2XIsGroupRing[102X ([14X65.1-5[114X)), where the magma is a group,[133X
  
  [30X    [33X[0;6YLaurent  polynomial  rings,  which are group rings of the free abelian
        groups generated by the indeterminates,[133X
  
  [30X    [33X[0;6Yfree  algebras  and  free  associative  algebras, with or without one,
        where  the  magma  is  a  free  magma  or  a free semigroup, or a free
        magma-with-one or a free monoid, respectively.[133X
  
  [33X[0;0YNote  that  formally,  polynomial  rings  in [5XGAP[105X are not constructed as free
  magma rings.[133X
  
  [33X[0;0YFurthermore,  a  free  Lie  algebra  is  [13Xnot[113X  a  magma  ring, because of the
  additional   relations   given  by  the  Jacobi  identity;  see [14X65.4[114X  for  a
  generalization of magma rings that covers such structures.[133X
  
  [33X[0;0YThe  coefficient ring [22XR[122X and the magma [22XM[122X cannot be regarded as subsets of [22XRM[122X,
  hence the natural [13Xembeddings[113X of [22XR[122X and [22XM[122X into [22XRM[122X must be handled via explicit
  embedding  maps  (see [14X65.3[114X).  Note  that  in  a  magma ring, the addition of
  elements  is  in  general  different  from  an  addition that may be defined
  already  for  the  elements  of  the magma; for example, the addition in the
  group  ring of a matrix group does in general [13Xnot[113X coincide with the addition
  of matrices.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= Algebra( GF(2), [ [ [ Z(2) ] ] ] );;  Size( a );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xrm:= FreeMagmaRing( GF(2), a );;[127X[104X
    [4X[25Xgap>[125X [27Xemb:= Embedding( a, rm );;[127X[104X
    [4X[25Xgap>[125X [27Xz:= Zero( a );;  o:= One( a );;[127X[104X
    [4X[25Xgap>[125X [27Ximz:= z ^ emb;  IsZero( imz );[127X[104X
    [4X[28X(Z(2)^0)*[ [ 0*Z(2) ] ][128X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xim1:= ( z + o ) ^ emb;[127X[104X
    [4X[28X(Z(2)^0)*[ [ Z(2)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xim2:= z ^ emb + o ^ emb;[127X[104X
    [4X[28X(Z(2)^0)*[ [ 0*Z(2) ] ]+(Z(2)^0)*[ [ Z(2)^0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xim1 = im2;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X65.1 [33X[0;0YFree Magma Rings[133X[101X
  
  [1X65.1-1 FreeMagmaRing[101X
  
  [29X[2XFreeMagmaRing[102X( [3XR[103X, [3XM[103X ) [32X function
  
  [33X[0;0Yis a free magma ring over the ring [3XR[103X, free on the magma [3XM[103X.[133X
  
  [1X65.1-2 GroupRing[101X
  
  [29X[2XGroupRing[102X( [3XR[103X, [3XG[103X ) [32X function
  
  [33X[0;0Yis the group ring of the group [3XG[103X, over the ring [3XR[103X.[133X
  
  [1X65.1-3 IsFreeMagmaRing[101X
  
  [29X[2XIsFreeMagmaRing[102X( [3XD[103X ) [32X Category
  
  [33X[0;0YA  domain lies in the category [2XIsFreeMagmaRing[102X if it has been constructed as
  a  free  magma  ring.  In  particular,  if  [3XD[103X lies in this category then the
  operations  [2XLeftActingDomain[102X  ([14X57.1-11[114X)  and  [2XUnderlyingMagma[102X  ([14X65.1-6[114X)  are
  applicable  to  [3XD[103X,  and  yield the ring [22XR[122X and the magma [22XM[122X such that [3XD[103X is the
  magma ring [22XRM[122X.[133X
  
  [33X[0;0YSo  being  a  magma  ring  in [5XGAP[105X includes the knowledge of the ring and the
  magma. Note that a magma ring [22XRM[122X may abstractly be generated as a magma ring
  by  a  magma  different  from the underlying magma [22XM[122X. For example, the group
  ring of the dihedral group of order [22X8[122X over the field with [22X3[122X elements is also
  spanned by a quaternion group of order [22X8[122X over the same field.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xd8:= DihedralGroup( 8 );[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xrm:= FreeMagmaRing( GF(3), d8 );[127X[104X
    [4X[28X<algebra-with-one over GF(3), with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xemb:= Embedding( d8, rm );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= List( GeneratorsOfGroup( d8 ), x -> x^emb );;[127X[104X
    [4X[25Xgap>[125X [27Xx1:= gens[1] + gens[2];;[127X[104X
    [4X[25Xgap>[125X [27Xx2:= ( gens[1] - gens[2] ) * gens[3];;[127X[104X
    [4X[25Xgap>[125X [27Xx3:= gens[1] * gens[2] * ( One( rm ) - gens[3] );;[127X[104X
    [4X[25Xgap>[125X [27Xg1:= x1 - x2 + x3;;[127X[104X
    [4X[25Xgap>[125X [27Xg2:= x1 + x2;;[127X[104X
    [4X[25Xgap>[125X [27Xq8:= Group( g1, g2 );;[127X[104X
    [4X[25Xgap>[125X [27XSize( q8 );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XForAny( [ d8, q8 ], IsAbelian );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XList( [ d8, q8 ], g -> Number( AsList( g ), x -> Order( x ) = 2 ) );[127X[104X
    [4X[28X[ 5, 1 ][128X[104X
    [4X[25Xgap>[125X [27XDimension( Subspace( rm, q8 ) );[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [1X65.1-4 IsFreeMagmaRingWithOne[101X
  
  [29X[2XIsFreeMagmaRingWithOne[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0Y[2XIsFreeMagmaRingWithOne[102X  is  just  a  synonym for the meet of [2XIsFreeMagmaRing[102X
  ([14X65.1-3[114X) and [2XIsMagmaWithOne[102X ([14X35.1-2[114X).[133X
  
  [1X65.1-5 IsGroupRing[101X
  
  [29X[2XIsGroupRing[102X( [3Xobj[103X ) [32X property
  
  [33X[0;0YA [13Xgroup ring[113X is a magma ring where the underlying magma is a group.[133X
  
  [1X65.1-6 UnderlyingMagma[101X
  
  [29X[2XUnderlyingMagma[102X( [3XRM[103X ) [32X attribute
  
  [33X[0;0Ystores the underlying magma of a free magma ring.[133X
  
  [1X65.1-7 AugmentationIdeal[101X
  
  [29X[2XAugmentationIdeal[102X( [3XRG[103X ) [32X attribute
  
  [33X[0;0Yis  the  augmentation  ideal  of  the group ring [3XRG[103X, i.e., the kernel of the
  trivial representation of [3XRG[103X.[133X
  
  
  [1X65.2 [33X[0;0YElements of Free Magma Rings[133X[101X
  
  [33X[0;0YIn  order  to  treat elements of free magma rings uniformly, also without an
  external   representation,   the   attributes   [2XCoefficientsAndMagmaElements[102X
  ([14X65.2-4[114X) and [2XZeroCoefficient[102X ([14X65.2-5[114X) were introduced that allow one to [21Xtake
  an element of an arbitrary magma ring into pieces[121X.[133X
  
  [33X[0;0YConversely, for constructing magma ring elements from coefficients and magma
  elements,  [2XElementOfMagmaRing[102X  ([14X65.2-6[114X) can be used. (Of course one can also
  embed  each  magma  element into the magma ring, see [14X65.3[114X, and then form the
  linear  combination,  but many unnecessary intermediate elements are created
  this way.)[133X
  
  [1X65.2-1 IsMagmaRingObjDefaultRep[101X
  
  [29X[2XIsMagmaRingObjDefaultRep[102X( [3Xobj[103X ) [32X Representation
  
  [33X[0;0YThe default representation of a magma ring element is a list of length 2, at
  first  position  the  zero  coefficient,  at second position a list with the
  coefficients  at  the  even  positions,  and  the  magma elements at the odd
  positions, with the ordering as defined for the magma elements.[133X
  
  [33X[0;0YIt  is  assumed  that  arithmetic  operations  on  magma  rings produce only
  normalized elements.[133X
  
  [1X65.2-2 IsElementOfFreeMagmaRing[101X
  
  [29X[2XIsElementOfFreeMagmaRing[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsElementOfFreeMagmaRingCollection[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YThe  category  of  elements  of  a  free  magma  ring  (See  [2XIsFreeMagmaRing[102X
  ([14X65.1-3[114X)).[133X
  
  [1X65.2-3 IsElementOfFreeMagmaRingFamily[101X
  
  [29X[2XIsElementOfFreeMagmaRingFamily[102X( [3XFam[103X ) [32X Category
  
  [33X[0;0YElements  of  families  in  this  category have trivial normalisation, i.e.,
  efficient methods for [10X\=[110X and [10X\<[110X.[133X
  
  [1X65.2-4 CoefficientsAndMagmaElements[101X
  
  [29X[2XCoefficientsAndMagmaElements[102X( [3Xelm[103X ) [32X attribute
  
  [33X[0;0Yis  a list that contains at the odd positions the magma elements, and at the
  even positions their coefficients in the element [3Xelm[103X.[133X
  
  [1X65.2-5 ZeroCoefficient[101X
  
  [29X[2XZeroCoefficient[102X( [3Xelm[103X ) [32X attribute
  
  [33X[0;0YFor  an  element  [3Xelm[103X of a magma ring (modulo relations) [22XRM[122X, [2XZeroCoefficient[102X
  returns the zero element of the coefficient ring [22XR[122X.[133X
  
  [1X65.2-6 ElementOfMagmaRing[101X
  
  [29X[2XElementOfMagmaRing[102X( [3XFam[103X, [3Xzerocoeff[103X, [3Xcoeffs[103X, [3Xmgmelms[103X ) [32X operation
  
  [33X[0;0Y[2XElementOfMagmaRing[102X  returns the element [22X∑_{i = 1}^n c_i m_i'[122X, where [22X[3Xcoeffs[103X =
  [ c_1, c_2, ..., c_n ][122X is a list of coefficients, [22X[3Xmgmelms[103X = [ m_1, m_2, ...,
  m_n  ][122X  is  a  list of magma elements, and [22Xm_i'[122X is the image of [22Xm_i[122X under an
  embedding  of a magma containing [22Xm_i[122X into a magma ring whose elements lie in
  the  family  [3XFam[103X.  [3Xzerocoeff[103X  must  be  the  zero  of  the  coefficient ring
  containing the [22Xc_i[122X.[133X
  
  
  [1X65.3 [33X[0;0YNatural Embeddings related to Magma Rings[133X[101X
  
  [33X[0;0YNeither  the  coefficient  ring [22XR[122X nor the magma [22XM[122X are regarded as subsets of
  the  magma  ring  [22XRM[122X, so one has to use [13Xembeddings[113X (see [2XEmbedding[102X ([14X32.2-10[114X))
  explicitly   whenever   one   needs  for  example  the  magma  ring  element
  corresponding to a given magma element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= Rationals;;  g:= SymmetricGroup( 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xfg:= FreeMagmaRing( f, g );[127X[104X
    [4X[28X<algebra-with-one over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XDimension( fg );[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfAlgebraWithOne( fg );[127X[104X
    [4X[28X[ (1)*(1,2,3), (1)*(1,2) ][128X[104X
    [4X[25Xgap>[125X [27X( 3*gens[1] - 2*gens[2] ) * ( gens[1] + gens[2] );[127X[104X
    [4X[28X(-2)*()+(3)*(2,3)+(3)*(1,3,2)+(-2)*(1,3)[128X[104X
    [4X[25Xgap>[125X [27XOne( fg );[127X[104X
    [4X[28X(1)*()[128X[104X
    [4X[25Xgap>[125X [27Xemb:= Embedding( g, fg );;[127X[104X
    [4X[25Xgap>[125X [27Xelm:= (1,2,3)^emb;  elm in fg;[127X[104X
    [4X[28X(1)*(1,2,3)[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xnew:= elm + One( fg );[127X[104X
    [4X[28X(1)*()+(1)*(1,2,3)[128X[104X
    [4X[25Xgap>[125X [27Xnew^2;[127X[104X
    [4X[28X(1)*()+(2)*(1,2,3)+(1)*(1,3,2)[128X[104X
    [4X[25Xgap>[125X [27Xemb2:= Embedding( f, fg );;[127X[104X
    [4X[25Xgap>[125X [27Xelm:= One( f )^emb2;  elm in fg;[127X[104X
    [4X[28X(1)*()[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X65.4 [33X[0;0YMagma Rings modulo Relations[133X[101X
  
  [33X[0;0YA  more  general  construction  than  that of free magma rings allows one to
  create  rings  that are not free [22XR[122X-modules on a given magma [22XM[122X but arise from
  the  magma  ring  [22XRM[122X  by factoring out certain identities. Examples for such
  structures  are  finitely  presented  (associative)  algebras  and  free Lie
  algebras (see [2XFreeLieAlgebra[102X ([14X64.2-4[114X)).[133X
  
  [33X[0;0YIn  [5XGAP[105X,  the  use  of magma rings modulo relations is limited to situations
  where  a  normal  form  of  the  elements  is  known  and where one wants to
  guarantee  that  all  elements  actually constructed are in normal form. (In
  particular,  the  computation  of  the  normal  form must be cheap.) This is
  because  the  methods for comparing elements in magma rings modulo relations
  via [10X\=[110X and [10X\<[110X just compare the involved coefficients and magma elements, and
  also   the  vector  space  functions  regard  those  monomials  as  linearly
  independent   over   the  coefficients  ring  that  actually  occur  in  the
  representation of an element of a magma ring modulo relations.[133X
  
  [33X[0;0YThus  only  very  special finitely presented algebras will be represented as
  magma  rings  modulo  relations,  in general finitely presented algebras are
  dealt with via the mechanism described in Chapter [14X63[114X.[133X
  
  [1X65.4-1 IsElementOfMagmaRingModuloRelations[101X
  
  [29X[2XIsElementOfMagmaRingModuloRelations[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsElementOfMagmaRingModuloRelationsCollection[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YThis category is used, e. g., for elements of free Lie algebras.[133X
  
  [1X65.4-2 IsElementOfMagmaRingModuloRelationsFamily[101X
  
  [29X[2XIsElementOfMagmaRingModuloRelationsFamily[102X( [3XFam[103X ) [32X Category
  
  [33X[0;0YThe  family  category  for  the category [2XIsElementOfMagmaRingModuloRelations[102X
  ([14X65.4-1[114X).[133X
  
  [1X65.4-3 NormalizedElementOfMagmaRingModuloRelations[101X
  
  [29X[2XNormalizedElementOfMagmaRingModuloRelations[102X( [3XF[103X, [3Xdescr[103X ) [32X operation
  
  [33X[0;0YLet  [3XF[103X  be  a  family of magma ring elements modulo relations, and [3Xdescr[103X the
  description   of   an   element   in   a   magma   ring   modulo  relations.
  [2XNormalizedElementOfMagmaRingModuloRelations[102X  returns  a  description  of the
  same element, but normalized w.r.t. the relations. So two elements are equal
  if  and only if the result of [2XNormalizedElementOfMagmaRingModuloRelations[102X is
  equal   for  their  internal  data,  that  is,  [2XCoefficientsAndMagmaElements[102X
  ([14X65.2-4[114X) will return the same for the corresponding two elements.[133X
  
  [33X[0;0Y[2XNormalizedElementOfMagmaRingModuloRelations[102X   is  allowed  to  return  [3Xdescr[103X
  itself, it need not make a copy. This is the case for example in the case of
  free magma rings.[133X
  
  [1X65.4-4 IsMagmaRingModuloRelations[101X
  
  [29X[2XIsMagmaRingModuloRelations[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YA  [5XGAP[105X object lies in the category [2XIsMagmaRingModuloRelations[102X if it has been
  constructed  as  a  magma ring modulo relations. Each element of such a ring
  has  a  unique  normal  form,  so  [2XCoefficientsAndMagmaElements[102X  ([14X65.2-4[114X) is
  well-defined for it.[133X
  
  [33X[0;0YThis  category  is  not inherited to factor structures, which are in general
  best described as finitely presented algebras, see Chapter [14X63[114X.[133X
  
  
  [1X65.5 [33X[0;0YMagma Rings modulo the Span of a Zero Element[133X[101X
  
  [1X65.5-1 IsElementOfMagmaRingModuloSpanOfZeroFamily[101X
  
  [29X[2XIsElementOfMagmaRingModuloSpanOfZeroFamily[102X( [3XFam[103X ) [32X Category
  
  [33X[0;0YWe  need  this  for  the normalization method, which takes a family as first
  argument.[133X
  
  [1X65.5-2 IsMagmaRingModuloSpanOfZero[101X
  
  [29X[2XIsMagmaRingModuloSpanOfZero[102X( [3XRM[103X ) [32X Category
  
  [33X[0;0YThe category of magma rings modulo the span of a zero element.[133X
  
  [1X65.5-3 MagmaRingModuloSpanOfZero[101X
  
  [29X[2XMagmaRingModuloSpanOfZero[102X( [3XR[103X, [3XM[103X, [3Xz[103X ) [32X function
  
  [33X[0;0YLet [3XR[103X be a ring, [3XM[103X a magma, and [3Xz[103X an element of [3XM[103X with the property that [22X[3Xz[103X *
  m  =  [3Xz[103X[122X holds for all [22Xm ∈ M[122X. The element [3Xz[103X could be called a [21Xzero element[121X of
  [3XM[103X,  but  note that in general [3Xz[103X cannot be obtained as [10XZero( [110X[22Xm[122X[10X )[110X for each [22Xm ∈
  M[122X, so this situation does not match the definition of [2XZero[102X ([14X31.10-3[114X).[133X
  
  [33X[0;0Y[2XMagmaRingModuloSpanOfZero[102X  returns  the  magma  ring  [22X[3XR[103X[3XM[103X[122X modulo the relation
  given  by  the  identification of [3Xz[103X with zero. This is an example of a magma
  ring modulo relations, see [14X65.4[114X.[133X
  
  
  [1X65.6 [33X[0;0YTechnical Details about the Implementation of Magma Rings[133X[101X
  
  [33X[0;0YThe  [13Xfamily[113X  containing  elements  in the magma ring [22XRM[122X in fact contains all
  elements with coefficients in the family of elements of [22XR[122X and magma elements
  in  the  family of elements of [22XM[122X. So arithmetic operations with coefficients
  outside [22XR[122X or with magma elements outside [22XM[122X might create elements outside [22XRM[122X.[133X
  
  [33X[0;0YIt  should  be  mentioned that each call of [2XFreeMagmaRing[102X ([14X65.1-1[114X) creates a
  new  family  of  elements, so for example the elements of two group rings of
  permutation  groups  over  the  same  ring  lie  in  different  families and
  therefore are regarded as different.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SymmetricGroup( 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xh:= AlternatingGroup( 3 );;[127X[104X
    [4X[25Xgap>[125X [27XIsSubset( g, h );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xf:= GF(2);;[127X[104X
    [4X[25Xgap>[125X [27Xfg:= GroupRing( f, g );[127X[104X
    [4X[28X<algebra-with-one over GF(2), with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xfh:= GroupRing( f, h );[127X[104X
    [4X[28X<algebra-with-one over GF(2), with 1 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubset( fg, fh );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xo1:= One( fh );  o2:= One( fg );  o1 = o2;[127X[104X
    [4X[28X(Z(2)^0)*()[128X[104X
    [4X[28X(Z(2)^0)*()[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xemb:= Embedding( g, fg );;[127X[104X
    [4X[25Xgap>[125X [27Xim:= Image( emb, h );[127X[104X
    [4X[28X<group of size 3 with 1 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubset( fg, im );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere  is  [13Xno[113X  generic  [13Xexternal representation[113X for elements in an arbitrary
  free magma ring. For example, polynomials are elements of a free magma ring,
  and  they have an external representation relying on the special form of the
  underlying  monomials.  On  the  other  hand,  elements in a group ring of a
  permutation group do not admit such an external representation.[133X
  
  [33X[0;0YFor   convenience,   magma  rings  constructed  with  [2XFreeAlgebra[102X  ([14X62.3-1[114X),
  [2XFreeAssociativeAlgebra[102X    ([14X62.3-3[114X),    [2XFreeAlgebraWithOne[102X    ([14X62.3-2[114X),   and
  [2XFreeAssociativeAlgebraWithOne[102X ([14X62.3-4[114X) support an external representation of
  their  elements,  which  is  defined  as a list of length 2, the first entry
  being  the  zero  coefficient,  the  second  being  a list with the external
  representations  of  the  magma  elements  at  the  odd  positions  and  the
  corresponding coefficients at the even positions.[133X
  
  [33X[0;0YAs  the  above  examples show, there are several possible representations of
  magma  ring  elements, the representations used for polynomials (see Chapter
   [14X66[114X) as well as the default representation [2XIsMagmaRingObjDefaultRep[102X ([14X65.2-1[114X)
  of  magma ring elements. The latter simply stores the zero coefficient and a
  list  containing  the  coefficients of the element at the even positions and
  the  corresponding magma elements at the odd positions, where the succession
  is compatible with the ordering of magma elements via [10X\<[110X.[133X
  
