  
  [1X6 [33X[0;0YIdentities Among Relators[133X[101X
  
  [33X[0;0YThe  identities  among  the  relators  for  a finitely presented group [22XG[122X are
  constructed as logged module polynomials. The procedure, described in [HW03]
  and  based  on  work in [BRS99], is to construct a full set of [13Xgroup relator
  sequences[113X  for the group; convert these into module polynomials (eliminating
  empty sequences); and then apply simplification rules (including the primary
  identity property) to eliminate obvious duplicates and conjugates.[133X
  
  [33X[0;0YWhen  a  reduced set of polynomials has been obtained, the relator sequences
  from  which  they  were formed are returned as the [13Xidentities among relators[113X
  for [22XG[122X.[133X
  
  
  [1X6.1 [33X[0;0YConstructing identities[133X[101X
  
  [1X6.1-1 RootIdentities[101X
  
  [33X[1;0Y[29X[2XRootIdentities[102X( [3Xgrp[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRootPositions[102X( [3Xgrp[103X ) [32X attribute[133X
  
  [33X[0;0YThe [13Xroot identities[113X of [22XG[122X are identities of the form [22XR^-1R^w[122X where [22XR = w^n[122X is
  a proper power relator and [22Xn>1[122X. (For equivalent forms invert, or permute the
  factors cyclically, or act with [22Xw^-1[122X.)[133X
  
  [33X[0;0YFor  [22XS_3  = ⟨ f,g ~|~ ρ=f^3, σ=g^2, τ=(fg)^2 ⟩[122X all three relators are proper
  powers:  [22X[1  ≡  ρ=f^3,  2 ≡ σ=g^2, 3 ≡ τ=(fg)^2][122X. So the root identities are
  [22Xρ^-1 ρ^a, σ^-1 σ^b[122X and [22Xτ^-1 τ^ab[122X.[133X
  
  [33X[0;0YFor [22XQ_8 = ⟨ a,b ~|~ q=a^4, r=b^4, s=abab^-1, t=a^2b^2 ⟩[122X only two of the four
  relators  are  proper  powers,  so the root identities are [22Xq^-1 q^a[122X and [22Xr^-1
  r^b[122X.[133X
  
  [33X[0;0YIn the example we see that the attribute [10XRootIdentities[110X returns a list which
  includes  [22XR^-1R^w^-1}[122X  as  well  as  [22XR^-1R^w[122X.  Relator  [22Xρ^-1ρ^f[122X is stored as
  [10X[[-1,id],[1,f]][110X, etc.[133X
  
  [33X[0;0YThe  [10XRootPositions[110X  attribute  is  a  boolean  list  specifying which of the
  relators are proper powers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xroots3 := RootIdentities(s3);[127X[104X
    [4X[28X[ [ [ -1, <identity ...> ], [ 1, s3_M1 ] ], [128X[104X
    [4X[28X  [ [ -1, <identity ...> ], [ 1, s3_M3 ] ], [128X[104X
    [4X[28X  [ [ -2, <identity ...> ], [ 2, s3_M2 ] ], [128X[104X
    [4X[28X  [ [ -2, <identity ...> ], [ 2, s3_M4 ] ], [128X[104X
    [4X[28X  [ [ -3, <identity ...> ], [ 3, s3_M1*s3_M2 ] ], [128X[104X
    [4X[28X  [ [ -3, <identity ...> ], [ 3, s3_M4*s3_M3 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XRootPositions(s3);[127X[104X
    [4X[28X[ true, true, true ][128X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( RootIdentities(q8), genfmq8, q8labs );[127X[104X
    [4X[28X[ [ [ -1, id ], [ 1, a ] ], [ [ -1, id ], [ 1, A ] ], [ [ -2, id ], [128X[104X
    [4X[28X[ 2, b ] ], [ [ -2, id ], [ 2, B ] ] ][128X[104X
    [4X[25Xgap>[125X [27XRootPositions(q8);[127X[104X
    [4X[28X[ true, true, false, false ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.1-2 IdentityRelatorSequences[101X
  
  [33X[1;0Y[29X[2XIdentityRelatorSequences[102X( [3Xgrp[103X ) [32X attribute[133X
  
  [33X[0;0YTo  construct  the  [13Xidentity  relator  sequences[113X for a group [22XG[122X we apply each
  relator [22XR[122X at each non-identity element [22Xx[122X, reducing the resulting words using
  the logged rewrite system.[133X
  
  [33X[0;0YWith  the  [10Xs3[110X  example, the monoid presentation has generators [22X{f,g,F,G}[122X and
  relators[133X
  
  
  [24X[33X[0;6Y[~ fF,~ gG,~ Ff,~ Gg,~ f^3,~ g^2,~ (fg)^2 ~]\ ,[133X
  
  [124X
  
  [33X[0;0Yand the elements are [22X{ id, f, g, F, fg, gf}[122X. The logged rewriting system has
  relations[133X
  
      ┌──────────────────────────────────────────────────────────────────────────────────────────┐
      │ [22Xf^-1 = F, quad g^-1 = [-2, id]g, quad F^-1 = f, quad G^-1 = g, quad G = [-2,id]g,[122X        │ 
      │ [22XfF = id, quad g^2 = [2, id] id, quad Ff = id, quad f^2 = [1, id]F, quad F^2 = [-1, id]f,[122X │ 
      │ [22XgF = [-3, id][2,FGF]fg, quad Fg = [-3,f][2,FG]gf,[122X                                        │ 
      │ [22Xfgf = [-2,FGF][3, id]g, quad gfg = [3,f]F[122X                                                │ 
      └──────────────────────────────────────────────────────────────────────────────────────────┘
  
  [33X[0;0YHere is the Cayley graphs of [22XS_3[122X, with the solid arrows forming the spanning
  tree:  Applying  [22XR  =  τ  =  (fg)^2[122X at [22Xx = f[122X gives the cycle (top right-hand
  quadrilateral):[133X
  
  
  [24X[33X[0;6Yf    \stackrel{f}{\longrightarrow}    F   \stackrel{g}{\longrightarrow}   gf
  \stackrel{f}{\longrightarrow} fg \stackrel{g}{\longrightarrow} f\ .[133X
  
  [124X
  
  [33X[0;0YEach of these edges has a non-trivial logged rewrite, particularly the third
  edge where [22Xgff -> gF -> fg[122X. Combining this log information we obtain:[133X
  
  
  [24X[33X[0;6Y[\tau,F]\,f   ~=~   f.\tau   ~=~   [1,{\rm   id}].[-3,f][2,FG].[1,G][-3,{\rm
  id}][2,FGF].[2,F]\,f .[133X
  
  [124X
  
  [33X[0;0YExpanding [22X[1, id][-3,f][2,FG][1,G][-3, id][2,FGF][2,F][-3,F][122X gives[133X
  
  
  [24X[33X[0;6Yfff.FGFGFf.gfggFG.gfffG.GFGF.fgfggFGF.fggF.fGFGFF[133X
  
  [124X
  
  [33X[0;0Ywhich  cancels  to  the  identity,  as expected. Converting this back to the
  group  presentation, we obtain the fourth identity given in the introduction
  ([14X1.1[114X):[133X
  
  
  [24X[33X[0;6Y\iota_{(\tau,f)}  ~=~ \rho\ (\tau^{-1})^f\ \sigma^{FG}\ \rho^G\ (\tau^{-1})\
  \sigma^{FGF}\ \sigma^F\ (\tau^{-1})^F\, .[133X
  
  [124X
  
  [33X[0;0YThe  operation  [10XIdentityRelatorSequences[110X  returns  a  list  which  omits any
  duplicates  or empty lists. For the [10Xs3[110X example, all of the possible [22X5*3 = 15[122X
  sequences are added to the root identities.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xms3 := MonoidPresentationFpGroup( s3 );;[127X[104X
    [4X[25Xgap>[125X [27Xfms3 := FreeGroupOfPresentation( ms3 );;[127X[104X
    [4X[25Xgap>[125X [27Xgenfms3 := GeneratorsOfGroup(fms3 );;[127X[104X
    [4X[25Xgap>[125X [27Xs3labs := ["f","g","F","G"];; [127X[104X
    [4X[25Xgap>[125X [27XSetMonoidPresentationLabels( ms3, s3labs );; [127X[104X
    [4X[25Xgap>[125X [27Xidss3 := IdentityRelatorSequences( s3 );;          [127X[104X
    [4X[25Xgap>[125X [27Xlenidss3 := Length( idss3 );                                   [127X[104X
    [4X[28X17[128X[104X
    [4X[25Xgap>[125X [27XList( idss3, L -> Length(L) );[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 6, 8, 8 ][128X[104X
    [4X[25Xgap>[125X [27X for i in [1..Length(idss3)] do                     [127X[104X
    [4X[25X>[125X [27X      PrintLnUsingLabels( idss3[i], genfms3, s3labs );[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[28X[ [ -3, id ], [ 3, f*g ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 3, G*F ] ][128X[104X
    [4X[28X[ [ -2, id ], [ 2, g ] ][128X[104X
    [4X[28X[ [ -2, id ], [ 2, G ] ][128X[104X
    [4X[28X[ [ -1, id ], [ 1, f ] ][128X[104X
    [4X[28X[ [ -1, id ], [ 1, F ] ][128X[104X
    [4X[28X[ [ 1, id ], [ -1, f ] ][128X[104X
    [4X[28X[ [ 1, id ], [ -1, F ] ][128X[104X
    [4X[28X[ [ 1, G ], [ -1, F*G ] ][128X[104X
    [4X[28X[ [ 2, id ], [ -2, G ] ][128X[104X
    [4X[28X[ [ 2, F ], [ -2, G*F ] ][128X[104X
    [4X[28X[ [ 3, f ], [ -3, G ] ][128X[104X
    [4X[28X[ [ -3, f ], [ 2, F*G ], [ 3, f ], [ -2, f ] ][128X[104X
    [4X[28X[ [ -2, F*G*F ], [ 3, id ], [ 2, id ], [ -3, G*F ] ][128X[104X
    [4X[28X[ [ -2, F*G*F ], [ 3, id ], [ 1, G ], [ -3, id ], [ 2, F*G*F ], [128X[104X
    [4X[28X[ -1, G*F ] ][128X[104X
    [4X[28X[ [ 1, id ], [ -3, f ], [ 2, F*G ], [ 1, G ], [ -3, id ], [128X[104X
    [4X[28X[ 2, F*G*F ], [ 2, F ], [ -3, F ] ][128X[104X
    [4X[28X[ [ 1, G ], [ -3, id ], [ 2, F*G*F ], [ 2, F ], [ 1, id ], [128X[104X
    [4X[28X[ -3, f ], [ 2, F*G ], [ -3, F*G ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.1-3 LogSequenceLessThan[101X
  
  [33X[1;0Y[29X[2XLogSequenceLessThan[102X( [3XJ[103X, [3XK[103X ) [32X operation[133X
  
  [33X[0;0YThis  is  an  operation  used to sort lists of identity sequences. First the
  lengths  of  sequences  [10XJ, K[110X are compared. If the lengths are equal then the
  sequences  are  compared  as  lists.  The  list  [10Xidss3[110X  is sorted using this
  function.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XLogSequenceLessThan( idss3[7], idss3[8] ); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.1-4 ExpandLogSequence[101X
  
  [33X[1;0Y[29X[2XExpandLogSequence[102X( [3XmG[103X, [3XL[103X ) [32X operation[133X
  
  [33X[0;0YThis  operation takes a log sequence [22XL[122X, writes each term as a conjugate of a
  relator,  takes  the  product  of all of these, and then cancels consecutive
  inverse  generators  to return a word in the free group of the presentation.
  This  is precisely what we did by hand with [22Xι_(τ,f)[122X on the previous page. If
  the  sequence  is  an  identity  sequence  the  identity  element  should be
  returned, so this provides a useful check.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XExpandLogSequence( ms3, idss3[17] );[127X[104X
    [4X[28X<identity ...>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YIdentities for [22XS_3[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  now  return  to  the  example considered in section [14X1.1[114X. In the previous
  section we have constructed [22X17[122X identity sequences, and we now wish to reduce
  this number to find a minimal set.[133X
  
  [1X6.2-1 ReduceLogSequences[101X
  
  [33X[1;0Y[29X[2XReduceLogSequences[102X( [3XG[103X, [3Xids[103X ) [32X operation[133X
  
  [33X[0;0YThis  operation  applies a collection of operations, which will be described
  in the following section, to reduce the list [10Xidss3[110X from [22X17[122X to [22X5[122X identities.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xridss3 := ReduceLogSequences( s3, idss3 );;[127X[104X
    [4X[25Xgap>[125X [27Xlenridss3 := Length( ridss3 );                                   [127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..lenridss3] do                                [127X[104X
    [4X[25X>[125X [27X       PrintLnUsingLabels( ridss3[i], genfms3, s3labs );   [127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28X[ [ -3, id ], [ 3, f*g ] ][128X[104X
    [4X[28X[ [ -2, id ], [ 2, g ] ][128X[104X
    [4X[28X[ [ -1, id ], [ 1, f ] ][128X[104X
    [4X[28X[ [ 1, id ], [ -3, f ], [ 2, F*G ], [ 1, G ], [ -3, id ], [128X[104X
    [4X[28X[ 2, F*G*F ], [ 2, F ], [ -3, F ] ][128X[104X
    [4X[28X[ [ 1, id ], [ -3, g ], [ 2, F*G*F*g ], [ 2, F*g ], [ 1, g ], [128X[104X
    [4X[28X[ -3, id ], [ 2, F ], [ -3, F ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  wish  to show that the fifth of these identities is a combination of the
  first four. Recall that the fourth identity was obtained by applying [22XR = τ =
  (fg)^2[122X  at  [22Xx = f[122X. The fifth comes from applying [22XR = τ[122X at [22Xx = gf[122X, so this is
  the same cycle but with a different start point.[133X
  
  [1X6.2-2 ConjugateByWordLogSequence[101X
  
  [33X[1;0Y[29X[2XConjugateByWordLogSequence[102X( [3XmG[103X, [3XK[103X, [3Xw[103X ) [32X operation[133X
  
  [33X[0;0YThis  operation  conjugates  every  term  in a log sequence by a word in the
  generators.  In the example we conjugate the fifth identity [10XK5[110X by [22XG[122X. It then
  becomes  apparent that the fourth identity [10XK4[110X has the form [10X[ A, B, [ -3, F ]
  ][110X while [10XK5[110X has the form [10X[ B, A, [ -3, FG ] ][110X, where the [10XF[110X and the [10XGF[110X are the
  inverses of the vertices where the cycle starts.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XK4 := ShallowCopy( ridss3[4] );; [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( K4, genfms3, s3labs ); [127X[104X
    [4X[28X[ [ 1, id ], [ -3, f ], [ 2, F*G ], [ 1, G ], [ -3, id ], [128X[104X
    [4X[28X[ 2, F*G*F ], [ 2, F ], [ -3, F ] ][128X[104X
    [4X[25Xgap>[125X [27XL5 := ShallowCopy( ridss3[5] );; [127X[104X
    [4X[25Xgap>[125X [27XK5 := ConjugateByWordLogSequence( ms3, L5, genfms3[4] );; [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( K5, genfms3, s3labs ); [127X[104X
    [4X[28X[ [ 1, G ], [ -3, id ], [ 2, F*G*F ], [ 2, F ], [ 1, id ], [128X[104X
    [4X[28X[ -3, f ], [ 2, F*G ], [ -3, F*G ] ][128X[104X
    [4X[25Xgap>[125X [27XA := K4{[1..3]};;              [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( A, genfms3, s3labs ); [127X[104X
    [4X[28X[ [ 1, id ], [ -3, f ], [ 2, F*G ] ][128X[104X
    [4X[25Xgap>[125X [27XB := K4{[4..7]};;              [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( B, genfms3, s3labs ); [127X[104X
    [4X[28X[ [ 1, G ], [ -3, id ], [ 2, F*G*F ], [ 2, F ] ][128X[104X
    [4X[25Xgap>[125X [27XPositionSublist( K5, A ); [127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XPositionSublist( K5, B ); [127X[104X
    [4X[28X1[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.2-3 ChangeStartLogSequence[101X
  
  [33X[1;0Y[29X[2XChangeStartLogSequence[102X( [3Xmon[103X, [3XK[103X, [3Xp[103X ) [32X operation[133X
  
  [33X[0;0YThe  start  point of an identity log sequence can be chosen at random (since
  every  conjugate of an identity is that identity). This operation permutes a
  given sequence [22XK[122X so as to start at the [22Xp[122X-th position.[133X
  
  [33X[0;0YIn  our  example  we  wish  to show that [10XK4[110X and [10XK5[110X are equivalent up to root
  identities. To do this we first replace [10XK4[110X by [10XJ4 = [ B, [ -3, F ], A ][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XJ4 := ChangeStartLogSequence( ms3, K4, 4 );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( J4, genfms3, s3labs ); [127X[104X
    [4X[28X[ [ 1, G ], [ -3, id ], [ 2, F*G*F ], [ 2, F ], [ -3, F ], [128X[104X
    [4X[28X[ 1, id ], [ -3, f ], [ 2, F*G ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.2-4 InverseLogSequence[101X
  
  [33X[1;0Y[29X[2XInverseLogSequence[102X( [3XK[103X ) [32X operation[133X
  
  [33X[0;0YTo  invert a log sequence we reverse the order of the terms and replace each
  [10X[m,w][110X by [10X[-m,w][110X.[133X
  
  [33X[0;0YWe continue our example by replacing [10XJ4[110X by its inverse.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XJ4 := InverseLogSequence( J4 );; [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( J4, genfms3, s3labs ); [127X[104X
    [4X[28X[ [ -2, F*G ], [ 3, f ], [ -1, id ], [ 3, F ], [ -2, F ], [ -2, F*G*F ], [128X[104X
    [4X[28X[ 3, id ], [ -1, G ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.2-5 CancelImmediateInversesLogSequence[101X
  
  [33X[1;0Y[29X[2XCancelImmediateInversesLogSequence[102X( [3XK[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XCancelInversesLogSequence[102X( [3XmG[103X, [3XK[103X ) [32X operation[133X
  
  [33X[0;0YConcatenating  [10XJ4[110X and [10XK5[110X, we get [10X[ A^-1, [ 3, F ], B^-1, B, A, [ -3, FG ] ][110X,
  with  length  [22X16[122X.  Cancelling  immediate  inverses  removes the [10X[ B^-1, B ][110X.
  Cancelling  inverses  gets  rid of the terms [10Xa^-1[110X and [10XA[110X, converting [10X[ 3, F ][110X
  into  [10X[  3,  fgFG ] = [ 3, FG ][110X. Conjugating with [10Xgf[110X produces the third root
  identity [10X[ [ 3, fg ], [ -3, id ] ][110X, which then cancels.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XJ4K5 := Concatenation( J4, K5 );; [127X[104X
    [4X[25Xgap>[125X [27XJ4K5 := CancelImmediateInversesLogSequence( J4K5 );; [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( J4K5, genfms3, s3labs ); [127X[104X
    [4X[28X[ [ -2, F*G ], [ 3, f ], [ -1, id ], [ 3, F ], [ 1, id ], [128X[104X
    [4X[28X[ -3, f ], [ 2, F*G ], [ -3, F*G ] ][128X[104X
    [4X[25Xgap>[125X [27XJ4K5 := CancelInversesLogSequence( ms3, J4K5 ); [127X[104X
    [4X[28X[ ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YReducing identities[133X[101X
  
  [33X[0;0YIn  this  section  we  list  some  further  operations  which may be used to
  simplify  the  list  of  identities returned by [10XIdentityRelatorSequences[110X. We
  will use our [22XQ_8[122X presentation in the examples.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xmq8 := MonoidPresentationFpGroup( q8 );;[127X[104X
    [4X[25Xgap>[125X [27Xfmq8 := FreeGroupOfPresentation( mq8 );;[127X[104X
    [4X[25Xgap>[125X [27Xgenfmq8 := GeneratorsOfGroup(fmq8 );;[127X[104X
    [4X[25Xgap>[125X [27Xq8labs := ["a","b","A","B"];; [127X[104X
    [4X[25Xgap>[125X [27XSetMonoidPresentationLabels( mq8, q8labs );; [127X[104X
    [4X[25Xgap>[125X [27Xidsq8 := IdentityRelatorSequences( q8 );;          [127X[104X
    [4X[25Xgap>[125X [27Xlenidsq8 := Length( idsq8 );                                   [127X[104X
    [4X[28X28[128X[104X
    [4X[25Xgap>[125X [27XList( idsq8, L -> Length(L) );[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 8, 8, [128X[104X
    [4X[28X  9, 10, 10 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.3-1 LogSequenceRewriteRules[101X
  
  [33X[1;0Y[29X[2XLogSequenceRewriteRules[102X( [3XmG[103X ) [32X attribute[133X
  
  [33X[0;0YThe root identity [22XR^-1R^w[122X may be converted into the rewrite rule [22XR^w -> R[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xrulesq8 := LogSequenceRewriteRules( mq8 );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..8] do [127X[104X
    [4X[25X>[125X [27X       PrintLnUsingLabels( rulesq8[i], genfmq8, q8labs );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28X[ [ 1, a ], [ 1, id ] ][128X[104X
    [4X[28X[ [ -1, a ], [ -1, id ] ][128X[104X
    [4X[28X[ [ 1, A ], [ 1, id ] ][128X[104X
    [4X[28X[ [ -1, A ], [ -1, id ] ][128X[104X
    [4X[28X[ [ 2, b ], [ 2, id ] ][128X[104X
    [4X[28X[ [ -2, b ], [ -2, id ] ][128X[104X
    [4X[28X[ [ 2, B ], [ 2, id ] ][128X[104X
    [4X[28X[ [ -2, B ], [ -2, id ] ][128X[104X
    [4X[28X[ [ 3, a*b*a*B ], [ 3, id ] ][128X[104X
    [4X[28X[ [ 3, b*A*B*A ], [ 3, id ] ][128X[104X
    [4X[28X[ [ -3, a*b*a*B ], [ -3, id ] ][128X[104X
    [4X[28X[ [ -3, b*A*B*A ], [ -3, id ] ][128X[104X
    [4X[28X[ [ 4, a^2*b^2 ], [ 4, id ] ][128X[104X
    [4X[28X[ [ 4, B^2*A^2 ], [ 4, id ] ][128X[104X
    [4X[28X[ [ -4, a^2*b^2 ], [ -4, id ] ][128X[104X
    [4X[28X[ [ -4, B^2*A^2 ], [ -4, id ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.3-2 OnePassReduceLogSequence[101X
  
  [33X[1;0Y[29X[2XOnePassReduceLogSequence[102X( [3XJ[103X, [3Xrules[103X ) [32X operation[133X
  
  [33X[0;0YThe  rewrite  rules  returned  by  [10XLogSequenceRewriteRules[110X  may  be  used to
  simplify  other  identity sequences. In the example the fourth rule [22X(q^-1)^A
  -> q^-1[122X, applied twice, reduces [22X(q^-1)^A^2[122X to [22Xq^-1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XJ7 := idsq8[7];[127X[104X
    [4X[28X[ [ 1, <identity ...> ], [ -1, q8_M3^2 ] ][128X[104X
    [4X[25Xgap>[125X [27XOnePassReduceLogSequence( J7, rulesq8 );[127X[104X
    [4X[28X[ [ 1, <identity ...> ], [ -1, <identity ...> ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  operation [10XReduceLogSequences[110X, described in subsection [14X6.2-1[114X, applied to
  the list [10Xidsq8[110X reduces the [22X28[122X identities to [22X15[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xridsq8 := ReduceLogSequences( q8, idsq8 );;[127X[104X
    [4X[25Xgap>[125X [27Xlenrids := Length( ridsq8 );                                   [127X[104X
    [4X[28X15[128X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..lenrids] do                                [127X[104X
    [4X[25X>[125X [27X       PrintLnUsingLabels( ridsq8[i], genfmq8, q8labs );   [127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28X[ [ -2, id ], [ 2, b ] ][128X[104X
    [4X[28X[ [ -1, id ], [ 1, a ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 2, A^2 ], [ 1, id ], [ -4, a^2 ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ 4, a ], [ -3, b ] ][128X[104X
    [4X[28X[ [ 1, id ], [ -4, id ], [ 2, A^2 ], [ -4, A^2 ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ 3, id ], [ 2, id ], [ -4, b ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A ], [ 1, id ], [ -3, a ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A^2 ], [ 1, id ], [ -3, B ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ -4, A ], [ 2, A^3 ], [ 1, id ], [128X[104X
    [4X[28X[ -3, b ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 4, B*A^2 ], [ -4, A^2 ], [ 1, id ], [ 2, id ], [128X[104X
    [4X[28X[ -4, b ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A ], [ 3, A^2 ], [ 4, id ], [128X[104X
    [4X[28X[ -4, B ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ 4, B*A ], [ -4, A ], [ 1, id ], [128X[104X
    [4X[28X[ -3, a ], [ 4, B ], [ -1, b ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A ], [ 3, A^2 ], [ 4, B*A^2 ], [128X[104X
    [4X[28X[ -4, A^2 ], [ 1, id ], [ -1, B ] ][128X[104X
    [4X[28X[ [ 4, id ], [ -4, b ], [ 1, b ], [ -3, a^2*b ], [ 4, B*a*b ], [128X[104X
    [4X[28X[ -4, a*b ], [ 3, b ], [ -1, id ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A ], [ 1, id ], [ -4, a ], [128X[104X
    [4X[28X[ 2, A ], [ 1, id ], [ -4, a^2 ], [ -3, B ] ] [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe now demonstrate that this list may be reduced further.[133X
  
  [1X6.3-3 MoveRightLogSequence[101X
  
  [33X[1;0Y[29X[2XMoveRightLogSequence[102X( [3XmG[103X, [3XJ[103X, [3XL[103X, [3Xq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XMoveLeftLogSequence[102X( [3XmG[103X, [3XJ[103X, [3XL[103X, [3Xq[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSwapLogSequence[102X( [3XmG[103X, [3XJ[103X, [3Xp[103X, [3Xq[103X ) [32X operation[133X
  
  [33X[0;0YThe terms in an identity sequence may be interchanged because[133X
  
  
  [24X[33X[0;6YR^w Q^v ~=~ Q^v R^{wQ^v} ~=~ Q^{v(R^w)^{-1}} R^w.[133X
  
  [124X
  
  [33X[0;0YIn  the first two of these three operations [10XL =[110X [22X[p..r][122X is a range specifying
  a  sublist  [10XK=J{[p..r]}[110X  of  [10XJ[110X,  and  [22Xl[122X  is  the  length of [10XJ[110X. The operation
  [10XMoveRightLogSequence(mG,J,L,q)[110X,  with [22X0[122X < [22Xp[122X < [22Xq[122X and [22Xq+r ≤ p+l[122X, moves sublist
  [10XK[110X  to  the  [22Xq[122X-th  position, conjugating entries in [22XJ{[p+1 ... q]}[122X and moving
  them all to the left.[133X
  
  [33X[0;0YSimilarly  [10XMoveLeftLogSequence(mG,J,L,q)[110X,  with  [22X0[122X  <  [22Xq[122X < [22Xp[122X nd [22Xr ≤ l[122X, moves
  sublist  [10XK[110X  to  the [22Xq[122X-th position, conjugating entries in [22XJ{[q ... p-1]}[122X and
  moving them all to the right.[133X
  
  [33X[0;0YThe  operation [10XSwapLogSequence(mG,J,p,q)[110X with [22Xp[122X < [22Xq[122X swaps a pair of terms in
  a sequence [10XJ[110X by calling the two previous commands.[133X
  
  [33X[0;0YIn   all   three  operations  the  procedure  is  completed  by  a  call  to
  [10XOnePassReduceLogSequence[110X.[133X
  
  [33X[0;0YIn  the example the third identity is converted into the fifth by moving the
  third  term  one place right and then changing the start position, so it may
  be omitted.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XJ3 := ShallowCopy( ridsq8[3] );;                            [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( J3, genfmq8, q8labs );   [127X[104X
    [4X[28X[ [ -4, id ], [ 2, A^2 ], [ 1, id ], [ -4, a^2 ] ][128X[104X
    [4X[25Xgap>[125X [27XK3 := MoveRightLogSequence( mq8, J3, [3], 4 );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( K3, genfmq8, q8labs );    [127X[104X
    [4X[28X[ [ -4, id ], [ 2, A^2 ], [ -4, A^2 ], [ 1, id ] ][128X[104X
    [4X[25Xgap>[125X [27XJ5 := ShallowCopy( ridsq8[5] );; [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( J5, genfmq8, q8labs );[127X[104X
    [4X[28X[ [ 1, id ], [ -4, id ], [ 2, A^2 ], [ -4, A^2 ] ][128X[104X
    [4X[25Xgap>[125X [27XJ5 = ChangeStartLogSequence( mq8, K3, 4 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.3-4 SubstituteLogSubsequence[101X
  
  [33X[1;0Y[29X[2XSubstituteLogSubsequence[102X( [3XmG[103X, [3XK[103X, [3XJ1[103X, [3XJ2[103X ) [32X operation[133X
  
  [33X[0;0YIf  we  move  the  second  term in [10XJ5[110X to the right, we find that sublist [10XU =
  [[1,id],[2,id]][110X  is  equal  to  [10XV = [[4,A^2],[4,id]][110X, with both expanding to
  [22Xa^4b^4[122X.[133X
  
  [33X[0;0YNow  [10XU[110X  appears  in the tenth identity, and if we replace it with [10XV[110X and then
  cancel, we obtain the empty list. So the tenth identity may be omitted.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XK5 := MoveRightLogSequence( mq8, J5, [2], 3 );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( K5, genfmq8, q8labs );       [127X[104X
    [4X[28X[ [ 1, id ], [ 2, id ], [ -4, id ], [ -4, A^2 ] ][128X[104X
    [4X[25Xgap>[125X [27XK5a := K5{[1..2]};; [127X[104X
    [4X[25Xgap>[125X [27XK5b := InverseLogSequence( K5{[3..4]} );;[127X[104X
    [4X[25Xgap>[125X [27XK5a;K5b;[127X[104X
    [4X[28X[ [ 1, <identity ...> ], [ 2, <identity ...> ] ][128X[104X
    [4X[28X[ [ 4, q8_M3^2 ], [ 4, <identity ...> ] ][128X[104X
    [4X[25Xgap>[125X [27XJ10 := ShallowCopy( ridsq8[10] );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( J10, genfmq8, q8labs );[127X[104X
    [4X[28X[ [ -4, id ], [ 4, B*A^2 ], [ -4, A^2 ], [ 1, id ], [ 2, id ], [128X[104X
    [4X[28X[ -4, b ] ][128X[104X
    [4X[25Xgap>[125X [27XK10 := SubstituteLogSubsequence( mq8, J10, K5a, K5b );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( K10, genfmq8, q8labs );            [127X[104X
    [4X[28X[ [ -4, id ], [ 4, B*A^2 ], [ -4, A^2 ], [ 4, A^2 ], [ 4, id ], [128X[104X
    [4X[28X[ -4, b ] ][128X[104X
    [4X[25Xgap>[125X [27XCancelInversesLogSequence( mq8, K10 );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSimilarly, we may reduce the ninth identity. Initially, [10XU[110X does not appear as
  a  sublist  of  [10XJ9[110X. Swapping the fourth and fifth terms and conjugating by [22XA[122X
  produces  [10XU[110X,  which is then replaced by [10XV[110X. After a cancellation, we obtain a
  conjugate of the fourth identity.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XJ9 := ShallowCopy( ridsq8[9] );; [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( J9, genfmq8, q8labs );            [127X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ -4, A ], [ 2, A^3 ], [ 1, id ], [128X[104X
    [4X[28X[ -3, b ] ][128X[104X
    [4X[25Xgap>[125X [27XK9 := MoveLeftLogSequence( mq8, J9, [5], 4 );; [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( K9, genfmq8, q8labs );            [127X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ -4, A ], [ 1, id ], [ 2, a ], [ -3, b ] ][128X[104X
    [4X[25Xgap>[125X [27XL9 := ConjugateByWordLogSequence( mq8, K9, genfmq8[3] );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( L9, genfmq8, q8labs );            [127X[104X
    [4X[28X[ [ -4, A ], [ 3, A^2 ], [ -4, A^2 ], [ 1, id ], [ 2, id ], [128X[104X
    [4X[28X[ -3, b*A ] ][128X[104X
    [4X[25Xgap>[125X [27XM9 := SubstituteLogSubsequence( mq8, L9, K5a, K5b);;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( M9, genfmq8, q8labs );            [127X[104X
    [4X[28X[ [ -4, A ], [ 3, A^2 ], [ -4, A^2 ], [ 4, A^2 ], [ 4, id ], [128X[104X
    [4X[28X[ -3, b*A ] ][128X[104X
    [4X[25Xgap>[125X [27XN9 := CancelInversesLogSequence( mq8, M9 );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( N9, genfmq8, q8labs );            [127X[104X
    [4X[28X[ [ -4, A ], [ 3, A^2 ], [ 4, id ], [ -3, b*A ] ][128X[104X
    [4X[25Xgap>[125X [27XP9 := ConjugateByWordLogSequence( mq8, N9, genfmq8[1] );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( P9, genfmq8, q8labs );            [127X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ 4, a ], [ -3, b ] ][128X[104X
    [4X[25Xgap>[125X [27XP9 = ridsq8[4];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe will not, for now, attempt to reduce the list of identities further.[133X
  
  
  [1X6.4 [33X[0;0YThe original approach[133X[101X
  
  [33X[0;0YThis section describes the approach used from the earliest versions of [5XIdRel[105X
  up  to version 2.38 in 2017. For version 2.39 the methods were revised so as
  to  produce  some  data  for  infinite  groups.  This  experimental  work is
  described in later sections.[133X
  
  [1X6.4-1 IdentitiesAmongRelators[101X
  
  [33X[1;0Y[29X[2XIdentitiesAmongRelators[102X( [3Xgrp[103X ) [32X attribute[133X
  
  [33X[0;0YIt  is [13Xnot[113X guaranteed that a minimal set of identities is obtained. For [10Xq8[110X a
  set  of  seven  identities  is returned, whereas a minimal set contains only
  six. See Example 5.1 of [HW03] for further details.[133X
  
  [33X[0;0YWhy [10Xidrelq8[110X in the following example is shorter than [10Xridsq8[110X above remains to
  be investigated![133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xidrelq8 := IdentitiesAmongRelators( q8 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( idrelq8 );[127X[104X
    [4X[28X14[128X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..14] do [127X[104X
    [4X[25X>[125X [27X      PrintLnUsingLabels( idrelq8[i], genfmq8, q8labs ); [127X[104X
    [4X[25X>[125X [27X   od; [127X[104X
    [4X[28X[ [ -1, id ], [ 1, a ] ][128X[104X
    [4X[28X[ [ -2, id ], [ 2, b ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ 3, id ], [ 2, id ], [ -4, b ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 2, A^2 ], [ 1, id ], [ -4, a^2 ] ][128X[104X
    [4X[28X[ [ 1, id ], [ -4, id ], [ 2, A^2 ], [ -4, A^2 ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A ], [ 1, id ], [ -3, a ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ 4, a ], [ -3, b ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A^2 ], [ 1, id ], [ -3, B ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 4, B*A^2 ], [ -4, A^2 ], [ 1, id ], [ 2, id ], [128X[104X
    [4X[28X[ -4, b ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A ], [ 3, A^2 ], [ 4, id ], [128X[104X
    [4X[28X[ -4, B ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ -4, A ], [ 2, A^3 ], [ 1, id ], [128X[104X
    [4X[28X[ -3, b ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A ], [ 1, id ], [ -4, a ], [128X[104X
    [4X[28X[ 2, A ], [ 1, id ], [ -4, a^2 ], [ -3, B ] ][128X[104X
    [4X[28X[ [ -4, id ], [ 3, A ], [ 4, B*A ], [ -4, A ], [ 1, id ], [128X[104X
    [4X[28X[ -3, a ], [ 4, B ], [ -1, b ] ][128X[104X
    [4X[28X[ [ -3, id ], [ 4, B*A ], [ -4, A ], [ 3, A^2 ], [ 4, B*A^2 ], [128X[104X
    [4X[28X[ -4, A^2 ], [ 1, id ], [ -1, B ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.4-2 IdentityYSequences[101X
  
  [33X[1;0Y[29X[2XIdentityYSequences[102X( [3Xgrp[103X ) [32X attribute[133X
  
  [33X[0;0YThese identities are then transformed into module polynomials[133X
  
  
  [24X[33X[0;6Y\rho(a + ba) + \sigma({\rm id} + ab + ba) - \tau({\rm id} + a + A)\ ,[133X
  
  [124X
  
  [33X[0;0Ywhere the monoid elements are transformed into their normal forms.[133X
  
  [33X[0;0YThe collection of saturated sets of these module polynomials is then reduced
  as   far  as  possible,  and  the  minimal  set  obtained  returned  as  the
  [10XIdentityYSequences[110X  of  the group. The group relator sequences corresponding
  to these module polynomials form the [10XIdentitiesAmongRelators[110X for the group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xidyseq8 := IdentityYSequences( q8 );; [127X[104X
    [4X[25Xgap>[125X [27Xfor y in idyseq8 do [127X[104X
    [4X[25X>[125X [27X       PrintLnYSequence( y, genfmq8, q8labs, genq8R, q8Rlabs ); [127X[104X
    [4X[25X>[125X [27X   od; [127X[104X
    [4X[28Xq8_Y2*(1*A), q^-1*(-1*A) + q*(1*id)) [128X[104X
    [4X[28Xq8_Y1*(1*B), r^-1*(-1*B) + r*(1*id)) [128X[104X
    [4X[28Xq8_Y6*(-1*id), r*(-1*id) + s*(-1*A + -1*id) + t^-1*(1*b + 1*id)) [128X[104X
    [4X[28Xq8_Y3*(-1*a), q*(-1*a) + r*(-1*A) + t^-1*(1*A + 1*a)) [128X[104X
    [4X[28Xq8_Y5*(-1*a), q*(-1*a) + r*(-1*A) + t^-1*(1*A + 1*a)) [128X[104X
    [4X[28Xq8_Y7*(1*a*b), q*(1*a*b) + s^-1*(-1*a*b + -1*B) + t^-1*(-1*b) + t*(1*id)) [128X[104X
    [4X[28Xq8_Y4*(1*A), s^-1*(-1*a*b) + s*(1*a^2) + t^-1*(-1*A) + t*(1*id)) [128X[104X
    [4X[28Xq8_Y8*(1*a*b), q*(1*a*b) + s^-1*(-1*a*b + -1*A) + t^-1*(-1*a*B) + t*(1*id)) [128X[104X
    [4X[28Xq8_Y10*(1*B), q*(1*B) + r*(1*B) + t^-1*(-1*B + -1*b + -1*id) + t*(1*id)) [128X[104X
    [4X[28Xq8_Y11*(1*b), s^-1*(-1*b) + s*(1*B) + t^-1*(-1*a*B + -1*id) + t*(1*b + 1*a)) [128X[104X
    [4X[28Xq8_Y9*(-1*a), q*(-1*a) + r*(-1*a^2) + s^-1*(1*a*B) + s*(-1*id) + t^-1*(1*a + [128X[104X
    [4X[28X1*id)) [128X[104X
    [4X[28Xq8_Y15*(1*a*b), q*(2*a*b) + r*(1*b) + s^-1*(-1*a*b + -1*A) + t^-1*(-1*a*B + [128X[104X
    [4X[28X-1*B + -1*b) + t*(1*id)) [128X[104X
    [4X[28Xq8_Y12*(1*b), q^-1*(-1*a^2) + q*(1*b) + s^-1*(-1*a*b) + s*(1*a*B) + t^-1*([128X[104X
    [4X[28X-1*a*B + -1*b) + t*(1*a + 1*id)) [128X[104X
    [4X[28Xq8_Y13*(1*a*b), q^-1*(-1*A) + q*(1*a*b) + s^-1*(-1*a*b) + s*(1*a*B) + t^-1*([128X[104X
    [4X[28X-1*a*B + -1*b) + t*(1*a + 1*id)) [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.5 [33X[0;0YPartial lists of elements[133X[101X
  
  [33X[0;0YAs  we  have  seen, the procedure for obtaining identities involves applying
  each  relator  at  each  element of the group. Since this will not terminate
  when the group is infinite, we include an operation to construct words up to
  a given length in the monoid representation of the group.[133X
  
  [1X6.5-1 PartialElementsOfMonoidRepresentation[101X
  
  [33X[1;0Y[29X[2XPartialElementsOfMonoidRepresentation[102X( [3XG[103X, [3Xlen[103X ) [32X operation[133X
  
  [33X[0;0YAs an example we take the group [22X⟨ u,v,w ~|~ u^3, v^2, w^2, (uv)^2, (vw)^2⟩[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup(3);;[127X[104X
    [4X[25Xgap>[125X [27Xu := F.1;;  v := F.2;;  w := F.3;;[127X[104X
    [4X[25Xgap>[125X [27Xrels := [ u^3, v^2, w^2, (u*v)^2, (v*w)^2 ];;[127X[104X
    [4X[25Xgap>[125X [27Xq0 := F/rels;;[127X[104X
    [4X[25Xgap>[125X [27XSetArrangementOfMonoidGenerators( q0, [1,-1,2,-2,3,-3] );[127X[104X
    [4X[25Xgap>[125X [27XSetName( q0, "q0" );[127X[104X
    [4X[25Xgap>[125X [27Xmq0 := MonoidPresentationFpGroup( q0 );;[127X[104X
    [4X[25Xgap>[125X [27Xfmq0 := FreeGroupOfPresentation( mq0 );;[127X[104X
    [4X[25Xgap>[125X [27Xgenfmq0 := GeneratorsOfGroup( fmq0 );;[127X[104X
    [4X[25Xgap>[125X [27Xq0labs := ["u","U","v","V","w","W"];; [127X[104X
    [4X[25Xgap>[125X [27XSetMonoidPresentationLabels( mq0, q0labs );; [127X[104X
    [4X[25Xgap>[125X [27Xlrws := LoggedRewritingSystemFpGroup( q0 );;[127X[104X
    [4X[25Xgap>[125X [27Xpe1 := PartialElementsOfMonoidPresentation( q0, 1 );; [127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( pe1, genfmq0, q0labs ); [127X[104X
    [4X[28X[ id, u, U, v, w ][128X[104X
    [4X[25Xgap>[125X [27Xpe2 := PartialElementsOfMonoidPresentation( q0, 2 );;[127X[104X
    [4X[25Xgap>[125X [27XPrintLnUsingLabels( pe2, genfmq0, q0labs ); [127X[104X
    [4X[28X[ id, u, U, v, w, u*v, u*w, U*v, U*w, v*w, w*u, w*U ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
