  
  [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
  
  [33X[0;0YHere  are  some  of  the details when working with the group [22XS_3 = ⟨ a,b ~|~
  ρ=a^3,   σ=b^2,   τ=(ab)^2   ⟩[122X.   The  monoid  presentation  has  generators
  [22X{a^+,b^+,a^-,b^-}[122X and relators[133X
  
  
  [24X[33X[0;6Y[~  1  =  a^+a^-,~  2  = b^+b^-,~ 3 = a^-a^+,~ 4 = b^-b^+,~ 5 = a^{+3},~ 6 =
  b^{+2},~ 7 = (a^+b^+)^2 ~]\ ,[133X
  
  [124X
  
  [33X[0;0Yand  the elements are [22X{ id, a^+,b^+,a^-,a^+b^-,b^+a^+}[122X. The logged rewriting
  system has relations[133X
  
      ┌──────────────────────────────────────────────────────────────────────────────────────────┐
      │ [22Xa^+a^- = id,quad b^+b^- = id,quad a^-a^+ = id,quad b^-b^+ = id,[122X                          │ 
      │ [22Xa^+2 = [5, id]a^-,quad a^-2 = [-5, id]a^+,quad b^+2 = [6, id] id,quad b^- = [-6, id]b^+,[122X │ 
      │ [22Xb^+a^- = [-7,a^+b^-][6, id]a^+b^+,quad a^-b^+ = [-7,a^+][6,a^-b^-]b^+a^+,[122X                │ 
      │ [22Xa^+b^+a^+ = [7, id][-6, id]b^+,quad b^+a^+b^+ = [7,a^+]a^-.[122X                              │ 
      └──────────────────────────────────────────────────────────────────────────────────────────┘
  
  [33X[0;0YTo  construct  the  [13Xidentity  monoid relator sequences[113X we follow in turn the
  relators [22X5=ρ,6=σ,7=τ[122X at each of the elements of [22XS_3[122X. For example, applying [22Xτ[122X
  at [22Xa^+[122X gives the cycle:[133X
  
  
  [24X[33X[0;6Ya^+   \stackrel{a^+}{\longrightarrow}   a^-  \stackrel{b^+}{\longrightarrow}
  b^+a^+                 \stackrel{a^+}{\longrightarrow}                a^+b^+
  \stackrel{b^+}{\longrightarrow} a^+\ .[133X
  
  [124X
  
  [33X[0;0YEach of these edges has a non-trivial logged rewrite, particularly the third
  edge  where [22Xb^+a^+a^+ -> b^+a^- -> a^+b^+[122X. Combining this log information we
  obtain the sequence:[133X
  
  
  [24X[33X[0;6Y[5,{\rm    id}].[-7,a^+][6,a^-b^-].   [5,{\rm   id}]^{b^-}[-7,a^+b^-][6,{\rm
  id}].[6,{\rm id}]^{a^-}\ .[133X
  
  [124X
  
  [33X[0;0YExpanding this gives:[133X
  
  
  [24X[33X[0;6Ya^+a^+a^+.a^-b^-a^-b^-a^-a^+.b^+a^+b^+b^+a^-b^-.b^+a^+a^+a^+b^-.
  b^+a^-b^-a^-b^-a^-a^+b^-.b^+b^+.a^+b^+b^+a^-[133X
  
  [124X
  
  [33X[0;0Ywhich  cancels to leave, as expected, [22Xa^+(a^+b^+a^+b^+)a^- = [7,a^-][122X. Adding
  the inverse [22X[-7,a^-][122X to the front of the log expression gives the identity[133X
  
  
  [24X[33X[0;6Y[-7,a^-].[5,{\rm      id}].[-7,a^+][6,a^-b^-].     [5,b^-][-7,a^+b^-][6,{\rm
  id}].[6,a^-]\ .[133X
  
  [124X
  
  [33X[0;0YConverting  this  back  to  the group presentation, and conjugating by [22Xa[122X, we
  obtain the [13Xidentity group relator sequence[113X given in the introduction ([14X1.1[114X):[133X
  
  
  [24X[33X[0;6Y\tau^{-1}\; \rho^a\; \left(\tau^{-1}\right)^{a^2}\; \sigma^{a^{-1}b^{-1}a}\;
  \rho^{b^{-1}a}\; \left(\tau^{-1}\right)^{ab^{-1}a}\; \sigma^a\; \sigma\ .[133X
  
  [124X
  
  [33X[0;0YThis is then transformed into the module polynomial[133X
  
  
  [24X[33X[0;6Y\rho(a^+ + b^+a^+) + \sigma({\rm id} + a^+ + a^+b^+) - \tau({\rm id} + a^- +
  b^+)\ ,[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
  
  
  [1X6.1 [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
  
  [1X6.1-1 IdentitiesAmongRelators[101X
  
  [33X[1;0Y[29X[2XIdentitiesAmongRelators[102X( [3Xgrp[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIdentityYSequences[102X( [3Xgrp[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIdentityRelatorSequences[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
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xgseq8 := IdentityRelatorSequences( q8 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( gseq8 );[127X[104X
    [4X[28X19[128X[104X
    [4X[25Xgap>[125X [27Xgseq8[1];[127X[104X
    [4X[28X[ 1, 9, [ [ q8_R1^-1, <identity ...> ], [ q8_R1, f1^-1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xidsq8 := IdentitiesAmongRelators( q8 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( idsq8 );[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27XDisplay( idsq8 );[127X[104X
    [4X[28X[ [ [ q8_R1^-1, <identity ...> ], [ q8_R1, f1^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R2^-1, <identity ...> ], [ q8_R4^-1, f2^-1 ], [ q8_R2, f1^-2*f2^-1 ],[128X[104X
    [4X[28X      [ q8_R4, f2^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R1^-1, <identity ...> ], [ q8_R4^-1, f2^-1 ], [ q8_R3, f1^-1*f2^-1 ],[128X[104X
    [4X[28X      [ q8_R3, f2^-1 ], [ q8_R3, f1*f2^-1 ], [ q8_R1^-1, f2^-1 ], [128X[104X
    [4X[28X      [ q8_R3, f1^-2*f2^-1 ], [ q8_R4, f2^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R3^-1, <identity ...> ], [ q8_R4^-1, f2^-1 ], [ q8_R3, f1^-1*f2^-1 ],[128X[104X
    [4X[28X      [ q8_R4, f1*f2^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R4^-1, <identity ...> ], [ q8_R4^-1, f2^-1 ], [ q8_R3, f1^-1*f2^-1 ],[128X[104X
    [4X[28X      [ q8_R3, f2^-1 ], [ q8_R4^-1, f2^-1 ], [ q8_R2, f1^-2*f2^-1 ], [128X[104X
    [4X[28X      [ q8_R4, f2^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R4^-1, <identity ...> ], [ q8_R3, f1*f2 ], [ q8_R1^-1, f2 ], [128X[104X
    [4X[28X      [ q8_R3, f1^-2*f2 ], [ q8_R4, f2 ] ], [128X[104X
    [4X[28X  [ [ q8_R4^-1, <identity ...> ], [ q8_R4^-1, f1^-2 ], [ q8_R2, f1^-4 ], [128X[104X
    [4X[28X      [ q8_R1, f1^-1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xidyseq8 := IdentityYSequences( q8 );[127X[104X
    [4X[28X[ ( q8_Y1*( -q8_M1), q8_R1*( q8_M1 - <identity ...>) ), [128X[104X
    [4X[28X  ( q8_Y5*( <identity ...>), q8_R2*( q8_M2 - <identity ...>) ), [128X[104X
    [4X[28X  ( q8_Y18*( q8_M2), q8_R1*( -q8_M2 - <identity ...>) + q8_R3*( q8_M1^2 + q8_M\[128X[104X
    [4X[28X3 + q8_M1 + <identity ...>) ), [128X[104X
    [4X[28X  ( q8_Y8*( q8_M2), q8_R3*( q8_M3 - q8_M2) + q8_R4*( q8_M1 - <identity ...>) )[128X[104X
    [4X[28X    , [128X[104X
    [4X[28X  ( q8_Y17*( -q8_M2), q8_R2*( -q8_M1^2) + q8_R3*( -q8_M3 - <identity ...>) + q\[128X[104X
    [4X[28X8_R4*( q8_M2 + <identity ...>) ), [128X[104X
    [4X[28X  ( q8_Y11*( <identity ...>), q8_R1*( -q8_M2) + q8_R3*( q8_M1*q8_M2 + q8_M4) +\[128X[104X
    [4X[28X q8_R4*( q8_M2 - <identity ...>) ), [128X[104X
    [4X[28X  ( q8_Y10*( -q8_M1), q8_R1*( -<identity ...>) + q8_R2*( -q8_M1) + q8_R4*( q8_\[128X[104X
    [4X[28XM3 + q8_M1) ) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X6.1-2 RootIdentities[101X
  
  [33X[1;0Y[29X[2XRootIdentities[102X( [3Xgrp[103X ) [32X attribute[133X
  
  [33X[0;0YThe  [13Xroot  identities[113X  are identities of the form [22Xr^wr^-1[122X where [22Xr = w^n[122X is a
  relator  and  [22Xn>1[122X.  (For  equivalent  forms  invert,  or permute the factors
  cyclically, or act with [22Xw^-1[122X.)[133X
  
  [33X[0;0YFor  [10Xq8[110X only two of the four relators are proper powers, [22Xπ=a^4[122X and [22Xρ=b^4[122X, so
  the root identities are [22Xπ^aπ^-1[122X and [22Xρ^bρ^-1[122X. In the listing below the second
  of  these  is  displayed  as  [22Xρ^-1(ξ^-1)^b^-1}ρ^a^-2b^-1}ξ^b^-1}[122X,  where [22Xξ =
  a^2b^2[122X. Because the second term is the inverse of the fourth, we may convert
  the final three terms into a conjugate of the third, so that the identity is
  equivalent to [22Xρ^-1ρ^b[122X:[133X
  
  
  [24X[33X[0;6Y\left(\rho^{a^{-2}b^{-1}}\right)^{\xi^{b^{-1}}}                          ~=~
  \left(\rho^{a^{-2}b^{-1}}\right)^{b(a^2b^2)b^{-1}} ~=~ \rho^b~.[133X
  
  [124X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XRootIdentities( q8 );[127X[104X
    [4X[28X[ [ [ q8_R1^-1, <identity ...> ], [ q8_R1, f1^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R2^-1, <identity ...> ], [ q8_R4^-1, f2^-1 ], [ q8_R2, f1^-2*f2^-1 ],[128X[104X
    [4X[28X      [ q8_R4, f2^-1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XRootIdentities(s3);[127X[104X
    [4X[28X[ [ [ s3_R1^-1, <identity ...> ], [ s3_R1, f1^-1 ] ], [128X[104X
    [4X[28X  [ [ s3_R2^-1, <identity ...> ], [ s3_R2, f2 ] ], [128X[104X
    [4X[28X  [ [ s3_R3^-1, <identity ...> ], [ s3_R3, f1*f2 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YKnuth-Bendix identities[133X[101X
  
  [33X[0;0YGiven  an  initial set of rules, the logged Knuth-Bendix procedure considers
  overlaps  between  pairs  of  logged rules [22Xl_1=L_1r_1[122X and [22Xl_2=L_2r_2[122X. In the
  case  [22Xu_1l_1=l_2v_2[122X for some words [22Xu_1,v_2[122X, the critical pair resulting from
  the  overlap  is  [22X(u_1r_1,r_2v_2)[122X. Logged reduction is then applied to these
  two words giving [22Xu_1r_1=M_1z_1[122X and [22Xr_2v_2=M_2z_2[122X, say. Then, if [22Xz_1>z_2[122X, the
  additional  rule  [22Xz_1  =  (M_1^-1L_1^-u_1^-1}L_2M_2)z_2[122X is added. There is a
  similar formula when [22Xz_2>z_1[122X.[133X
  
  [1X6.2-1 IdentitiesAmongRelatorsKB[101X
  
  [33X[1;0Y[29X[2XIdentitiesAmongRelatorsKB[102X( [3Xgrp[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIdentityYSequencesKB[102X( [3Xgrp[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIdentityRelatorSequencesKB[102X( [3Xgrp[103X ) [32X attribute[133X
  
  [33X[0;0YThe  third  possibility  is  that  [22Xz_1=z_2[122X,  and  in  this  case there is no
  reduction    rule    to    be    added.    However    the   log   expression
  [22XM_1^-1L_1^-u_1^-1}L_2M_2[122X  must reduce to the identity, and so is an identity
  relator   sequence.  Since  version  2.41  of  this  package,  the  function
  [10XLoggedRewritingSystemFpGroup[110X  returns two lists: a complete set of reduction
  rules,  and a set of identity relator sequences produced in this way. In the
  case  of  the  quaternion  group  [10Xq8[110X a total of [22X42[122X sequences are produced by
  logged Knuth-Bendix, and these are then reduced to [22X8[122X identities.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XgseqKB8 := IdentityRelatorSequencesKB( q8 );;[127X[104X
    [4X[25Xgap>[125X [27XLength(last);[127X[104X
    [4X[28X42[128X[104X
    [4X[25Xgap>[125X [27XidrelsKB8 := IdentitiesAmongRelatorsKB( q8 );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( idrelsKB8 );[127X[104X
    [4X[28X[ [ [ q8_R1^-1, <identity ...> ], [ q8_R1, f1^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R3^-1, f1^-1 ], [ q8_R3^-1, f1^-2 ], [ q8_R1, <identity ...> ], [128X[104X
    [4X[28X      [ q8_R3^-1, f1 ], [ q8_R1, f2^-1 ], [ q8_R3^-1, <identity ...> ] ], [128X[104X
    [4X[28X  [ [ q8_R3, f1*f2 ], [ q8_R4^-1, <identity ...> ], [ q8_R3^-1, f1^-2 ], [128X[104X
    [4X[28X      [ q8_R4, f1^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R2^-1, f1^-3 ], [ q8_R4, f1^-1 ], [ q8_R3, <identity ...> ], [128X[104X
    [4X[28X      [ q8_R3^-1, f1*f2^-1 ], [ q8_R4^-1, <identity ...> ], [ q8_R2, f1^-2 ] ][128X[104X
    [4X[28X    , [ [ q8_R4^-1, f2 ], [ q8_R2, f1^-2*f2 ], [ q8_R4^-1, <identity ...> ], [128X[104X
    [4X[28X      [ q8_R3, f1^-1 ], [ q8_R3, <identity ...> ] ], [128X[104X
    [4X[28X  [ [ q8_R4^-1, <identity ...> ], [ q8_R3^-1, f1^-2 ], [128X[104X
    [4X[28X      [ q8_R1, <identity ...> ], [ q8_R3^-1, f1 ], [ q8_R4, f2^-1 ] ], [128X[104X
    [4X[28X  [ [ q8_R4^-1, <identity ...> ], [ q8_R4^-1, f1^-2 ], [ q8_R2, f1^-4 ], [128X[104X
    [4X[28X      [ q8_R1, <identity ...> ] ], [128X[104X
    [4X[28X  [ [ q8_R4, f1^-1 ], [ q8_R3, f1*f2 ], [ q8_R1^-1, f2 ], [128X[104X
    [4X[28X      [ q8_R2^-1, f1^-2*f2 ], [ q8_R4, f2 ], [ q8_R3, f1 ], [ q8_R3^-1, f2 ], [128X[104X
    [4X[28X      [ q8_R4^-1, <identity ...> ], [ q8_R3, f1^-1 ], [ q8_R4^-1, f1^-1 ], [128X[104X
    [4X[28X      [ q8_R2, f1^-3 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [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
  
  [1X6.3-1 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 [27Xmonq0 := MonoidPresentationFpGroup( q0 );[127X[104X
    [4X[28Xmonoid presentation with group relators [128X[104X
    [4X[28X[ q0_M1^3, q0_M3^2, q0_M5^2, (q0_M1*q0_M3)^2, (q0_M3*q0_M5)^2 ][128X[104X
    [4X[25Xgap>[125X [27Xlrws := LoggedRewritingSystemFpGroup( q0 );; [127X[104X
    [4X[25Xgap>[125X [27XPartialElementsOfMonoidPresentation( q0, 1 ); [127X[104X
    [4X[28X[ <identity ...>, q0_M1, q0_M2, q0_M3, q0_M5 ][128X[104X
    [4X[25Xgap>[125X [27XPartialElementsOfMonoidPresentation( q0, 2 ); [127X[104X
    [4X[28X[ <identity ...>, q0_M1, q0_M2, q0_M3, q0_M5, q0_M1*q0_M3, q0_M1*q0_M5, [128X[104X
    [4X[28X  q0_M2*q0_M3, q0_M2*q0_M5, q0_M3*q0_M5, q0_M5*q0_M1, q0_M5*q0_M2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X6.4 [33X[0;0YIdentities for infinite groups[133X[101X
  
  [33X[0;0YBecause  the  list  of elements is incomplete, it is not guaranteed that the
  list of identities is complete.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xidsq0 := IdentitiesAmongRelators( q0 );[127X[104X
    [4X[28X[ [ [ q0_R1^-1, <identity ...> ], [ q0_R1, f1^-1 ] ], [128X[104X
    [4X[28X  [ [ q0_R2^-1, <identity ...> ], [ q0_R2, f2 ] ], [128X[104X
    [4X[28X  [ [ q0_R3^-1, <identity ...> ], [ q0_R3, f3 ] ], [128X[104X
    [4X[28X  [ [ q0_R4^-1, <identity ...> ], [ q0_R2^-1, f1^-1 ], [ q0_R4, f1*f2 ], [128X[104X
    [4X[28X      [ q0_R2, f1*f2 ] ], [128X[104X
    [4X[28X  [ [ q0_R5^-1, <identity ...> ], [ q0_R3^-1, f2^-1 ], [ q0_R5, f2*f3 ], [128X[104X
    [4X[28X      [ q0_R3, f2*f3 ] ], [128X[104X
    [4X[28X  [ [ q0_R1^-1, <identity ...> ], [ q0_R2^-1, f1^-1 ], [ q0_R4, f1*f2 ], [128X[104X
    [4X[28X      [ q0_R2^-1, f1^-1*f2^-1*f1*f2 ], [ q0_R4, f1^2*f2 ], [ q0_R1^-1, f2 ], [128X[104X
    [4X[28X      [ q0_R2^-1, f1^-1*f2^-1*f1^-1*f2 ], [ q0_R4, f2 ] ], [128X[104X
    [4X[28X  [ [ q0_R2^-1, <identity ...> ], [ q0_R3^-1, f2^-1 ], [ q0_R5, f2*f3 ], [128X[104X
    [4X[28X      [ q0_R2^-1, f3 ], [ q0_R3^-1, f2^-1*f3^-1*f2^-1*f3 ], [ q0_R5, f3 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y [133X
  
