ApCoCoA-1:Symbolic data Computations

Computation Examples for Non-abelian Groups

Computations of Baumslag groups

Recall that the Baumslag-Solitar groups have the following presentation

BS(m,n)<a, b | b*a^m = a^n*b> where m, n are natural numbers

We enumerate partial Groebner bases for the Baumslag-Solitar groups as follows.

/*Use the ApCoCoA package ncpoly.*/
Use ZZ/(2)[a[1..2],b[1..2]];
NC.SetOrdering("LLEX");
A1:=[[a[1],a[2]],[1]];
A2:=[[a[2],a[1]],[1]];
B1:=[[b[1],b[2]],[1]];
B2:=[[b[2],b[1]],[1]];
-- Relation ba^2=a^3b. Change 2 and 3 in "()" to make another relation
R:=[[b[1],a[1]^(2)],[a[1]^(3),b[1]]];
G:=[A1,A2,B1,B2,R];
-- Enumerate a partial Groebner basis (see NC.GB for more details)
NC.GB(G,31,1,100,1000);

/*Use the ApCoCoA package gbmr.*/
Define Baumslag1(...)
  If Not Len(ARGV) = 2 And Not Len(ARGV) = 4 Then
    Error("Error in Baumslag1(...). There have to be two argument (n and m for exponents)
      or four arguments (n, m, degree, loops)");
  EndIf;
  For I:= 1 To Len(ARGV) Do
    If Not Type(ARGV[I]) = INT Then
      Error("Error in Baumslag1(...). The Type of the Arguments must be INT");
    ElIf ARGV[I] < 1 Then
      Error("Error in Baumslag1(...). The integer arguments must be positive");
    EndIf;
  EndFor;
  X:= "ab";
  Ordering:= "LLEX";
  R:= [];
  AM:= "";
  BN:= "";
  For I:= 1 To ARGV[1] Do
    AM:= AM + "a"; 
  EndFor;
  For I:= 1 To ARGV[2] Do
    BN:= BN + "b"; 
  EndFor;
  F:= [[[1, AM], [-1, ""]], [[1, BN], [-1, ""]]];
  If Len(ARGV) = 1 Then
    S:= $apcocoa/gbmr.MRBP(X, Ordering, R, F);
  Else
    S:= $apcocoa/gbmr.MRBP(X, Ordering, R, F, ARGV[2], ARGV[3], 1);
  EndIf;
Return S;
EndDefine;

Baumslag2(m, n, [DegreeBound, LoopBound]) (optional parameters in "[ ]")

Baumslag-Solitar group with the following presentation < a, b | b * a^m = a^n * b >

Define Baumslag2(...)
  If Not Len(ARGV) = 2 And Not Len(ARGV) = 4 Then
    Error("Error in Baumslag1(...). There have to be two argument (n and m for exponents) or four arguments (n, m, degree, loops)");
  EndIf;
  For I:= 1 To Len(ARGV) Do
    If Not Type(ARGV[I]) = INT Then
      Error("Error in Baumslag1(...). The Type of the Arguments must be INT");
    ElIf ARGV[I] < 1 Then
      Error("Error in Baumslag1(...). The integer arguments must be positive");
    EndIf;
  EndFor;
  X:= "abcdABCD";
  Ordering:= "LLEX";
  R:= [];
  AM:= "";
  AN:= "";
  For I:= 1 To ARGV[1] Do
    AM:= AM + "a"; 
  EndFor;
  For I:= 1 To ARGV[2] Do
    AN:= AN + "a"; 
  EndFor;
  F1 := [[1, "aA"], [-1, ""]];
  F2 := [[1, "bB"], [-1, ""]];
  F3 := [[1, "cC"], [-1, ""]];
  F4 := [[1, "dD"], [-1, ""]];
  F5 := [[1, "Aa"], [-1, ""]];
  F6 := [[1, "Bb"], [-1, ""]];
  F7 := [[1, "Cc"], [-1, ""]];
  F8 := [[1, "Dd"], [-1, ""]];
  F:= [F1, F2, F3, F4, F5, F6, F7, F8, [[1, "a"], [-1, "c"]], [[1, "b"], [-1, "d"]], [[1, "b" + AN], [-1, AM + "b"]]];
  If Len(ARGV) = 1 Then
    S:= $apcocoa/gbmr.MRBP(X, Ordering, R, F);
  Else
    S:= $apcocoa/gbmr.MRBP(X, Ordering, R, F, ARGV[2], ARGV[3], 1);
  EndIf;
  Return S;
EndDefine;