source: git/Singular/LIB/fpalgebras.lib @ fe1b64

////////////////////////////////////////////////////////////////
version="version fpalgebras.lib 4.1.1.0 Feb_2018 ";
category="Noncommutative";
info="
LIBRARY: fpalgebras.lib
AUTHORS: Karim Abou Zeid,       karim.abou.zeid at rwth-aachen.de

Support: Project II.6 in the transregional collaborative research centre
SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German DFG

OVERVIEW:
Generation of various groups and algebras in the Letterplace ring

PROCEDURES:
baumslagSolitar(int n, int m, int d, list #);
baumslag(int m, int n, int d);
crystallographicGroupP1(int d);
crystallographicGroupPM(int d);
crystallographicGroupPG(int d);
crystallographicGroupP2MM(int d);
crystallographicGroupP2(int d);
crystallographicGroupP2GG(int d);
crystallographicGroupCM(int d);
crystallographicGroupC2MM(int d);
crystallographicGroupP4(int d);
crystallographicGroupP4MM(int d);
crystallographicGroupP4GM(int d);
crystallographicGroupP3(int d);
crystallographicGroupP31M(int d);
crystallographicGroupP3M1(int d);
crystallographicGroupP6(int d);
crystallographicGroupP6MM(int d);
dyckGroup1(int n, int d, intvec P);
dyckGroup2(int n, int d, intvec P);
dyckGroup3(int n, int d, intvec P);
fibonacciGroup(int m, int d);
tetrahedronGroup(int g, int d);
triangularGroup(int g, int d);
";

LIB "freegb.lib";
LIB "general.lib";
////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////
// Baumslag ////////////////////////////////////////////////////////
// from Grischa Studzinski /////////////////////////////////////////
////////////////////////////////////////////////////////////////////

proc baumslagSolitar(int n, int m, int d, list #)
"USAGE: baumslagSolitar(m,n,d[,IsGroup]); n an integer, m an integer, d an integer, IsGroup an optional integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - in the group case: A = a^(-1), B = b^(-1)
@*    - negativ input is only allowed in the group case!
@*    - d gives a degreebound and must be >m,n
"
{
  int isGroup = 0;
  if (size(#) > 0) {isGroup = #[1];}

  if (isGroup != 0)
  {
   int baseringdef;
   if (defined(basering)) // if a basering is defined, it should be saved for later use
   {
    def save = basering;
    baseringdef = 1;
   }
   if (m < 0 || n < 0) {ERROR("Exponent can't be negativ in monoid rings!");}
   if (d < 1 || d < m || d < n) {ERROR("Degree bound must be positiv and greater then m,n!");}
   int i;
   ring mr = 0,(a,b),Dp;
   def Mr = makeLetterplaceRing(d);
   setring Mr;
   poly p,q;
   if (n==0) {p = b(1);}
   else
   {
    p = a(1)*b(2);
    for (i = 1; i < n; i++) {p = lpMult(a(1),p);}
   }
   if (m==0) {q = b(1);}
   else
   {
    q = b(1)*a(2);
    for (i = 1; i < m; i++) {q = lpMult(q,a(1));}
   }
   ideal I = p - q;
   export(I);
   if (baseringdef == 1) {setring save;}
   return(Mr); 
  }
  else
  {
   int baseringdef;
   if (defined(basering)) // if a basering is defined, it should be saved for later use
   {
    def save = basering;
    baseringdef = 1;
   }
   int i;
   if (d < 1 || d < absValue(m) || d < absValue(n)) {ERROR("Degree bound must be positiv and greater then |m|,|n|!");}
   ring gr = 0,(a,b,A,B),Dp;
   def Gr = makeLetterplaceRing(d);
   setring Gr;
   poly p,q;
   if (n==0) {p = b(1);}
   else
   {if (n > 0) 
    {
     p = a(1)*b(2);
     for (i = 1; i < n; i++) {p = lpMult(a(1),p);}
    }
    else
    {
     p = A(1)*b(2);
     for (i = 1; i < -n; i++) {p = lpMult(A(1),p);}
    }
   }
   if (m==0) {q = b(1);}
   else
   {if (m > 0) 
    {
     q = b(1)*a(2);
     for (i = 1; i < m; i++) {q = lpMult(q,a(1));}
    }
    else
    {
     q = A(1)*b(2);
     for (i = 1; i < -m; i++) {q = lpMult(q,A(1));}
    }
   }
   ideal I = p - q, a(1)*A(2) - 1, b(1)*B(2) - 1, a(1)*A(2) - A(1)*a(2), b(1)*B(2) - B(1)*b(2);
   export(I);
   if (baseringdef == 1) {setring save;}
   return(Gr); 
  }
}


proc baumslagGroup(int m, int n, int d)
"USAGE: baumslagGroup(m,n,d); m an integer, n an integer, d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - Baumslag group with the following presentation
@*      < a, b | a^m = b^n = 1 >
@*    -d gives the degreebound for the Letterplace ring
"
{
 if (m < 0 || n < 0 ) {ERROR("m,n must be non-negativ integers!");}
 if (d < 1 || d < m || d < n) {ERROR("degreebound must be positiv and larger than n and m!");}
 int i;
 ring r = 0,(a,b),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 poly p,q;
 p = 1; q = 1;
 for (i = 1; i <= m; i++){p = lpMult(p,a(1));}
 for (i = 1; i <= n; i++){q = lpMult(q,b(1));}
 ideal I = p-1,q-1;
 export(I);
 return(R);
}


////////////////////////////////////////////////////////////////////
// Crystallographic Groups //////////////////////////////////////////
// from Grischa Studzinski /////////////////////////////////////////
////////////////////////////////////////////////////////////////////

proc crystallographicGroupP1(int d)
"USAGE: crystallographicGroupP1(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p1 group with the following presentation
@*      < x, y | [x, y] = 1 >
@*    -d gives the degreebound for the Letterplace ring
"
{
 if (d < 2){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, X(1)*x(2)-1, x(1)*X(2)-1, y(1)*Y(2)-1, Y(1)*y(2)-1; 
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

// old? there is already another crystallographicGroupP2 proc
/* proc crystallographicGroupP2(int d) */
/* " */
/* p2 group with the following presentation */
/* < x, y, r | [x, y] = r^2 = 1, r^-1*x*r = x^-1, r^-1*y*r = y^-1 > */
/* Note: r = r^-1 */
/* " */
/* { */
/*  if (d < 3){ERROR("Degreebound is to small for choosen example!");} */
 
/*  int baseringdef; */
/*  if (defined(basering)) // if a basering is defined, it should be saved for later use */
/*  { */
/*   def save = basering; */
/*   baseringdef = 1; */
/*  } */
/*  ring r = 2,(x,y,r,X,Y),dp; */
/*  def R = makeLetterplaceRing(d); */
/*  setring R; */
/*  ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2), r(1)*r(2)-1, r(1)*x(2)*r(3)-X(1), r(1)*y(2)*r(3)-Y(1),x(1)*X(2)-1, */  
/* X(1)*x(2)-1, Y(1)*y(2)-1,  y(1)*Y(2)-1; */
/*  I = simplify(I,2); */
/*  export(I); */
/*  if (baseringdef == 1) {setring save;} */
/*  return(R); */
/* } */

proc crystallographicGroupPM(int d)
"USAGE: crystallographicGroupPM(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - pm group with the following presentation
@*      < x, y, m | [x, y] = m^2 = 1, m^-1*x*m = x, m^-1*y*m = y^-1 >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 3){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,m,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-m(1)*m(2), m(1)*m(2)-1, m(1)*x(2)*m(3)-x(1), m(1)*y(2)*m(3)-Y(1),x(1)*X(2)-1,  
X(1)*x(2)-1, Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupPG(int d)
"USAGE: crystallographicGroupPG(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - pg group with the following presentation
@*      < x, y, t | [x, y] = 1, t^2 = x, t^-1*y*t = y^-1 >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 3){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,t,X,Y,T),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, t(1)*t(2) - x(1), T(1)*y(2)*t(3)-Y(1), X(1)*x(2)-1, x(1)*X(2)-1,  
Y(1)*y(2)-1,  y(1)*Y(2)-1, t(1)*T(2)-1, T(1)*t(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}


proc crystallographicGroupP2MM(int d)
"USAGE: crystallographicGroupP2MM(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p2mm group with the following presentation
@*      < x, y, p, q | [x, y] = [p, q] = p^2 = q^2 = 1, p^-1*x*p = x, q^-1*x*q = x^-1, p^-1*y*p = y^-1, q^-1*y*q = y >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 3){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,p,q,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, p(1)*q(2)-q(1)*p(2)-1, p(1)*p(2) - 1, q(1)*q(2) - 1, p(1)*y(2)*p(3)-Y(1), p(1)*x(2)*p(3)-x(1), 
 q(1)*y(2)*q(3)-y(1), q(1)*x(2)*q(3)-X(1), X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1,  x(1)*y(2)-y(1)*x(2)- p(1)*p(2),
 x(1)*y(2)-y(1)*x(2)- q(1)*q(2), p(1)*p(2)-q(1)*q(2);
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP2(int d)
"USAGE: crystallographicGroupP2(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p2 group with the following presentation
@*      < x, y, m, t | [x, y] = t^2 = 1, m^2 = y, t^-1*x*t = x, m^-1*x*m = x^-1, t^-1*y*t = y^-1, t^-1*m*t = m^-1 >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 3){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,m,t,X,Y,M),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2), m(1)*m(2)-y(1), t(1)*t(2) - 1, t(1)*x(2)*t(3)-x(1), 
M(1)*x(2)*m(3)-X(1), t(1)*y(2)*t(3)-Y(1), t(1)*m(2)*t(3)-M(1), X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1, 
m(1)*M(2)-1, M(1)*m(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP2GG(int d)
"USAGE: crystallographicGroupP2GG(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p2gg group with the following presentation
@*      < x, y, u, v | [x, y] = (u*v)^2 = 1, u^2 = x, v^2 = y, v^-1*x*v = x^-1, u^-1*y*u = y^-1 >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 4){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,u,v,X,Y,u,v),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-u(1)*v(2)*u(3)*v(4), u(1)*v(2)*u(3)*v(4)-1, u(1)*u(2)-x(1), v(1)*v(2) - y, 
V(1)*x(2)*v(3)-X(1), U(1)*y(2)*u(3)-Y(1), 
X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1, u(1)*U(2)-1, U(1)*u(2)-1, v(1)*V(2)-1, V(1)*v(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupCM(int d)
"USAGE: crystallographicGroupCM(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - cm group with the following presentation
@*      < x, y, t | [x, y] = t^2 = 1, t^-1*x*t = x*y, t^-1*y*t = y^-1 >   
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 3){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,t,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2), t(1)*t(2)-1,  
t(1)*x(2)*t(3)-x(1)*y(2), t(1)*y(2)*t(3)-Y(1), 
X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupC2MM(int d)
"USAGE: crystallographicGroupC2MM(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - c2mm group with the following presentation
@*      < x, y, m, r | [x, y] = m^2 = r^2 = 1, m^-1*y*m = y^-1, m^-1*x*m = x*y, r^-1*y*r = y^-1, r^-1*x*r = x^-1, m^-1*r*m = r^-1 >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 3){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,m,r,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-m(1)*m(2), x(1)*y(2)-y(1)*x(2)-r(1)*r(2), m(1)*m(2)-1,  r(1)*r(2)-1, 
 m(1)*m(2)-r(1)*r(2), m(1)*y(2)*m(3)-Y(1), m(1)*x(2)*m(3)-x(1)*y(2), (1)*y(2)*r(3)-Y(1), r(1)*x(2)*r(3)-X(1), m(1)*r(2)*m(3)-r(1), 
X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP4(int d)
"USAGE: crystallographicGroupP4(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p4 group with the following presentation
@*      < x, y, r | [x, y] = r^4 = 1, r^-1*x*r = x^-1, r^-1*x*r = y >   
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 5){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,r,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4), r(1)*r(2)*r(3)*r(4)-1, 
 r(1)*r(2)*r(3)*x(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), 
X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP4MM(int d)
"USAGE: crystallographicGroupP4MM(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p4mm group with the following presentation
@*      < x, y, r, m | [x, y] = r^4 = m^2 = 1, r^-1*y*r = x^-1, r^-1*x*r = y, m^-1*x*m = y, m^-1*r*m = r^-1 >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 5){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,r,m,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4),  r(1)*r(2)*r(3)*r(4)-1, 
 r(1)*r(2)*r(3)*x(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), 
X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP4GM(int d)
"USAGE: crystallographicGroupP4GM(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p4gm group with the following presentation
@*      < x, y, r, t | [x, y] = r^4 = t^2 = 1, r^-1*y*r = x^-1, r^-1*x*r = y, t^-1*x*t = y, t^-1*r*t = x^-1*r^-1>
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 5){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,r,t,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4),  r(1)*r(2)*r(3)*r(4)-1, x(1)*y(2)-y(1)*x(2)-t(1)*t(2),
 t(1)*t(2)-1,  r(1)*r(2)*r(3)*r(4)-t(1)*t(2),  r(1)*r(2)*r(3)*y(4)*r(5)-X(1), r(1)*r(2)*r(3)*x(4)*r(5)-y(1), 
 t(1)*r(2)*t(3)-X(1)*r(2)*r(3)*r(4), X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP3(int d)
"USAGE: crystallographicGroupP3(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p3 group with the following presentation
@*      < x, y, r | [x, y] = r^3 = 1, r^-1*x*r = x^-1*y, r^-1*y*r = x^-1>
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 4){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,r,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3),  r(1)*r(2)*r(3)-1, 
  r(1)*r(2)*x(3)*r(4)-X(1)*y(2),  r(1)*r(2)*y(3)*r(4)-X(1), X(1)*x(2)-1, x(1)*X(2)-1,  Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP31M(int d)
"USAGE: crystallographicGroupP31M(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p31m group with the following presentation
@*      < x, y, r, t | [x, y] = r^2 = t^2 = (t*r)^3 = 1, r^-1*x*r = x, t^-1*y*t = y, t^-1*x*t = x^-1*y, r^-1*y*r = x*y^-1 >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 6){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,r,t,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2), x(1)*y(2)-y(1)*x(2)-t(1)*t(2), r(1)*r(2)-1, t(1)*t(2)-1,
 t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-1, r(1)*r(2)-t(1)*t(2),  x(1)*y(2)-y(1)*x(2)-t(1)*r(2)*t(3)*r(4)*t(5)*r(6), 
 t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-r(1)*r(2), t(1)*r(2)*t(3)*r(4)*t(5)*r(6)-t(1)*t(2),
 r(1)*x(2)*r(3)-x(1),  t(1)*y(2)*t(3)-y(1), t(1)*x(2)*t(3)-X(1)*y(2), r(1)*y(2)*r(3)-x(1)*Y(2), 
 X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP3M1(int d)
"USAGE: crystallographicGroupP3M1(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p3m1 group with the following presentation
@*      < x, y, r, m | [x, y] = r^3 = m^2 = 1, m^-1*r*m = r^2, r^-1*x*r = x^-1*y, r^-1*y*r = x^-1, m^-1*x*m = x^-1, m^-1*y*m = x^-1*y >
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 4){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,r,m,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3), x(1)*y(2)-y(1)*x(2)-m(1)*m(2), r(1)*r(2)*r(3)-1, m(1)*m(2)-1,
 r(1)*r(2)*r(3)-m(1)*m(2),  m(1)*r(2)*m(3)-r(1)*r(2), r(1)*r(2)*x(3)*r(4)-X(1)*y(2), r(1)*r(2)*y(3)*r(4)-X(1),m(1)*x(2)*m(3)-X(1),
 m(1)*y(2)*m(3)-X(1)*y(2), X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP6(int d)
"USAGE: crystallographicGroupP6(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p6 group with the following presentation
@*      < x, y, r | [x, y] = r^6 = 1, r^-1*x*r = y, r^-1*y*r = x^-1*y>
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 7){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,r,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4)*r(5)*r(6), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-1,
 r(1)*r(2)*r(3)*r(4)*r(5)*x(6)*r(7)-y(1), r(1)*r(2)*r(3)*r(4)*r(5)*y(6)*r(7)-X(1)*y(2),
 X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

proc crystallographicGroupP6MM(int d)
"USAGE: crystallographicGroupP6MM(d); d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - p6mm group with the following presentation
@*      < x, y, r, m | [x, y] = r^6 = m^2 = 1, r^-1*y*r = x^-1*y, r^-1*x*r = y, m^-1*x*m = x^-1, m^-1*y*m = x^-1*y, m^-1*r*m = r^-1*y>
@*    - d gives the degreebound for the Letterplace ring
"
{
 if (d < 7){ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,r,m,X,Y),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I = x(1)*y(2)-y(1)*x(2)-1, x(1)*y(2)-y(1)*x(2)-r(1)*r(2)*r(3)*r(4)*r(5)*r(6), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-1, 
 x(1)*y(2)-y(1)*x(2)-m(1)*m(2), r(1)*r(2)*r(3)*r(4)*r(5)*r(6)-m(1)*m(2), m(1)*m(2)-1, m(1)*x(2)*m(3)-X(1),  m(1)*y(2)*m(3)-X(1)*y(2), 
 r(1)*r(2)*r(3)*r(4)*r(5)*x(6)*r(7)-y(1), r(1)*r(2)*r(3)*r(4)*r(5)*y(6)*r(7)-X(1)*y(2), M(1)*r(2)*m(3)- r(1)*r(2)*r(3)*r(4)*r(5)*y(6)
 X(1)*x(2)-1, x(1)*X(2)-1, Y(1)*y(2)-1,  y(1)*Y(2)-1;
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

////////////////////////////////////////////////////////////////////
// Dyck Group //////////////////////////////////////////////////////
// from Grischa Studzinski /////////////////////////////////////////
////////////////////////////////////////////////////////////////////

proc dyckGroup1(int n, int d, intvec P)
"USAGE: dyckGroup1(n,d,P); n an integer, d an integer, P an intvec
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - The Dyck group with the following presentation
@*      < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 >
@*    - negative exponents are allowed
@*    - representation in the form x_i^p_i - x_(i+1)^p_(i+1)
@*    - d gives the degreebound for the Letterplace ring
"
{
 int baseringdef,i,j;
 if (n < 1) {ERROR("There must be at least one variable!");}
 if (d < n) {ERROR("Degreebound is to small!");}
 for (i = 1; i <= size(P); i++) {if (d < absValue(P[i])){ERROR("Degreebound is to small!");}}

 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x(1..n),Y(1..n)),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I; poly p,q;
 p = 1; q = 1;
 for (i = 1; i<= n; i++) {p = lpMult(p,var(i));}
 I = p-1;
 for (i = n; i > 0; i--)
 {
  if (P[i] >= 0) {for (j = 1; j <= P[i]; j++){q = lpMult(q,var(i));}} 
  else {for (j = 1; j <= -P[i]; j++){q = lpMult(q,var(i+n));}}
  I = p - q,I;
  p = q; q = 1;
 }
 
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}


proc dyckGroup2(int n, int d, intvec P)
"USAGE: dyckGroup2(n,d,P); n an integer, d an integer, P an intvec
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - The Dyck group with the following presentation
@*      < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 >
@*    - negative exponents are allowed
@*    - representation in the form x_i^p_i - 1
@*    - d gives the degreebound for the Letterplace ring
"
{
 int baseringdef,i,j;
 if (n < 1) {ERROR("There must be at least one variable!");}
 if (d < n) {ERROR("Degreebound is to small!");}
 for (i = 1; i <= size(P); i++) {if (d < absValue(P[i])){ERROR("Degreebound is to small!");}}

 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x(1..n),Y(1..n)),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I; poly p;
 p = 1;
 for (i = 1; i<= n; i++) {p = lpMult(p,var(i));}
 I = p-1;
 for (i = n; i > 0; i--)
 {
  p = 1;
  if (P[i] >= 0) {for (j = 1; j <= P[i]; j++){p = lpMult(p,var(i));}} 
  else {for (j = 1; j <= -P[i]; j++){p = lpMult(p,var(i+n));}}
  I = p - 1,I;
 }
 
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}



proc dyckGroup3(int n, int d, intvec P)
"USAGE: dyckGroup2(n,d,P); n an integer, d an integer, P an intvec
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - The Dyck group with the following presentation
@*      < x_1, x_2, ... , x_n | (x_1)^p1 = (x_2)^p2 = ... = (x_n)^pn = x_1 * x_2 * ... * x_n = 1 >
@*    - only positive exponents are allowed
@*    - no inverse generators needed
@*    - d gives the degreebound for the Letterplace ring
"
{
 int baseringdef,i,j;
 if (n < 1) {ERROR("There must be at least one variable!");}
 if (d < n) {ERROR("Degreebound is to small!");}
 for (i = 1; i <= size(P); i++) {if (P[i] < 0){ERROR("Exponents must be positive!");}}
 for (i = 1; i <= size(P); i++) {if (d < P[i]){ERROR("Degreebound is to small!");}}
 

 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,x(1..n),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I; poly p;
 p = 1;
 for (i = 1; i<= n; i++) {p = lpMult(p,var(i));}
 I = p-1;
 for (i = n; i > 0; i--)
 {
  p = 1;
  for (j = 1; j <= P[i]; j++){p = lpMult(p,var(i));} 
  I = p - 1,I;
 }
 
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}

////////////////////////////////////////////////////////////////////
// Fibonacci Group /////////////////////////////////////////////////
// from Grischa Studzinski /////////////////////////////////////////
////////////////////////////////////////////////////////////////////

proc fibonacciGroup(int m, int d)
"USAGE: fibonacciGroup(m,d); m an integer, d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - The Fibonacci group F(2, m) with the following presentation
@*      < x_1, x_2, ... , x_m | x_i * x_(i + 1) = x_(i + 2) >
@*    - d gives the degreebound for the Letterplace ring
"
// TODO: basefield Q oder F2?
// TODO: inverse Elemente!
{
 if (m < 3) {ERROR("At least three generators are required!");}
 if (d < 2) {ERROR("Degree bound must be at least 2!");}
 int baseringdef,i;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x(1..m),Y(1..m)),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I; poly p;
 for (i = 1; i < m-1; i++)
 {
  p = lpMult(var(i),var(i+1))-var(i+2);
  I = I,p;
 }
 for (i = 1; i <= m; i++)
 {
  p = lpMult(var(i),var(i+m))-1;
  I = I,p;
  p = lpMult(var(i+m),var(i))-1;
  I = I,p;
 } 
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}


////////////////////////////////////////////////////////////////////
// Tetrahedon Groups ///////////////////////////////////////////////
// from Grischa Studzinski /////////////////////////////////////////
////////////////////////////////////////////////////////////////////

proc tetrahedronGroup(int g, int d)
"USAGE: tetrahedronGroup(g,d); g an integer, d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - g gives the number of the example
@*    - d gives the degreebound for the Letterplace ring
@*
The examples are found in
Classification of the finite generalized tetrahedron groups
by Gerhard Rosenberger and Martin Scheer.
The 5 examples are denoted in Proposition 1.9 and concern
finite generalized tetrahedron group in the Tsarnarov-case, which are
not equivalent to a presentation for an ordinary tetrahedron group.
@*
"
{
 if (g < 1 || g > 5) {ERROR("There are only 5 examples!");}
 if ((g == 1 && d < 6)||(g == 2 && d < 6)||(g == 3 && d < 5)||(g == 4 && d < 4)||(g == 5 && d < 5))
 {ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef,i,j;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(x,y,z),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I;
 if (g == 1) 
 {I = x(1)*x(2)*x(3)*x(4)*x(5)-1, y(1)*y(2)-1, z(1)*z(2)*z(3)-1, x(1)*y(2)*x(3)*y(4)*x(5)*y(6)-1, x(1)*x(2)*z(3)*x(4)*x(5)*z(6)-1,
      y(1)*z(2)*y(3)*z(4)-1;
 }
 if (g == 2)
 {I = x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)*z(5)-1,x(1)*y(2)*x(3)*y(4)-1,x(1)*z(2)*x(3)*z(4)-1, 
      y(1)*z(2)*z(3)*y(4)*z(5)*z(6)-1;
 }
 if (g == 3)
 {I =  x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)-1, x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1;
 }
 if (g == 4)
 {I =  x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)-1,x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1;
 }
 if (g ==5)
 {I =  x(1)*x(2)*x(3)-1, y(1)*y(2)*y(3)-1, z(1)*z(2)*z(3)*z(4)*z(5)-1,x(1)*y(2)*x(3)*y(4)-1, x(1)*z(2)*x(3)*z(4)-1, y(1)*z(2)*y(3)*z(4)-1;
 }
  
 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}


////////////////////////////////////////////////////////////////////
// Triangular Groups ///////////////////////////////////////////////
// from Grischa Studzinski /////////////////////////////////////////
////////////////////////////////////////////////////////////////////

proc triangularGroup(int g, int d)
"USAGE: triangularGroup(g,d); g an integer, d an integer
RETURN: ring
NOTE: - the ring contains the ideal I, which contains the required relations
@*    - g gives the number of the example
@*    - d gives the degreebound for the Letterplace ring
@*
The examples are found in
Classification of the finite generalized tetrahedron groups
by Gerhard Rosenberger and Martin Scheer.
The 14 examples are denoted in theorem 2.12
@*
"
{
 if (g < 1 || g > 14) {ERROR("There are only 14 examples!");}
 if ((g == 1 && d < 20)||(g == 2 && d < 21)||(g == 3 && d < 10)||(g == 4 && d < 12)||(g == 5 && d < 10)||(g == 6 && d < 18)||(g == 7 && d < 20)||(g == 8 && d < 16)||(g == 9 && d < 10)||(g == 10 && d < 14)||(g == 11 && d < 16)||(g == 12 && d < 24)||(g == 13 && d < 28)||(g == 14 && d < 37))
 {ERROR("Degreebound is to small for choosen example!");}
 
 int baseringdef;
 if (defined(basering)) // if a basering is defined, it should be saved for later use
 {
  def save = basering;
  baseringdef = 1;
 }
 ring r = 2,(a,b),dp;
 def R = makeLetterplaceRing(d);
 setring R;
 ideal I;
 
 if (g == 1) 
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1,
  a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*b(10)*a(11)*b(12)*a(13)*b(14)*a(15)*b(16)*b(17)*a(18)*b(19)*b(20)-1;
 }
 if (g == 2)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1,
  a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*a(12)*b(13)*b(14)*a(15)*b(16)*a(17)*b(18)*a(19)*b(20)*b(21)-1;
 }
 if (g == 3)
 {I = a(1)*a(2)*a(3)-1, b(1)*b(2)*b(3)-1,
  a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1;
 }
 if (g == 4)
 {I = a(1)*a(2)*a(3)-1, b(1)*b(2)*b(3)-1,
  a(1)*b(2)*a(3)*a(4)*b(5)*b(6)*a(7)*b(8)*a(9)*a(10)*b(11)*b(12)-1;
 }
 if (g == 5)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1,
   a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1;
 }
 if (g == 6)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1,
   a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*b(8)*b(9)*a(10)*b(11)*a(12)*b(13)*a(14)*b(15)*b(16)*b(17)*b(18)-1;
 }
 if (g == 7)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)*b(5)-1,
   a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*b(8)*b(9)*b(10)*a(11)*b(12)*a(13)*b(14)*b(15)*a(16)*b(17)*b(18)*b(19)*b(20)-1;
 }
 if (g == 8)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)*b(4)-1,
   a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*b(8)*a(9)*b(10)*a(11)*b(12)*a(13)*b(14)*b(15)*b(16)-1;
 }
 if (g == 9)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1,
   a(1)*b(2)*a(3)*b(4)*b(5)*a(6)*b(7)*a(8)*b(9)*b(10)-1;
 }
 if (g == 10)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1,
  a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*a(12)*b(13)*b(14)-1;
 }
 if (g == 11)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1,
  a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*b(9)*a(10)*b(11)*a(12)*b(13)*a(14)*b(15)*b(16)-1;
 }
 if (g == 12)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1,
  a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*b(7)*a(8)*b(9)*a(10)*b(11)*b(12)*a(13)*b(14)*a(15)*b(16)*a(17)*b(18)*b(19)*a(20)*b(21)*a(22)*b(23)*b(24)-1;
 }
 if (g == 13)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1,
a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*a(9)*b(10)*b(11)*a(12)*b(13)*b(14)*a(15)*b(16)*a(17)*b(18)*a(19)*b(20)*a(21)*b(22)*a(23)*b(24)*b(25)*a(26)*b(27)*b(28)-1;
 }
 if (g == 14)
 {I = a(1)*a(2)-1, b(1)*b(2)*b(3)-1,
a(1)*b(2)*a(3)*b(4)*a(5)*b(6)*a(7)*b(8)*b(9)*a(10)*b(11)*b(12)*a(13)*b(14)*a(15)*b(16)*b(17)*a(18)*b(19)*b(20)*a(21)*b(22)*a(23)*b(24)*a(25)*b(26)*a(27)*b(28)*b(29)*a(30)*b(31)*a(32)*b(33)*b(34)*a(35)*b(36)*b(37)-1;
 }

 I = simplify(I,2);
 export(I);
 if (baseringdef == 1) {setring save;}
 return(R);
}