  
  [1X5 [33X[0;0YGroebner Basis[133X[101X
  
  [33X[0;0YThis  chapter  contains  the  declarations  and  implementations  needed for
  Groebner  basis. Currently, we do not provide algorithms to actually compute
  Groebner  basis;  instead, the declarations and implementations are provided
  here  for [5XGAP[105X objects and the actual elements of Groebner basis are computed
  by the [5XGBNP[105X package.[133X
  
  
  [1X5.1 [33X[0;0YConstructing a Groebner Basis[133X[101X
  
  [1X5.1-1 InfoGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XInfoGroebnerBasis[102X [32X info class[133X
  
  [33X[0;0Yis the info class for functions dealing with Groebner basis.[133X
  
  [1X5.1-2 GroebnerBasis[101X
  
  [33X[1;0Y[29X[2XGroebnerBasis[102X( [3XI[103X, [3Xrels[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XI[103X -- an ideal, [3Xrels[103X -- a list of relations generating [3XI[103X.[133X
  
  [6XReturns:[106X  [33X[0;10Yan   object  [3XGB[103X  in  the  [2XIsGroebnerBasis[102X  ([14X5.2-3[114X)  category  with
            [2XIsCompleteGroebnerBasis[102X ([14X5.2-2[114X) property set on true.[133X
  
  [33X[0;0YSets also [3XGB[103X as a value of the attribute [2XGroebnerBasisOfIdeal[102X ([14X4.10-1[114X) for [3XI[103X
  (so one has an access to it by calling GroebnerBasisOfIdeal([3XI[103X)).[133X
  [33X[0;0YThere  are  absolutely no computations and no checks for correctness in this
  function.  Giving a set of relations that does not form a Groebner basis may
  result  in incorrect answers or unexpected errors. This function is intended
  to  be used by packages providing access to external Groebner basis programs
  and should be invoked before further computations on Groebner basis or ideal
  I (cf. also [2XIsCompleteGroebnerBasis[102X ([14X5.2-2[114X)).[133X
  
  
  [1X5.2 [33X[0;0YCategories and Properties of Groebner Basis[133X[101X
  
  [1X5.2-1 IsCompletelyReducedGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XIsCompletelyReducedGroebnerBasis[102X( [3Xgb[103X ) [32X property[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue when [3XGB[103X is a Groebner basis which is completely reduced.[133X
  
  [1X5.2-2 IsCompleteGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XIsCompleteGroebnerBasis[102X( [3Xgb[103X ) [32X property[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue when [3XGB[103X is a complete Groebner basis.[133X
  
  [33X[0;0YWhile philosophically something that isn't a complete Groebner basis isn't a
  Groebner  basis  at all, this property can be used in conjunction with other
  properties  to see if the the Groebner basis contains enough information for
  computations.  An example of a system that creates incomplete Groebner bases
  is `Opal'.[133X
  [33X[0;0Y[13XNote:[113X The current package used for creating Groebner bases is [5XGBNP[105X, and this
  package does not create incomplete Groebner bases.[133X
  
  [1X5.2-3 IsGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XIsGroebnerBasis[102X( [3Xobject[103X ) [32X category[133X
  
  [33X[0;0YArguments: [3Xobject[103X -- any object in [5XGAP[105X.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue when [3Xobject[103X is a Groebner basis and false otherwise.[133X
  
  [33X[0;0YThe  function only returns true for Groebner bases that has been set as such
  using the [10XGroebnerBasis[110X function, as illustrated in the following example.[133X
  
  [1X5.2-4 IsHomogeneousGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XIsHomogeneousGroebnerBasis[102X( [3Xgb[103X ) [32X property[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue when [3XGB[103X is a Groebner basis which is homogeneous.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQ := Quiver( 3, [ [1,2,"a"], [2,3,"b"] ] );[127X[104X
    [4X[28X<quiver with 3 vertices and 2 arrows>[128X[104X
    [4X[25Xgap>[125X [27XPA := PathAlgebra( Rationals, Q );[127X[104X
    [4X[28X<Rationals[<quiver with 3 vertices and 2 arrows>]>[128X[104X
    [4X[25Xgap>[125X [27Xrels := [ PA.a*PA.b ];[127X[104X
    [4X[28X[ (1)*a*b ][128X[104X
    [4X[25Xgap>[125X [27Xgb := GBNPGroebnerBasis( rels, PA );[127X[104X
    [4X[28X[ (1)*a*b ][128X[104X
    [4X[25Xgap>[125X [27XI := Ideal( PA, gb );[127X[104X
    [4X[28X<two-sided ideal in <Rationals[<quiver with 3 vertices and 2 arrows>]>[128X[104X
    [4X[28X    , (1 generators)>[128X[104X
    [4X[25Xgap>[125X [27Xgrb := GroebnerBasis( I, gb );[127X[104X
    [4X[28X<complete two-sided Groebner basis containing 1 elements>[128X[104X
    [4X[25Xgap>[125X [27Xalg := PA/I;[127X[104X
    [4X[28X<Rationals[<quiver with 3 vertices and 2 arrows>]/[128X[104X
    [4X[28X<two-sided ideal in <Rationals[<quiver with 3 vertices and 2 arrows>]>[128X[104X
    [4X[28X    , (1 generators)>>[128X[104X
    [4X[25Xgap>[125X [27XIsGroebnerBasis(gb);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsGroebnerBasis(grb);[127X[104X
    [4X[28Xtrue [128X[104X
  [4X[32X[104X
  
  [1X5.2-5 IsTipReducedGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XIsTipReducedGroebnerBasis[102X( [3Xgb[103X ) [32X property[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner Basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue when [3XGB[103X is a Groebner basis which is tip reduced.[133X
  
  
  [1X5.3 [33X[0;0YAttributes and Operations for Groebner Basis[133X[101X
  
  [1X5.3-1 AdmitsFinitelyManyNontips[101X
  
  [33X[1;0Y[29X[2XAdmitsFinitelyManyNontips[102X( [3XGB[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a complete Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue  if  the Groebner basis admits only finitely many nontips and
            false otherwise.[133X
  
  [1X5.3-2 CompletelyReduce[101X
  
  [33X[1;0Y[29X[2XCompletelyReduce[102X( [3XGB[103X, [3Xa[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis, [3Xa[103X -- an element in a path algebra.[133X
  
  [6XReturns:[106X  [33X[0;10Y[3Xa[103X reduced by [3XGB[103X.[133X
  
  [33X[0;0YIf [3Xa[103X is already completely reduced, the original element [3Xa[103X is returned.[133X
  
  [1X5.3-3 CompletelyReduceGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XCompletelyReduceGroebnerBasis[102X( [3XGB[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  completely  reduced  Groebner basis of the ideal generated by
            [3XGB[103X.[133X
  
  [33X[0;0YThe  operation modifies a Groebner basis [3XGB[103X such that each relation in [3XGB[103X is
  completely     reduced.     The     [10XIsCompletelyReducedGroebnerBasis[110X     and
  [10XIsTipReducedGroebnerBasis[110X  properties are set as a result of this operation.
  The  resulting  relations  will  be  placed in sorted order according to the
  ordering of [3XGB[103X.[133X
  
  [1X5.3-4 Enumerator[101X
  
  [33X[1;0Y[29X[2XEnumerator[102X( [3XGB[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Yan  enumerat  that enumerates the relations making up the Groebner
            basis.[133X
  
  [33X[0;0YThese  relations should be enumerated in ascending order with respect to the
  ordering for the family the elements are contained in.[133X
  
  [1X5.3-5 IsPrefixOfTipInTipIdeal[101X
  
  [33X[1;0Y[29X[2XIsPrefixOfTipInTipIdeal[102X( [3XGB[103X, [3XR[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis, [3XR[103X -- a relation.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue if the tip of the relation [3XR[103X is in the tip ideal generated by
            the tips of [3XGB[103X.[133X
  
  [33X[0;0YThis  is  used  mainly  for the construction of right Groebner basis, but is
  made   available  for  general  use  in  case  there  are  other  unforeseen
  applications.[133X
  
  [1X5.3-6 Iterator[101X
  
  [33X[1;0Y[29X[2XIterator[102X( [3XGB[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Yan  iterator  (in  the  IsIterator  category,  see the [5XGAP[105X manual,
            chapter 28.7).[133X
  
  [33X[0;0YCreates  an iterator that iterates over the relations making up the Groebner
  basis.  These relations are iterated over in ascending order with respect to
  the ordering for the family the elements are contained in.[133X
  
  [1X5.3-7 Nontips[101X
  
  [33X[1;0Y[29X[2XNontips[102X( [3XGB[103X ) [32X attribute[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ya list of nontip elements for [3XGB[103X.[133X
  
  [33X[0;0YIn order to compute the nontip elements, the Groebner basis must be complete
  and  tip reduced, and there must be a finite number of nontips. If there are
  an infinite number of nontips, the operation returns `fail'.[133X
  
  [1X5.3-8 NontipSize[101X
  
  [33X[1;0Y[29X[2XNontipSize[102X( [3XGB[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a complete Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe number of nontips admitted by [3XGB[103X.[133X
  
  [1X5.3-9 TipReduce[101X
  
  [33X[1;0Y[29X[2XTipReduce[102X( [3XGB[103X, [3Xa[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis, [3Xa[103X - an element in a path algebra.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe element [3Xa[103X tip reduced by the Groebner basis.[133X
  
  [33X[0;0YIf [3Xa[103X is already tip reduced, then the original [3Xa[103X is returned.[133X
  
  [1X5.3-10 TipReduceGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XTipReduceGroebnerBasis[102X( [3XGB[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XGB[103X -- a Groebner basis.[133X
  
  [6XReturns:[106X  [33X[0;10Ya tip reduced Groebner basis.[133X
  
  [33X[0;0YThe  returned  Groebner  basis  is  equivalent  to  [3XGB[103X  If [3XGB[103X is already tip
  reduced,  this  function  returns  the original object [3XGB[103X, possibly with the
  addition of the `IsTipReduced`' property set.[133X
  
  
  [1X5.4 [33X[0;0YRight Groebner Basis[133X[101X
  
  [33X[0;0YIn  this  section  we support right Groebner basis for two-sided ideals with
  Groebner basis. More general cases may be supported in the future.[133X
  
  [1X5.4-1 IsRightGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XIsRightGroebnerBasis[102X( [3Xobject[103X ) [32X property[133X
  
  [33X[0;0YArguments: [3Xobject[103X -- any object in [5XGAP[105X.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue when [3Xobject[103X is a right Groebner basis.[133X
  
  [1X5.4-2 RightGroebnerBasis[101X
  
  [33X[1;0Y[29X[2XRightGroebnerBasis[102X( [3XI[103X ) [32X operation[133X
  
  [33X[0;0YArguments: [3XI[103X -- a right ideal.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  right Groebner basis for [3XI[103X, which must support a right Groebner
            basis  theory.  Right  now,  this  requires  that [3XI[103X has a complete
            Groebner basis.[133X
  
  [1X5.4-3 RightGroebnerBasisOfIdeal[101X
  
  [33X[1;0Y[29X[2XRightGroebnerBasisOfIdeal[102X( [3XI[103X ) [32X attribute[133X
  
  [33X[0;0YArguments: [3XI[103X -- a right ideal.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  right  Groebner  basis  of  a  right  ideal, [3XI[103X, if one has been
            computed.[133X
  
