public QuiverWithPotential(Quiver <TVertex> quiver, Potential <TVertex> potential)
{
if (quiver == null)
{
throw new ArgumentNullException(nameof(quiver));
}
if (potential == null)
{
throw new ArgumentNullException(nameof(potential));
}
foreach (var cycle in potential.Cycles)
{
if (!quiver.Vertices.Contains(cycle.CanonicalPath.StartingPoint))
{
throw new ArgumentException($"The starting point {cycle.CanonicalPath.StartingPoint} of the canonical path of one of the cycles in the potential is not a vertex in the quiver.");
}
foreach (var arrow in cycle.CanonicalPath.Arrows.Skip(1))
{
if (!quiver.Vertices.Contains(arrow.Source))
{
throw new ArgumentException(String.Format("The vertex {0} is present in one of the cycles in the potential but is not a vertex in the quiver.", arrow.Source));
}
}
}
Quiver = quiver;
Potential = potential;
}
public SelfInjectiveQP <int> GetSelfInjectiveTriangleQP(int numRows, int firstVertex = DefaultFirstVertex)
{
if (numRows < 2)
{
throw new ArgumentOutOfRangeException(nameof(numRows));
}
var qp = UsefulQPs.GetTriangleQP(numRows);
var potential = new Potential <int>();
// Rotation "once" clockwise to get the Nakayama permutation
var nakayamaPermutation = new Dictionary <int, int>();
// Start with the right-most "column" (1, 3, 6, 10, ...), which is mapped to the bottom row,
// and go left to (2, 5, 9, ...), which is mapped to the second last row, and so on.
var columnVertices = Enumerable.Range(1, numRows).Select(k => Utility.TriangularNumber(k));
for (int rowIndex = numRows; rowIndex >= 1; rowIndex--)
{
var rowVertices = Enumerable.Range(Utility.TriangularNumber(rowIndex - 1) + 1, rowIndex).Reverse();
var inputOutputPairs = columnVertices.Zip(rowVertices, (x, y) => (x, y));
foreach (var(x, y) in inputOutputPairs)
{
nakayamaPermutation[x] = y;
}
columnVertices = columnVertices.Select(x => x - 1).Skip(1);
}
return(new SelfInjectiveQP <int>(qp, nakayamaPermutation));
}
/// <summary>
/// Gets the "classic" non-cancellative QP, which is not even weakly cancellative.
/// </summary>
/// <returns>The "classic" non-cancellative QP.</returns>
public static QuiverWithPotential <int> GetClassicNonCancellativeQP()
{
var potential = new Potential <int>();
potential = potential.AddCycle(new DetachedCycle <int>(1, 2, 4, 5, 1), -1);
potential = potential.AddCycle(new DetachedCycle <int>(1, 3, 4, 5, 1), +1);
var qp = new QuiverWithPotential <int>(potential);
return(qp);
}
public static QuiverWithPotential <int> GetCobwebQP(int numVerticesInCenterPolygon, int firstVertex = DefaultFirstVertex)
{
if (!CobwebParameterIsValid(numVerticesInCenterPolygon))
{
throw new ArgumentOutOfRangeException(nameof(numVerticesInCenterPolygon));
}
int numLayers = (numVerticesInCenterPolygon - 1) / 2;
int numVerticesInFullLayer = 2 * numVerticesInCenterPolygon;
var potential = new Potential <int>();
// Center polygon
potential = potential.AddCycle(new DetachedCycle <int>(Enumerable.Range(firstVertex, numVerticesInCenterPolygon).AppendElement(firstVertex)), +1);
if (numLayers > 1)
{
// First layer (the squares and triangles between the first and second layers)
var curLayer = GetLayerVertices(0);
var nextLayer = GetLayerVertices(1);
for (int indexInLayer = 0; indexInLayer < numVerticesInCenterPolygon; indexInLayer++)
{
var triangleVertices = new int[] { curLayer[indexInLayer], nextLayer[2 * indexInLayer - 1], nextLayer[2 * indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(triangleVertices), +1);
var squareVertices = new int[] { curLayer[indexInLayer], curLayer[indexInLayer + 1], nextLayer[2 * indexInLayer + 1], nextLayer[2 * indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(squareVertices), -1);
}
// Remaining layers (only squares)
for (int layerIndex = 1; layerIndex < numLayers - 1; layerIndex++) // 0-based layer index
{
curLayer = GetLayerVertices(layerIndex);
nextLayer = GetLayerVertices(layerIndex + 1);
for (int indexInLayer = 0; indexInLayer < numVerticesInFullLayer; indexInLayer++)
{
int sign = (layerIndex + indexInLayer).Modulo(2) == 1 ? +1 : -1;
var squareVertices = sign == +1 ?
new int[] { curLayer[indexInLayer + 1], curLayer[indexInLayer], nextLayer[indexInLayer], nextLayer[indexInLayer + 1], curLayer[indexInLayer + 1] } :
new int[] { curLayer[indexInLayer], curLayer[indexInLayer + 1], nextLayer[indexInLayer + 1], nextLayer[indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(squareVertices), sign);
}
}
}
var qp = new QuiverWithPotential <int>(potential);
return(qp);
CircularList <int> GetLayerVertices(int layerIndex) => new CircularList <int>(GetVerticesInCobwebQPLayer(numVerticesInCenterPolygon, layerIndex, firstVertex));
}
public QuiverWithPotential(Potential <TVertex> potential)
{
if (potential == null)
{
throw new ArgumentNullException(nameof(potential));
}
var vertices = new HashSet <TVertex>(potential.Cycles.SelectMany(c => c.CanonicalPath.Vertices));
var arrows = new HashSet <Arrow <TVertex> >(potential.Cycles.SelectMany(c => c.CanonicalPath.Arrows));
var quiver = new Quiver <TVertex>(vertices, arrows);
Quiver = quiver;
Potential = potential;
}
public static QuiverWithPotential <int> GetTriangleQP(int numRows, int firstVertex = DefaultFirstVertex)
{
if (!TriangleParameterIsValid(numRows))
{
throw new ArgumentOutOfRangeException(nameof(numRows));
}
if (numRows == 1)
{
var quiver = UsefulQuivers.GetTriangleQuiver(numRows, firstVertex);
return(new QuiverWithPotential <int>(quiver, new Potential <int>()));
}
var potential = new Potential <int>();
var curRowVertices = new List <int>()
{
firstVertex
};
int nextVertex = firstVertex + 1;
for (int rowIndex = 1; rowIndex <= numRows - 1; rowIndex++) // 1-based row index
{
var nextRowVertices = Enumerable.Range(nextVertex, rowIndex + 1).ToList();
nextVertex += rowIndex + 1;
// (2*rowIndex - 1) triangles to add. Draw a small triangle QP to realize that
// the cycles below are the ones to add.
for (int indexInRow = 0; indexInRow < rowIndex; indexInRow++)
{
var cycleVertices = new int[] { curRowVertices[indexInRow], nextRowVertices[indexInRow + 1], nextRowVertices[indexInRow], curRowVertices[indexInRow] };
potential = potential.AddCycle(new DetachedCycle <int>(cycleVertices), +1); // Positive for clockwise cycles.
}
for (int indexInRow = 0; indexInRow < rowIndex - 1; indexInRow++)
{
var cycleVertices = new int[] { curRowVertices[indexInRow + 1], curRowVertices[indexInRow], nextRowVertices[indexInRow + 1], curRowVertices[indexInRow + 1] };
potential = potential.AddCycle(new DetachedCycle <int>(cycleVertices), -1); // Negative for counterclockwise cycles.
}
curRowVertices = nextRowVertices;
}
var qp = new QuiverWithPotential <int>(potential);
return(qp);
}
/// <summary>
///
/// </summary>
/// <param name="numRows">The number of rows of vertices.</param>
/// <param name="firstVertex"></param>
/// <returns></returns>
public static QuiverWithPotential <int> GetSquareQP(int numRows, int firstVertex = DefaultFirstVertex)
{
if (!SquareParameterIsValid(numRows))
{
throw new ArgumentOutOfRangeException(nameof(numRows));
}
int numVerticesInRow = numRows;
int numVertices = numVerticesInRow * numRows;
if (numRows == 1)
{
var quiver = UsefulQuivers.GetSquareQuiver(numRows, firstVertex);
return(new QuiverWithPotential <int>(quiver, new Potential <int>()));
}
var potential = new Potential <int>();
for (int rowIndex = 0; rowIndex < numRows - 1; rowIndex++)
{
var curRow = GetVerticesInSquareQPRow(numRows, rowIndex, firstVertex).ToList();
var nextRow = GetVerticesInSquareQPRow(numRows, rowIndex + 1, firstVertex).ToList();
for (int indexInRow = 0; indexInRow < numVerticesInRow - 1; indexInRow++)
{
int sign = (rowIndex + indexInRow).Modulo(2) == 0 ? +1 : -1;
var cycleVertices = sign == +1 ?
new int[] { curRow[indexInRow], curRow[indexInRow + 1], nextRow[indexInRow + 1], nextRow[indexInRow], curRow[indexInRow] } :
new int[] { curRow[indexInRow + 1], curRow[indexInRow], nextRow[indexInRow], nextRow[indexInRow + 1], curRow[indexInRow + 1] };
potential = potential.AddCycle(new DetachedCycle <int>(cycleVertices), sign);
}
}
var qp = new QuiverWithPotential <int>(potential);
return(qp);
}
public static QuiverWithPotential <int> GetPointedFlowerQP(int numPeriods, int firstVertex = DefaultFirstVertex)
{
if (!PointedFlowerParameterIsValid(numPeriods))
{
throw new ArgumentOutOfRangeException(nameof(numPeriods));
}
int numLayers = UsefulQuivers.GetNumberOfLayersInPointedFlowerQuiver(numPeriods);
int numVerticesInFullInnerLayer = 2 * numPeriods;
int numVerticesInOuterLayer = 3 * numPeriods;
var potential = new Potential <int>();
if (numLayers == 2)
{
int centerVertex = GetLayerVertices(0).Single();
var nextLayer = GetLayerVertices(1);
for (int periodIndex = 0; periodIndex < numPeriods; periodIndex++)
{
int indexInNextLayer = 3 * periodIndex;
// Square
var positiveCycleVertices = new int[] { centerVertex, nextLayer[indexInNextLayer], nextLayer[indexInNextLayer + 1], nextLayer[indexInNextLayer + 2], centerVertex };
// Triangle
var negativeCycleVertices = new int[] { centerVertex, nextLayer[indexInNextLayer + 3], nextLayer[indexInNextLayer + 2], centerVertex };
potential = potential.AddCycle(new DetachedCycle <int>(positiveCycleVertices), +1);
potential = potential.AddCycle(new DetachedCycle <int>(negativeCycleVertices), -1);
}
return(new QuiverWithPotential <int>(potential));
}
// Polygons between first and second layer
{
int centerVertex = GetLayerVertices(0).Single();
var nextLayer = GetLayerVertices(1);
for (int periodIndex = 0; periodIndex < numPeriods; periodIndex++)
{
int indexInNextLayer = 2 * periodIndex;
// Triangles
var positiveCycleVertices = new int[] { centerVertex, nextLayer[indexInNextLayer], nextLayer[indexInNextLayer + 1], centerVertex };
var negativeCycleVertices = new int[] { centerVertex, nextLayer[indexInNextLayer + 2], nextLayer[indexInNextLayer + 1], centerVertex };
potential = potential.AddCycle(new DetachedCycle <int>(positiveCycleVertices), +1);
potential = potential.AddCycle(new DetachedCycle <int>(negativeCycleVertices), -1);
}
}
// Cobweb layers
for (int layerIndex = 1; layerIndex < numLayers - 2; layerIndex++) // 0-based layer index
{
var curLayer = GetLayerVertices(layerIndex);
var nextLayer = GetLayerVertices(layerIndex + 1);
for (int indexInLayer = 0; indexInLayer < numVerticesInFullInnerLayer; indexInLayer++)
{
int sign = (layerIndex + indexInLayer).Modulo(2) == 0 ? +1 : -1;
var squareVertices = sign == +1 ?
new int[] { curLayer[indexInLayer + 1], curLayer[indexInLayer], nextLayer[indexInLayer], nextLayer[indexInLayer + 1], curLayer[indexInLayer + 1] } :
new int[] { curLayer[indexInLayer], curLayer[indexInLayer + 1], nextLayer[indexInLayer + 1], nextLayer[indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(squareVertices), sign);
}
}
// Outermost layer of polygons
{
int layerIndex = numLayers - 2;
var curLayer = GetLayerVertices(layerIndex);
var nextLayer = GetLayerVertices(layerIndex + 1);
for (int indexInPeriod = 0; indexInPeriod < numPeriods; indexInPeriod++)
{
// Add pentagon
int indexInPenultimateLayer = 2 * indexInPeriod;
int indexInUltimateLayer = 3 * indexInPeriod;
int pentagonSign = layerIndex.Modulo(2) == 0 ? +1 : -1; // Remember that layerIndex is the index of the *penultimate* layer
var pentagonVertices = pentagonSign == +1 ?
new int[] { curLayer[indexInPenultimateLayer + 1], curLayer[indexInPenultimateLayer], nextLayer[indexInUltimateLayer], nextLayer[indexInUltimateLayer + 1], nextLayer[indexInUltimateLayer + 2], curLayer[indexInPenultimateLayer + 1] } :
new int[] { curLayer[indexInPenultimateLayer], curLayer[indexInPenultimateLayer + 1], nextLayer[indexInUltimateLayer + 2], nextLayer[indexInUltimateLayer + 1], nextLayer[indexInUltimateLayer], curLayer[indexInPenultimateLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(pentagonVertices), pentagonSign);
// Add square
int squareSign = -pentagonSign;
indexInPenultimateLayer += 1;
indexInUltimateLayer += 2;
var squareVertices = squareSign == +1 ?
new int[] { curLayer[indexInPenultimateLayer + 1], curLayer[indexInPenultimateLayer], nextLayer[indexInUltimateLayer], nextLayer[indexInUltimateLayer + 1], curLayer[indexInPenultimateLayer + 1] } :
new int[] { curLayer[indexInPenultimateLayer], curLayer[indexInPenultimateLayer + 1], nextLayer[indexInUltimateLayer + 1], nextLayer[indexInUltimateLayer], curLayer[indexInPenultimateLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(squareVertices), squareSign);
}
}
var qp = new QuiverWithPotential <int>(potential);
return(qp);
CircularList <int> GetLayerVertices(int layerIndex)
=> new CircularList <int>(GetVerticesInPointedFlowerQPLayer(numPeriods, layerIndex, firstVertex));
}
public static QuiverWithPotential <int> GetEvenFlowerType2QP(int numVerticesInCenterPolygon, int firstVertex = DefaultFirstVertex)
{
if (!EvenFlowerType2ParameterIsValid(numVerticesInCenterPolygon))
{
throw new ArgumentOutOfRangeException(nameof(numVerticesInCenterPolygon));
}
int numLayers = UsefulQuivers.GetNumberOfLayersInEvenFlowerType2Quiver(numVerticesInCenterPolygon);
int numVerticesInFullInnerLayer = 2 * numVerticesInCenterPolygon;
int numVerticesInOuterLayer = 3 * numVerticesInCenterPolygon;
var potential = new Potential <int>();
// Center polygon
potential = potential.AddCycle(new DetachedCycle <int>(Enumerable.Range(firstVertex, numVerticesInCenterPolygon).AppendElement(firstVertex)), +1);
// Full inner layers
if (numLayers > 2)
{
// First layer (the squares and triangles between the first and second layers)
var curLayer = GetLayerVertices(0);
var nextLayer = GetLayerVertices(1);
for (int indexInLayer = 0; indexInLayer < numVerticesInCenterPolygon; indexInLayer++)
{
var triangleVertices = new int[] { curLayer[indexInLayer], nextLayer[2 * indexInLayer - 1], nextLayer[2 * indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(triangleVertices), +1);
var squareVertices = new int[] { curLayer[indexInLayer], curLayer[indexInLayer + 1], nextLayer[2 * indexInLayer + 1], nextLayer[2 * indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(squareVertices), -1);
}
// Remaining layers (only squares)
for (int layerIndex = 1; layerIndex < numLayers - 2; layerIndex++) // 0-based layer index
{
curLayer = GetLayerVertices(layerIndex);
nextLayer = GetLayerVertices(layerIndex + 1);
for (int indexInLayer = 0; indexInLayer < numVerticesInFullInnerLayer; indexInLayer++)
{
int sign = (layerIndex + indexInLayer).Modulo(2) == 1 ? +1 : -1;
var squareVertices = sign == +1 ?
new int[] { curLayer[indexInLayer + 1], curLayer[indexInLayer], nextLayer[indexInLayer], nextLayer[indexInLayer + 1], curLayer[indexInLayer + 1] } :
new int[] { curLayer[indexInLayer], curLayer[indexInLayer + 1], nextLayer[indexInLayer + 1], nextLayer[indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(squareVertices), sign);
}
}
}
// Outer layer
{
int layerIndex = numLayers - 2;
var curLayer = GetLayerVertices(layerIndex);
var nextLayer = GetLayerVertices(layerIndex + 1);
if (numLayers == 2)
{
// Add squares and diamonds in this degenerate case
for (int indexInLayer = 0; indexInLayer < curLayer.Count; indexInLayer++)
{
var diamondVertices = new int[] { curLayer[indexInLayer], nextLayer[3 * indexInLayer - 2], nextLayer[3 * indexInLayer - 1], nextLayer[3 * indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(diamondVertices), +1);
var squareVertices = new int[] { curLayer[indexInLayer], curLayer[indexInLayer + 1], nextLayer[3 * indexInLayer + 1], nextLayer[3 * indexInLayer], curLayer[indexInLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(squareVertices), -1);
}
}
else
{
// Add squares and pentagons in the non-degenerate case
for (int indexInCenterLayer = 0; indexInCenterLayer < numVerticesInCenterPolygon; indexInCenterLayer++)
{
// Add square
int indexInPenultimateLayer = 2 * indexInCenterLayer;
int indexInUltimateLayer = 3 * indexInCenterLayer;
int squareSign = layerIndex.Modulo(2) == 1 ? +1 : -1; // Remember that layerIndex is the index of the *penultimate* layer
var squareVertices = squareSign == +1 ?
new int[] { curLayer[indexInPenultimateLayer + 1], curLayer[indexInPenultimateLayer], nextLayer[indexInUltimateLayer], nextLayer[indexInUltimateLayer + 1], curLayer[indexInPenultimateLayer + 1] } :
new int[] { curLayer[indexInPenultimateLayer], curLayer[indexInPenultimateLayer + 1], nextLayer[indexInUltimateLayer + 1], nextLayer[indexInUltimateLayer], curLayer[indexInPenultimateLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(squareVertices), squareSign);
// Add pentagon
int pentagonSign = -squareSign;
indexInPenultimateLayer += 1;
indexInUltimateLayer += 1;
var pentagonVertices = pentagonSign == +1 ?
new int[] { curLayer[indexInPenultimateLayer + 1], curLayer[indexInPenultimateLayer], nextLayer[indexInUltimateLayer], nextLayer[indexInUltimateLayer + 1], nextLayer[indexInUltimateLayer + 2], curLayer[indexInPenultimateLayer + 1] } :
new int[] { curLayer[indexInPenultimateLayer], curLayer[indexInPenultimateLayer + 1], nextLayer[indexInUltimateLayer + 2], nextLayer[indexInUltimateLayer + 1], nextLayer[indexInUltimateLayer], curLayer[indexInPenultimateLayer] };
potential = potential.AddCycle(new DetachedCycle <int>(pentagonVertices), pentagonSign);
}
}
}
var qp = new QuiverWithPotential <int>(potential);
return(qp);
CircularList <int> GetLayerVertices(int layerIndex)
=> new CircularList <int>(GetVerticesInEvenFlowerType2QPLayer(numVerticesInCenterPolygon, layerIndex, firstVertex));
}
/// <summary>
/// Creates a new instance of the <see cref="SemimonomialIdeal{TVertex}"/> class that
/// represents the Jacobian ideal of the given potential (in some unspecified quiver
/// containing all the arrows in the potential).
/// </summary>
/// <param name="potential">The potential whose semimonomial Jacobian ideal to create.</param>
/// <returns>The semimonomial Jacobian ideal of <paramref name="potential"/>.</returns>
/// <exception cref="NotSupportedException"><paramref name="potential"/> has a cycle with
/// coefficient not equal to either of -1 and +1,
/// or some arrow occurs multiple times in a single cycle of <paramref name="potential"/>.</exception>
/// <exception cref="ArgumentException">For some arrow in <paramref name="potential"/> and
/// sign, the arrow is contained in more than one cycle of that sign.</exception>
/// <remarks>
/// <para>The preconditions on <paramref name="potential"/> as of this writing is that the
/// the scalars are <c>-1</c> or <c>+1</c>, every arrow occurs in at most one cycle per
/// sign, and every arrow occurs at most once per cycle.</para>
/// <para>The reasoning behind the differentiation between <see cref="NotSupportedException"/>
/// and <see cref="ArgumentException"/> is that <see cref="NotSupportedException"/> is
/// thrown when things might work out in theory (e.g., all scalars could be -2 or +2
/// instead of -1 or +1, or an arrow could appear more than once in a cycle if the cycle is
/// a suitable power of some other cycle (and stars align if the arrow is also contained in
/// a cycle of the opposite sign)), while <see cref="ArgumentException"/> is thrown when
/// things do not work out even in theory (i.e., semimonomiality fails to hold). That is,
/// <see cref="NotSupportedException"/> is thrown in cases that might be "fixed"
/// in the future.</para>
/// </remarks>
public static SemimonomialIdeal <TVertex> CreateSemimonomialIdealFromPotential <TVertex>(Potential <TVertex> potential)
where TVertex : IEquatable <TVertex>, IComparable <TVertex>
{
// Validation
var cycleCoefficients = new HashSet <int>(potential.LinearCombinationOfCycles.ElementToCoefficientDictionary.Values);
if (!cycleCoefficients.IsSubsetOf(new int[] { -1, +1 }))
{
throw new NotSupportedException("Only potentials with all coefficients equal to +1 or -1 are supported.");
}
if (potential.Cycles.Any(cycle => cycle.Arrows.HasDuplicate()))
{
throw new NotSupportedException("Potentials with an arrow occurring more than once in a cycle are not supported.");
}
var signedCycles = potential.LinearCombinationOfCycles.ElementToCoefficientDictionary;
var signedArrows = signedCycles.SelectMany(pair =>
{
var cycle = pair.Key;
var sign = pair.Value;
return(cycle.Arrows.Select(arrow => (arrow, sign)));
});
if (signedArrows.TryGetDuplicate(out var duplicate))
{
throw new ArgumentException($"The potential has a signed arrow {duplicate} occurring in more than one cycle.", nameof(potential));
}
var monomialGenerators = new List <Path <TVertex> >();
var nonMonomialGenerators = new List <DifferenceOfPaths <TVertex> >();
var arrows = potential.Cycles.SelectMany(cycle => cycle.Arrows).WithoutDuplicates();
foreach (var arrow in arrows)
{
// Guaranteed by the validation to have only terms with coefficients +1 or -1
// Also guaranteed to have at most 2 terms
var linCombOfPaths = potential.DifferentiateCyclically(arrow);
var numTerms = linCombOfPaths.NumberOfTerms;
if (numTerms == 1)
{
monomialGenerators.Add(linCombOfPaths.Elements.Single());
}
else if (numTerms == 2)
{
var paths = linCombOfPaths.Elements.ToList();
var differenceOfPaths = new DifferenceOfPaths <TVertex>(paths[0], paths[1]);
nonMonomialGenerators.Add(differenceOfPaths);
}
else
{
System.Diagnostics.Debug.Fail($"{numTerms} terms in the cyclic derivative with respect to {arrow}. Expected at most two terms.");
}
}
var semimonomialIdeal = new SemimonomialIdeal <TVertex>(monomialGenerators, nonMonomialGenerators);
return(semimonomialIdeal);
}