ProperPair Class

class stdpairs.ProperPair(monomial, face, ambient_ideal, properness=False)

Bases: object

A class representing a proper pair with respect to given monomial ideal \(I\), which is a pair \((a,F)\) where \(a\) is a monomial in the ambient affine monoid, and \(F\) denotes a face such that \(a+\mathbb{N}F \cap I = \emptyset\). Or, abusively, this class also generated without checking the condition \(a+\mathbb{N}F \cap I = \emptyset\) using properness argument.

INPUT:

  • monomial – A NumPy.ndarray object with 2-dimensional shape and integer elements or sage.matrix.matrix_integer_dense type variable. In any cases, this argument should have only one column.

  • face – A tuple object representing a face in ambient_ideal.ambient_monoid().

  • ambient_ideal – A MonomialIdeal object.

  • properness=False – A bool object. If this argument is True, then this class does not give an error even if \(a+\mathbb{N}F \cap I\) is nonempty.

OUTPUT:

  • ProperPair object \((a,F)\) whose \(a\) is monomial argument and \(F\) is face argument.

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair
sage: import numpy
sage: A = matrix(ZZ,[[1,2],[0,2]])
sage: Q = AffineMonoid(A)
sage: M = matrix(ZZ,[[4,6],[4,6]])
sage: I = MonomialIdeal(M,Q)
sage: #Using ``NumPy``
sage: # Below ``P`` is a proper pair.
sage: P = ProperPair(numpy.array([[2],[2]]), (0,), I)
sage: # But below ``Not_P`` raise error that this is not a proper pair.
sage: # Not_P =ProperPair(matrix(ZZ,[[2],[2]]), (1,), I)
sage: # However, using ``properness`` as ``True`` the same construction does not give you error;
sage: # Basically the module does not check whether the given pair is proper or not.
sage: Not_P =ProperPair(matrix(ZZ,[[2],[2]]), (1,), I,True)

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair
sage: A = matrix(ZZ,[[1,2],[0,2]])
sage: Q = AffineMonoid(A)
sage: M = matrix(ZZ,[[4,6],[4,6]])
sage: I = MonomialIdeal(M,Q)
sage: #Using ``NumPy``
sage: # Below ``P`` is a proper pair.
sage: P = ProperPair(matrix(ZZ,[[2],[2]]), (0,), I)

TESTS:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair
sage: A = matrix(ZZ,[[0,1,1,0],[0,0,1,1],[1,1,1,1]])
sage: Q = AffineMonoid(A)
sage: I = MonomialIdeal(matrix(ZZ,[[2,2,2],[0,1,2],[2,2,2]]),Q)
sage: P = ProperPair(matrix(ZZ,[[1], [1], [1]]), (0,), I)
sage: if (P.is_maximal() != False):
....:     raise SyntaxError("Method is_maximal() is problematic; please report it to the developer.")
sage: P = ProperPair(matrix(ZZ,[[1], [1], [1]]), (3,), I)
sage: if (P.is_maximal() != False):
....:     raise SyntaxError("Method is_maximal() is problematic; please report it to the developer.")
sage: P = ProperPair(matrix(ZZ,[[1], [1], [1]]), (0,3), I)
sage: if (P.is_maximal() != True):
....:     raise SyntaxError("Method is_maximal() is problematic; please report it to the developer.")
ambient_ideal()

returns an ideal which decides whether self is proper (i.e., a translation of submonoid does not intersect with the ideal) or not. This was given when self was created.

OUTPUT:

  • ambient_ideal – A MonomialIdeal object.

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair
sage: Q=AffineMonoid(matrix(ZZ,[[1,2],[0,2]]))
sage: I=MonomialIdeal(matrix(ZZ,[[4,6],[4,6]]),Q)
sage: P=ProperPair(matrix(ZZ,[[2],[2]]),(0,),I)
sage: P.ambient_ideal()
An ideal whose generating set is
[[4]
 [4]]
face()

returns the face of self as a tuple object.

OUTPUT:

  • gens – A tuple object.

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair
sage: Q=AffineMonoid(matrix(ZZ,[[1,2],[0,2]]))
sage: I=MonomialIdeal(matrix(ZZ,[[4,6],[4,6]]),Q)
sage: P=ProperPair(matrix(ZZ,[[2],[2]]),(0,),I)
sage: P.face()
(0,)
sage: Q.face((0,))
array([[1],
       [0]])
is_element(a_monomial)

Given a_monomial monomial, say \(b\), in the ambient monoid, find a matrix whose row \(u\) is a solution of \(F*u =b-a\) where \((a,F)\) is the give proper pair self. In other words, if a_monomial in \(a+\mathbb{N}F\), return a matrix whose row \(u\) satisfies \(a+F*u=b\). Otherwise, return an empty matrix.

INPUT:

  • a_monomial – A NumPy.ndarray object with 2-dimensional shape, integer elements, and only one vector, or sage.matrix.matrix_integer_dense object with one column.

OUTPUT:

  • A sage.matrix.matrix_integer_dense object.

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair
sage: Q=AffineMonoid(matrix(ZZ,[[1,2],[0,2]]))
sage: I=MonomialIdeal(matrix(ZZ,[[3,7],[2,0]]),Q)
sage: P= ProperPair(matrix(ZZ,[[2],[2]]),(1,),I)
sage: P.is_element(matrix(ZZ,[[8],[8]]))
[3]
sage: # This means that [[8],[8]]- [[2],[2]] = 3*[[2],[2]]
sage: P.is_element(matrix(ZZ,[[7],[7]]))
[]
sage: # This does not have solution since [[7],[7]] cannot be obtained by integer.
is_maximal()

returns True if and only if a pair is maximal with respect to divisibility. A proper pair \((a,F)\) is maximal with respect to divisibility if for any proper pair \((b,G)\) there is no such element \(c\) in the ambient affine monoid such that \(a+c+\mathbb{N}F \subseteq b+\mathbb{N}G\). Notes that this function is not properly working if you generated the given pair with properness=`False` argument.

OUTPUT:

  • A bool object.

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair
sage: A = matrix(ZZ,[[0,1,1,0],[0,0,1,1],[1,1,1,1]])
sage: Q = AffineMonoid(A)
sage: I = MonomialIdeal(matrix(ZZ,[[2,2,2],[0,1,2],[2,2,2]]),Q)
sage: P = ProperPair(matrix(ZZ,[[1], [1], [1]]), (0,), I)
sage: P.is_maximal()
False
sage: P = ProperPair(matrix(ZZ,[[1], [1], [1]]), (0,3), I)
sage: P.is_maximal()
True
monomial()

returns the monomial of self.

OUTPUT:

  • monomial – A NumPy.ndarray object with 2-dimensional shape, integer elements, and only one column.

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair
sage: Q=AffineMonoid(matrix(ZZ,[[1,2],[0,2]]))
sage: I=MonomialIdeal(matrix(ZZ,[[4,6],[4,6]]),Q)
sage: P=ProperPair(matrix(ZZ,[[2],[2]]),(0,),I)
sage: P.monomial()
array([[2],
       [2]])