Global functions¶

Decompose a difference between submonoids as submonoids¶

If we have two translations of submonoids \(b+\mathbb{N}G\) and \(b+\mathbb{N}G'\) where \(G' \supseteq G\) as a face, then there is an algorithm to represent the difference \((b+\mathbb{N}G) \setminus (b+\mathbb{N}G)'\) as a union of finitely many translations of submonoids \(\bigcup_{i=1}^{m}a_{i}+\mathbb{N}F_{i}\) where \(G \subseteq F_{i}\) for any \(i \in \{1,2, \cdots, m\}\). This algorithm was implemented as pair_difference(). More details on this algorithm can be found in Yu, 2020 or Matusevich and Yu, 2020.

stdpairs.pair_difference(pair_a, pair_b)¶

Given two pairs pair_a and pair_b representing \((a,F)\) and \((b,G)\) such that \(F \subseteq G\), return a collection \(C\) such that \((a+\mathbb{N}F) \setminus (b+\mathbb{N}G) = \bigcup_{(c,H) \in C}(c+\mathbb{N}H)\). This is an implementation of Theorem 4.1 in https://arxiv.org/abs/2005.10968

INPUT:

pair_a, pair_b – ProperPair objects. pair_a.face() must contain pair_b.face() as a subtuple.

OUTPUT:

A dictionary object whose keys are face, and whose values are ProperPair objects having the same face with its key.

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair, pair_difference
sage: Q = AffineMonoid(matrix(ZZ, [[2,0,1],[0,1,1]]))
sage: I = MonomialIdeal(matrix(ZZ,0),Q)
sage: P= ProperPair(matrix(ZZ,[[0],[0]]), (0,1,2), I )
sage: T= ProperPair(matrix(ZZ,[[0],[2]]), (0,1,2), I )
sage: diff_PT = pair_difference(P,T)
sage: diff_PT.keys()
dict_keys([(0,)])
sage: sorted(diff_PT[(0,)],key=str)
[([[0], [0]]^T,[[2], [0]]),
 ([[0], [1]]^T,[[2], [0]]),
 ([[1], [1]]^T,[[2], [0]]),
 ([[1], [2]]^T,[[2], [0]])]

Find a prime ideal corresponding to a face¶

stdpairs.prime_ideal(face, affine_monoid)¶

Given a tuple object face representing face of affine_monoid, return monomial ideal other, return the prime monomial ideal corresponding to the face.

OUTPUT:

A MonomialIdeal object representing a prime monomial ideal corresponding to the face.

EXAMPLE:

sage: from stdpairs import AffineMonoid, prime_ideal
sage: Q=AffineMonoid(matrix(ZZ,[[1,2],[0,2]]))
sage: prime_ideal((0,),Q)
An ideal whose generating set is
[[2]
 [2]]
sage: prime_ideal((1,),Q)
An ideal whose generating set is
[[1]
 [0]]

Division of pairs¶

stdpairs.div_pairs(pair_a, pair_b)¶

Given two pairs pair_a and pair_b representing \((a,F)\) and \((b,G)\) respectively, this method returns a set of monomials \(c\) such that \(a+c+\mathbb{N}F \subseteq b+\mathbb{N}G\) as a matrix.

INPUT:

pair_a,``pair_b`` – ProperPair objects.

OUTPUT:

A sage.matrix.matrix_integer_dense object whose columns are minimal solutions of \(a+c+\mathbb{N}F \subseteq b+\mathbb{N}G\).

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, ProperPair, div_pairs
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: T = ProperPair(matrix(ZZ,[[1], [1], [1]]), (3,), I)
sage: div_pairs(P,T)
[]
sage: TT = ProperPair(matrix(ZZ,[[1], [1], [1]]), (0,3), I)
sage: div_pairs(P,TT)
[0]
[0]
[0]

Interface using `Macaulay2`¶

Using MonomialSubalgebra class in Macualay2, one can translate MonomialIdeal object in this package into MonomialSubalgebra object in Macualay2 via methods below.

stdpairs.from_macaulay2(var_name)¶

Given a macaulay2 type object mac2_mon_subalg which saves the MonomialSubalgebra object, this method returns AffineMonoid object corresponding to the MonomialSubalgebra.

INPUT:

var_name – A string object which equal to the variable name of a MonomialSubalgebra class variable created by Normaliz package in Macaulay2.

OUTPUT:

An AffineMonoid object.

EXAMPLE:

sage: from stdpairs import from_macaulay2
sage: R = macaulay2.eval('ZZ[x,y,z]')
sage: temp=macaulay2.needsPackage('"Normaliz"')
sage: temp=macaulay2.eval('S=createMonomialSubalgebra {x^5*y, y*z^2, z^3}')
sage: Q = from_macaulay2('S')
sage: Q
An affine semigroup whose generating set is
[[5 0 0]
 [1 1 0]
 [0 2 3]]

stdpairs.to_macaulay2(monomial_ideal, ring_name='R', ideal_name='I', std_cover_name='SC')¶

Given a MonomialIdeal object monomial_ideal, this method returns a dictionary object containing the ambient monoid, the given monomial ideal, and its standard cover as string objects. Users can access those Macaulay2 variables using ring_name, ideal_name, and std_cover_name. Notes that the ambient monoid and the monomial ideal will be delivered as macaulay2 type object, especially with MonomialSubalgebra type. See MonomialSubalgebra for detail. The standard cover will be delivered as a list object in Macaulay 2. Also, this method will executes monomial_ideal.standard_cover() if it was not executed previously.

INPUT:

monomial_ideal – A MonomialIdeal type object.

OUTPUT:

A dictionary object as below.

{ 'AffineSemigroupRing':mac2_mon_subalg, 'MonomialIdeal':mac2_mon_sub_alg_ideal, 'StandardCover':mac2_std_cover}

where mac2_mon_subalg, mac2_mon_sub_alg_ideal, and mac2_std_cover are string object.

EXAMPLE:

sage: from stdpairs import AffineMonoid,MonomialIdeal, to_macaulay2
sage: Q=AffineMonoid(matrix(ZZ,[[0,1,1,0],[0,0,1,1],[1,1,1,1]]))
sage: I=MonomialIdeal(matrix(ZZ,[[2,2,2],[0,1,2],[2,2,2]]),Q)
sage: S=to_macaulay2(I)
sage: S
{'AffineSemigroupRing': ZZ[c, a*c, a*b*c, b*c]
<BLANKLINE>
monomial subalgebra of PolyRing,
'MonomialIdeal':   2 2   2   2   2 2 2
{a c , a b*c , a b c }
<BLANKLINE>
List,
'StandardCover': {{1, {c, b*c}}, {a*c, {c, b*c}}, {a*b*c, {c, b*c}}}
<BLANKLINE>
List}
sage: # To access values in dictionary via Macaulay2,
sage: mac_ring=macaulay2.eval("R")
sage: mac_ideal=macaulay2.eval("I")
sage: mac_sc=macaulay2.eval("SC")

Save and load via `string`¶

SageMath provide a global function writing and reading objects as a binary file by pickling. However, it may not work in case Sage code for the package is dramatically changed. To avoid such a catastrophy, we provides some global functions which may save and load your calculation about the monomial ideal objects easily.

stdpairs.txt_to_affinemonoid(text_info)¶

Given a string text_info generated by a method save_txt() of an AffineMonoid object, this method returns an AffineMonoid object which is the same as the original AffineMonoid object.

INPUT:

text_info – A string objct generated by the method AffineMonoid.save_txt().

OUTPUT:

An AffineMonoid object.

EXAMPLE:

sage: from stdpairs import AffineMonoid, txt_to_affinemonoid
sage: R = AffineMonoid(matrix(ZZ, [[2,0,1],[0,1,1]]))
sage: R.save_txt()
'Q\n2,0,1|0,1,1\n'
sage: L = txt_to_affinemonoid('Q\n2,0,1|0,1,1\n')
sage: R == L
True

stdpairs.txt_to_monomialideal(text_info)¶

Given a string text_info generated by MonomialIdeal.save_txt() method, returns an MonomialIdeal object which is the same as the original MonomialIdeal object.

INPUT:

text_info – A string objct generated by the method MonomialIdeal.save_txt().

OUTPUT:

An MonomialIdeal object.

EXAMPLE:

sage: from stdpairs import AffineMonoid, MonomialIdeal, txt_to_monomialideal
sage: Q = AffineMonoid(matrix(ZZ, [[1,2],[0,2]]))
sage: I = MonomialIdeal(matrix(ZZ,[[3],[2]]),Q)
sage: I.irreducible_decomposition()
[An ideal whose generating set is
 [[1]
  [0]],
 An ideal whose generating set is
 [[2]
  [2]]]
sage: str_I = I.save_txt()
sage: J = txt_to_monomialideal(str_I)
sage: J == I
True
sage: J.standard_cover()
{(0,): [([[0], [0]]^T,[[1], [0]])], (1,): [([[0], [0]]^T,[[2], [2]])]}
sage: #This does not require re-calculation.

Global functions¶

Decompose a difference between submonoids as submonoids¶

Find a prime ideal corresponding to a face¶

Division of pairs¶

Interface using Macaulay2¶

Save and load via string¶

Interface using `Macaulay2`¶

Save and load via `string`¶