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MonomialIdeal -- the class of all monomial ideals handled by the engine

Description

Monomial ideals are kinds of ideals, but many algorithms are much faster. Generally, any routines available for ideals are also available for monomial ideals.
i1 : R = QQ[a..d];
i2 : I = monomialIdeal(a*b*c,b*c*d,a^2*d,b^3*c)

                            3    2
o2 = monomialIdeal (a*b*c, b c, a d, b*c*d)

o2 : MonomialIdeal of R
i3 : I^2

                     2 2 2     4 2   6 2   3        2 3        2 2    4 2  
o3 = monomialIdeal (a b c , a*b c , b c , a b*c*d, a b c*d, a*b c d, b c d,
     ------------------------------------------------------------------------
      4 2   2     2   2 2 2
     a d , a b*c*d , b c d )

o3 : MonomialIdeal of R
i4 : I + monomialIdeal(b*c)

                          2
o4 = monomialIdeal (b*c, a d)

o4 : MonomialIdeal of R
i5 : I : monomialIdeal(b*c)

                        2
o5 = monomialIdeal (a, b , d)

o5 : MonomialIdeal of R
i6 : radical I

o6 = monomialIdeal (b*c, a*d)

o6 : MonomialIdeal of R
i7 : associatedPrimes I

o7 = {monomialIdeal (a, b), monomialIdeal (a, c), monomialIdeal (b, d),
     ------------------------------------------------------------------------
     monomialIdeal (c, d), monomialIdeal (a, b, d)}

o7 : List
i8 : primaryDecomposition I

                      2                      2                           
o8 = {monomialIdeal (a , b), monomialIdeal (a , c), monomialIdeal (b, d),
     ------------------------------------------------------------------------
                                              3
     monomialIdeal (c, d), monomialIdeal (a, b , d)}

o8 : List

Specialized functions only available for monomial ideals

i9 : borel I

                     3   2      2   3   2           2      2     2   2  
o9 = monomialIdeal (a , a b, a*b , b , a c, a*b*c, b c, a*c , b*c , a d,
     ------------------------------------------------------------------------
             2
     a*b*d, b d, a*c*d, b*c*d)

o9 : MonomialIdeal of R
i10 : isBorel I

o10 = false
i11 : I - monomialIdeal(b^3*c,b^4)

                             2
o11 = monomialIdeal (a*b*c, a d, b*c*d)

o11 : MonomialIdeal of R
i12 : standardPairs I

                                                                           
o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
      -----------------------------------------------------------------------
                           2
      {b, a}}, {b, {c}}, {b , {c}}}

o12 : List
i13 : independentSets I

o13 = {a*b, a*c, b*d, c*d}

o13 : List
i14 : dual I

                        3        2      3
o14 = monomialIdeal (a*b , a*c, a b*d, b d, c*d)

o14 : MonomialIdeal of R
The ring of a monomial ideal must be a commutative polynomial ring. This ring must not be a skew commuting ring, and/or a quotient ring.

Menu

Functions and methods returning a monomial ideal:

  • MonomialIdeal * MonomialIdeal -- see * -- a binary operator, usually used for multiplication
  • MonomialIdeal * Ring -- see * -- a binary operator, usually used for multiplication
  • Ring * MonomialIdeal -- see * -- a binary operator, usually used for multiplication
  • RingElement * MonomialIdeal -- see * -- a binary operator, usually used for multiplication
  • MonomialIdeal + MonomialIdeal -- see + -- a unary or binary operator, usually used for addition
  • borel(MonomialIdeal) -- see borel(Matrix) -- make a Borel fixed submodule
  • MonomialIdeal ^ Array -- see Ideal ^ Array -- bracket power of an ideal
  • MonomialIdeal ^ ZZ -- see Ideal ^ ZZ -- power of an ideal
  • mingens(MonomialIdeal) -- see mingens(Module) -- minimal generator matrix
  • monomialIdeal -- make a monomial ideal
  • MonomialIdeal - MonomialIdeal -- monomial ideal difference
  • MonomialIdeal : RingElement
  • monomialIdeal(String) (missing documentation)
  • quotient(MonomialIdeal,RingElement)
  • saturate(MonomialIdeal,RingElement) -- see saturate -- saturation of ideal or submodule
  • trim(MonomialIdeal) -- see trim -- minimize generators and relations

Methods that use a monomial ideal:

  • ZZ % MonomialIdeal -- see % -- a binary operator, usually used for remainder and reduction
  • ZZ // MonomialIdeal -- see // -- a binary operator, usually used for quotient
  • codim(MonomialIdeal) -- compute the codimension
  • dim(MonomialIdeal) -- see dim(Ideal) -- compute the Krull dimension
  • dual(MonomialIdeal) -- the Alexander dual of a monomial ideal
  • dual(MonomialIdeal,List) -- the Alexander dual
  • dual(MonomialIdeal,RingElement) -- the Alexander dual
  • generators(MonomialIdeal) -- see generators(Ideal) -- the generator matrix of an ideal
  • ideal(MonomialIdeal) -- convert a monomial ideal to an ideal
  • independentSets(MonomialIdeal) -- see independentSets -- some size-maximal independent subsets of variables modulo an ideal
  • isBorel(MonomialIdeal) -- see isBorel -- whether an ideal is fixed by upper triangular changes of coordinates
  • isMonomialIdeal(MonomialIdeal) -- see isMonomialIdeal -- whether something is a monomial ideal
  • isSquareFree(MonomialIdeal) -- see isSquareFree -- whether something is square free monomial ideal
  • lcm(MonomialIdeal) -- least common multiple of all minimal generators
  • Matrix % MonomialIdeal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
  • RingElement % MonomialIdeal -- see methods for normal forms and remainder -- normal form of ring elements and matrices
  • monomialIdeal(MonomialIdeal) -- see monomialIdeal(Ideal) -- monomial ideal of lead monomials of a Gröbner basis
  • numgens(MonomialIdeal) -- see numgens(Ring) -- number of generators of a polynomial ring
  • poincare(MonomialIdeal) -- see poincare -- assemble degrees of a ring, module, or ideal into a polynomial
  • polarize(MonomialIdeal) -- see polarize -- compute the polarization of a monomial ideal
  • Matrix // MonomialIdeal -- see quotient(Matrix,Matrix) -- factor a map through another with the same target
  • RingElement // MonomialIdeal -- see quotient(Matrix,Matrix) -- factor a map through another with the same target
  • standardPairs(MonomialIdeal) -- see standardPairs -- find the standard pairs of a monomial ideal
  • standardPairs(MonomialIdeal,List) -- see standardPairs -- find the standard pairs of a monomial ideal

For the programmer

The object MonomialIdeal is a type, with ancestor classes Ideal < HashTable < Thing.


The source of this document is in Macaulay2Doc/functions/monomialIdeal-doc.m2:57:0.