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Macaulay2Doc :: MonomialIdeal

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, {b, d}}, {a, {b, d}}, {1, {a, c}}, {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.

Functions and methods returning a monomial ideal :

Methods that use a monomial ideal :

For the programmer

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