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

borel(MonomialIdeal) -- make a Borel fixed submodule
isBorel(MonomialIdeal) -- whether an ideal is fixed by upper triangular changes of coordinates
MonomialIdeal - MonomialIdeal -- monomial ideal difference
dual(MonomialIdeal) -- the Alexander dual of a monomial ideal
independentSets -- some size-maximal independent subsets of variables modulo an ideal
irreducibleDecomposition -- express a monomial ideal as an intersection of irreducible monomial ideals
standardPairs -- find the standard pairs of a monomial ideal

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, {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.

Functions and methods returning a monomial ideal :

MonomialIdeal * 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 ^ ZZ, see Ideal ^ ZZ -- power
monomialIdeal -- make a monomial ideal
MonomialIdeal - MonomialIdeal -- monomial ideal difference
monomialIdeal(Ideal) -- monomial ideal of lead monomials of a Gröbner basis
monomialIdeal(Module), see monomialIdeal(Ideal) -- monomial ideal of lead monomials of a Gröbner basis
monomialIdeal(List), see monomialIdeal(Matrix) -- monomial ideal of lead monomials
monomialIdeal(Matrix) -- monomial ideal of lead monomials
monomialIdeal(RingElement), see monomialIdeal(Matrix) -- monomial ideal of lead monomials
monomialIdeal(String) -- make a monomial ideal using classic Macaulay syntax
MonomialIdeal : MonomialIdeal, see quotient(Ideal,Ideal) -- ideal or submodule quotient
quotient(MonomialIdeal,MonomialIdeal), see quotient(Ideal,Ideal) -- ideal or submodule quotient
radical(MonomialIdeal), see radical -- the radical of an ideal
saturate(MonomialIdeal,MonomialIdeal), see saturate -- saturation of ideal or submodule

Methods that use a monomial ideal :

ZZ % MonomialIdeal, see % -- a binary operator, usually used for remainder and reduction
MonomialIdeal * Module, 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
ZZ // MonomialIdeal, see // -- a binary operator, usually used for quotient
Ideal == MonomialIdeal, see == -- equality
MonomialIdeal == Ideal, see == -- equality
MonomialIdeal == MonomialIdeal, see == -- equality
MonomialIdeal == Ring, see == -- equality
MonomialIdeal == ZZ, see == -- equality
Ring == MonomialIdeal, see == -- equality
ZZ == MonomialIdeal, see == -- equality
associatedPrimes(MonomialIdeal), see associatedPrimes(Ideal) -- find the associated primes of an ideal
betti(MonomialIdeal), see betti(Ideal) -- gives the degrees of generators.
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
MonomialIdeal _ ZZ, see generators of ideals and modules
generators(MonomialIdeal), see generators(Ideal) -- the generator matrix of an ideal
Ideal * MonomialIdeal, see Ideal * Ideal -- product of ideals
MonomialIdeal * Ideal, see Ideal * Ideal -- product of ideals
Ideal + MonomialIdeal, see Ideal + Ideal -- sum of ideals
MonomialIdeal + Ideal, see Ideal + Ideal -- sum of ideals
ideal(MonomialIdeal) -- converts a monomial ideal to an ideal
independentSets(MonomialIdeal), see independentSets -- some size-maximal independent subsets of variables modulo an ideal
irreducibleDecomposition(MonomialIdeal) -- express a monomial ideal as an intersection of irreducible monomial ideals
isBorel(MonomialIdeal), see isBorel -- whether an ideal is fixed by upper triangular changes of coordinates
isIdeal(MonomialIdeal), see isIdeal -- whether something is an ideal
isMonomialIdeal(MonomialIdeal), see isMonomialIdeal -- whether something is a monomial ideal
isSquareFree(MonomialIdeal), see isSquareFree -- whether something is square free monomial ideal
jacobian(MonomialIdeal), see jacobian(Ideal) -- the Jacobian matrix of the generators of an ideal
lcm(MonomialIdeal) -- least common multiple of all minimal generators
Matrix // MonomialIdeal, see Matrix // Matrix -- factor a map through another
RingElement // MonomialIdeal, see Matrix // Matrix -- factor a map through another
Matrix % MonomialIdeal, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
RingElement % MonomialIdeal, see methods for normal forms and remainder -- calculate the normal form of ring elements and matrices
decompose(MonomialIdeal), see minimalPrimes -- minimal associated primes of an ideal
minimalPrimes(MonomialIdeal), see minimalPrimes -- minimal associated primes of an ideal
module(MonomialIdeal), see module(Ideal) -- turn an ideal into a module
numgens(MonomialIdeal), see numgens(Ring) -- number of generators of a polynomial ring
primaryDecomposition(MonomialIdeal), see primaryDecomposition -- irredundant primary decomposition of an ideal
resolution(MonomialIdeal), see resolution(Ideal) -- compute a projective resolution of (the quotient ring corresponding to) an ideal
ring(MonomialIdeal), see ring -- get the associated ring of an object
Ring / MonomialIdeal, see Ring / Ideal -- make a quotient ring
saturate(MonomialIdeal,RingElement), see saturate -- saturation of ideal or submodule
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.