MonomialOrder -- monomial ordering

Description

MonomialOrder -- an optional argument used with polynomial rings and monoids to indicate a monomial ordering other than the default (graded reverse lexicographic).

In Macaulay2, each polynomial ring (and also each monoid) is equipped with a monomial order, which is used for display of polynomials (terms are listed in descending monomial order), and also for Gröbner basis computations.

In the most general setting, a monomial ordering is given by a list of ordering tests, of various types listed and described below, each of which provides a partial ordering on the monomials. The ordering tests are applied sequentially, starting with the first one, until one monomial is judged greater than the other. At the end, if necessary, the graded reverse lexicographic order is used to compare the monomials. For examples, see below, or see monomial orderings.

Permissible elements:

GRevLex => n -- A graded reverse lexicographic block of variables
Lex => n
Weights => {...}
Position => Up or Position => Down
RevLex => n
GroupLex => n
GroupRevLex => n
MonomialSize => n, n being 8,16,32, or 64. Set the packing size for exponents for further variables

Some examples of monomial orders. Note that if only one item is in the list, we can dispense with the list.

MonomialOrder => {GRevLex=>2, GRevLex=>3} -- a product order
MonomialOrder => {2, 3} -- same
MonomialOrder => {Weights=>{1,13,6,2}} -- a weight order
MonomialOrder => Weights=>{1,13,6,2} -- same

If any monomials will be less than 1 in the ordering, then the option Global => false should be used.

i1 : QQ[x,y, Weights => {-1,1}, Global => false]

o1 = QQ[x..y]

o1 : PolynomialRing

i2 : x<1

o2 = true

i3 : y<1

o3 = false

Functions with optional argument named MonomialOrder :

graphIdeal(...,MonomialOrder=>...) -- see graphIdeal(RingMap) -- the ideal of the graph of the regular map corresponding to a ring map
graphRing(...,MonomialOrder=>...) -- see graphRing(RingMap) -- the coordinate ring of the graph of the regular map corresponding to a ring map
monoid(...,MonomialOrder=>...) -- specify the monomial ordering
newRing(...,MonomialOrder=>...) -- see newRing -- make a copy of a ring, with some features changed
pushForward(...,MonomialOrder=>...) -- see pushForward(RingMap,Module)
rsort(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
sort(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
sortColumns(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
symmetricAlgebra(...,MonomialOrder=>...) -- see symmetricAlgebra -- the symmetric algebra of a module
tensor(Monoid,Monoid,MonomialOrder=>...) -- see tensor(Monoid,Monoid) -- tensor product of monoids
tensor(Ring,Ring,MonomialOrder=>...) -- see tensor(Monoid,Monoid) -- tensor product of monoids

For the programmer

The object MonomialOrder is a symbol.

MonomialOrder -- monomial ordering

Description

See also

Functions with optional argument named MonomialOrder :

For the programmer