macaulayOrders -- finds all orders with respect to which the poset is Macaulay

Usage:

macaulayOrders P

macaulayOrders S

macaulayOrders I
Inputs:
- P, an instance of the type Poset,
- S, a quotient ring, a homogeneous level linearly independent quotient of a polynomial ring over a field
- I, an ideal, a homogeneous ideal of a polynomial ring such that the quotient is level linearly independent
Optional inputs:
- TikZ => a Boolean value, default value false, whether to print TikZ code
- Visual => a Boolean value, default value false, whether to visualize the Macaulay orders
- AllOrders => a Boolean value, default value true, whether to return all orders instead of at most one order
Outputs:
- a list, whose elements are lists representing orders with respect to which the poset is Macaulay

Description

Given a poset with rank function $r$, this method returns all Macaulay orders $<$ on the poset such that $r(p) < r(q)$ implies $p < q$.

Given a quotient $S$ of a polynomial ring, this method returns Macaulay orders on the monomial poset of $S$.

A total order $<$ on a poset $P$ is represented by a list. It is the permutation of the of the ground set of $P$ that is increasing with respect to $<$.

i1 : macaulayOrders(product(chain 3, chain 3))

o1 = {{{1, 1}, {1, 2}, {2, 1}, {1, 3}, {2, 2}, {3, 1}, {2, 3}, {3, 2}, {3,
     ------------------------------------------------------------------------
     3}}, {{1, 1}, {2, 1}, {1, 2}, {3, 1}, {2, 2}, {1, 3}, {3, 2}, {2, 3},
     ------------------------------------------------------------------------
     {3, 3}}}

o1 : List

The option TikZ can print a Hasse diagram of the poset for each order $<$ such that the vertices in each level are ordered left to right according to $<$.

i2 : R = (ZZ/2)[x,y]

o2 = R

o2 : PolynomialRing

i3 : I = ideal(x^4, x^3-y^3, y^4)

             4   3    3   4
o3 = ideal (x , x  + y , y )

o3 : Ideal of R

i4 : macaulayOrders(R/I, TikZ=>true)

        \draw (0) -- (1);
        \draw (0) -- (2);
        \draw (1) -- (4);
        \draw (1) -- (3);
        \draw (3) -- (8);
        \draw (3) -- (6);
        \draw (6) -- (9);
        \draw (4) -- (7);
        \draw (4) -- (6);
        \draw (7) -- (9);
        \draw (2) -- (5);
        \draw (2) -- (4);
        \draw (5) -- (8);
        \draw (5) -- (7);
\end{tikzpicture}

        \draw (0) -- (2);
        \draw (0) -- (1);
        \draw (2) -- (4);
        \draw (2) -- (5);
        \draw (5) -- (8);
        \draw (5) -- (7);
        \draw (7) -- (9);
        \draw (4) -- (6);
        \draw (4) -- (7);
        \draw (6) -- (9);
        \draw (1) -- (3);
        \draw (1) -- (4);
        \draw (3) -- (8);
        \draw (3) -- (6);
\end{tikzpicture}

                 2        2   2      2   3   2 2              2        2 
o4 = {{1, x, y, x , x*y, y , x y, x*y , y , x y }, {1, y, x, y , x*y, x ,
     ------------------------------------------------------------------------
        2   2    3   2 2
     x*y , x y, y , x y }}

o4 : List

Caveat

Given a QuotientRing or an Ideal, this method will not verify level linear independence.

Ways to use macaulayOrders:

macaulayOrders(Ideal)
macaulayOrders(Poset)
macaulayOrders(QuotientRing)

For the programmer

The object macaulayOrders is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MacaulayPosets.m2:1624:0.

macaulayOrders -- finds all orders with respect to which the poset is Macaulay

Description

Caveat

See also

Ways to use macaulayOrders:

For the programmer