isMacaulay -- whether a poset or ring is Macaulay

Usage:

isMacaulay P

isMacaulay S

isMacaulay 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
Outputs:
- a Boolean value,

Description

Suppose $P$ is a ranked poset and $<$ is a total order on the ground set of $P$. Let $\operatorname{Seg}_d n$ denote the largest $n$ elements of rank $d$ with respect to $<$. Suppose that for every integer $d$ between $0$ and the rank of $P$, and for every subset $A$ of the $d$th level of $P$, we have $\lvert\nabla_P\operatorname{Seg}_d\lvert A\rvert\rvert \leq \lvert\nabla_P(A)\rvert$ and $\nabla_P\operatorname{Seg}_d\lvert A\rvert = \operatorname{Seg}_{d+1}\lvert\nabla_P(A)\rvert$. Then, we say $P$ is Macaulay with respect to $<$. A Macaulay poset is a poset for which there exists an order with respect to which it is Macaulay.

There is an analogous property for rings, not to be confused with the Cohen-Macaulay property.

Products of chains are Macaulay by the Clements-Lindstrom Theorem.

i1 : isMacaulay booleanLattice 3

o1 = true

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

o2 = R

o2 : PolynomialRing

i3 : I = monomialIdeal(x^2, y^3, z^3)

                     2   3   3
o3 = monomialIdeal (x , y , z )

o3 : MonomialIdeal of R

i4 : isMacaulay(R/I)

o4 = true

It is possible for a monomial poset to not be Macaulay.

i5 : S = (ZZ/2)[w,x,y,z]

o5 = S

o5 : PolynomialRing

i6 : J = monomialIdeal(w^4, x^2, y^2, z^2, x*w, y*w, z*w)

                     4        2        2        2
o6 = monomialIdeal (w , w*x, x , w*y, y , w*z, z )

o6 : MonomialIdeal of S

i7 : isMacaulay(S/J)

o7 = false

If the poset is Macaulay, the option TikZ can provide a Hasse diagram in which the vertices are horizontally ordered according to a Macaulay order.

i8 : R = (ZZ/2)[x,y]

o8 = R

o8 : PolynomialRing

i9 : I = monomialIdeal(x^2, y^3)

                     2   3
o9 = monomialIdeal (x , y )

o9 : MonomialIdeal of R

i10 : isMacaulay(R/I, TikZ=>true)

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

o10 = true

Caveat

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

Ways to use isMacaulay:

isMacaulay(Ideal)
isMacaulay(Poset)
isMacaulay(QuotientRing)

For the programmer

The object isMacaulay 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.

isMacaulay -- whether a poset or ring is Macaulay

Description

Caveat

See also

Ways to use isMacaulay:

For the programmer