isWellDefined(Matroid) -- whether the input is a well-defined matroid

Synopsis

Function: isWellDefined
Usage:

isWellDefined M
Inputs:
- M, a matroid,
Outputs:
- a Boolean value, whether or not a set of subsets satisfies the circuit elimination axiom

Description

If E is a set and C is a collection of subsets of E such that (i) no two elements of C are comparable, and (ii): for C1, C2 in C and $e \in C1 \cap C2$, there exists $C3 \in C$ with $C \subseteq (C1 \cup C2) - e$, then C is the set of circuits of a matroid on E. Property (ii) is called the circuit elimination axiom, and these characterize the collections of subsets of E which can be circuits for a matroid on E. This method verifies if the circuit elimination axiom holds for the given input, and additionally whether the input has the correct keys and data types that an object of type Matroid has.

i1 : isWellDefined matroid({a,b,c,d},{{a,b},{c,d}})

o1 = false

i2 : isWellDefined matroid({a,b,c,d},{{a,b},{a,c}})

o2 = true

i3 : isWellDefined matroid({{1,2,3},{1,4,5},{2,3,4,5},{2,3,6,7},{4,5,6,7}}, EntryMode =>"circuits") -- the Escher "matroid"

o3 = false

i4 : isWellDefined matroid({{1,2,3},{1,4,5},{1,6,7},{2,3,4,5},{2,3,6,7},{4,5,6,7}}, EntryMode =>"circuits")

o4 = true

i5 : isWellDefined matroid random(ZZ^3, ZZ^5)

o5 = true

i6 : isWellDefined matroid completeGraph 4

o6 = true

i7 : isWellDefined uniformMatroid(4, 5)

o7 = true

A theorem of Terai and Trung states that a monomial ideal is the Stanley-Reisner ideal for (the independence complex of) a matroid iff all symbolic powers is Cohen-Macaulay (indeed, this happens iff the 3rd symbolic power is Cohen-Macaulay). This can be verified as follows:

i8 : R = QQ[x_0..x_3]

o8 = R

o8 : PolynomialRing

i9 : I = monomialIdeal(x_0*x_1, x_0*x_2, x_3)

o9 = monomialIdeal (x x , x x , x )
                     0 1   0 2   3

o9 : MonomialIdeal of R

i10 : isWellDefined matroid I

o10 = false

i11 : symbolicCube = intersect apply(irreducibleDecomposition I, P -> P^3)

                      3 3   3 2     3   2   3 3   2 2     2         2 2   
o11 = monomialIdeal (x x , x x x , x x x , x x , x x x , x x x x , x x x ,
                      0 1   0 1 2   0 1 2   0 2   0 1 3   0 1 2 3   0 2 3 
      -----------------------------------------------------------------------
           2       2   3
      x x x , x x x , x )
       0 1 3   0 2 3   3

o11 : MonomialIdeal of R

i12 : (codim symbolicCube, pdim betti res symbolicCube)

o12 = (2, 3)

o12 : Sequence

Ways to use this method: