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

## Synopsis

• Function: isWellDefined
• Usage:
isWellDefined M
• Inputs:
• M, ,
• Outputs:
• , 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 $e \in &nbsp; C1 \cap&nbsp;C2$, there exists $C3 \in &nbsp; C$ with $C \subseteq&nbsp;(C1 \cup&nbsp;C2) - e$, 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