MATROID THEORY NOTES 1. Axiom Systems The axioms for an ...

MATROID THEORY NOTES
MATH 777, SPRING 2008, COOPER
1. Axiom Systems
The axioms for an independence system (E, I):
(I1): ∅ ∈ I.
(I2): If I ∈ I and I ′ ⊆ I, then I ′ ∈ I.
(I3): If I 1 , I 2 ∈ I and |I 1 | < |I 2 |, then ∃e ∈ I 2 − I 1 such that I 1 ∪ e ∈ I.
The axioms for a basis system (E, B):
(B1): B ̸= ∅.
(B2): If B 1 , B 2 ∈ B and x ∈ B 1 − B 2 , then ∃y ∈ B 2 − B 1 such that
(B 1 − x) ∪ y ∈ B.
The axioms for a circuit system (E, C):
(C1): ∅ ∉ C.
(C2): If C 1 , C 2 ∈ C, and C 1 ⊆ C 2 , then C 1 = C 2 .
(C3): If C 1 , C 2 ∈ C, C 1 ≠ C 2 , and e ∈ C 1 ∩ C 2 , then ∃C 3 ∈ C such that
C 3 ⊆ (C 1 ∪ C 2 ) − e.
The axioms for a rank function r : 2 E → Z + :
(R1): For X ⊆ E, 0 ≤ r(X) ≤ |X|.
(R2): If X ⊆ Y ⊆ E, then r(X) ≤ r(Y ).
(R3): If X, Y ⊆ E, then
r(X ∪ Y ) + r(X ∩ Y ) ≤ r(X) + r(Y ).
The axioms for a closure function · : 2 E → 2 E :
(CL1): For X ⊆ E, X ⊆ X.
(CL2): If X ⊆ Y ⊆ E, then X ⊆ Y .
(CL3): If X ⊆ E, then X = X.
(CL4): If X ⊆ E, x ∈ E and y ∈ X ∪ x − X, then x ∈ X ∪ y.
The axioms for a system F ⊂ 2 E of flats:
(F1): E ∈ F.
(F2): If F 1 , F 2 ⊆ F, then F 1 ∩ F 2 ∈ F.
(F3): If F ∈ F and {F 1 , . . . , F k } is the set of minimal members of F that
properly contain F , then the sets F 1 \ F, . . . , F k \ F partition E \ F .
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2 MATH 777, SPRING 2008, COOPER
The axioms for a spanning set system S ⊂ 2 E :
(S1): S ≠ ∅.
(S2): If S 1 ∈ S and S 2 ⊇ S 1 , then S 2 ∈ S.
(S3): If S 1 , S 2 ∈ S and |S 1 | > |S 2 |, then ∃e ∈ S 1 − S 2 such that S 1 − e ∈ S.
2. Translations
The following set of “cryptomorphisms” between definitions of a matroid is strongly
connected as a digraph, and therefore provides a complete (though not always maximally
efficient) translation mechanism between any two.
I → B: B is the set of maximal elements of I.
B → I: I = {I : I ⊆ B, B ∈ B}.
I → C: C is the set of minimal elements of 2 E \ I.
C → I: I = {I : C ⊈ I, ∀C ∈ C}.
r → I: I = {I ⊆ E : r(I) = |I|}.
I → r: r(X) = max{|I| : I ⊆ X, I ∈ I}.
r → · : X = {x ∈ E : r(X ∪ x) = r(X)}.
· → I: I = {X ⊆ E : ∀x ∈ X, x ∉ X − x}.
· → F: F = {X : X ⊆ E}.
F → · : X = ⋂ {F : F ∈ F, F ⊇ X}.
r → S: S = {S : r(S) = r(E)}.
S → B: B = {B ⊆ E : ∀X ⊆ E, B ∪ X ∈ S}.
3. Examples
Let X ⊆ E be any subset throughout the following.
1. Graphic Matroids
E
I
B
C
r(X)
X
E(G)
acyclic sets
spanning forests
cycles
|X|− number of components of X as a subgraph
X∪ any e ∈ E so that X ∪ e has more cycles than X

MATROID THEORY NOTES 3
F no broken cycles, i.e., if C is a cycle of G, then |C \ X| ≠ 1.
S
spanning subgraphs
2. Linear Matroids, aka Vector Matroids
E
I
B
C
r(X)
X
F
S
elements of a (finite) vector space V
independent sets
bases
minimally dependent sets
dim(span(X))
span(X)
subspaces
sets spanning V
3. Algebraic Matroids
E
finite subset of a field extension K/F
I algebraically independent sets (i.e., if X = {x 1 , . . . , x n }, and
f(x 1 , . . . , x n ) = 0 for f ∈ F [t 1 , . . . , t n ], then f ≡ 0).
B
C
r(X)
X
F
S
transcendence bases for F (E), i.e., minimal sets X so that F (E) is
algebraic over F (X)
minimally dependent sets
transcendence degree of F (X)/F
F (X) alg ∩ E, where ·alg denotes algebraic closure
F (X) alg ∩ E, for various X ⊆ E
sets X so that F (E) is algebraic over F (X)
4. Uniform Matroids

4 MATH 777, SPRING 2008, COOPER
E
I
B
C
any finite set
( E
≤k)
( E
k)
( E
)
k+1
r(X) min{|X|, k}
X
F
S
X if |X| < k, E otherwise
( E
)