On the relationship between the notions of independence in

matroids, lattices, and Boolean algebras

Fabio Cuzzolin INRIA Rhone-Alpes, Grenoble, France

7/12/2007

21st British Combinatorial Conference

Reading, UK, July 9-13 2007

independence can be defined in different ways in Boolean algebras, semi-modular lattices, and matroids

for the partitions of a finite set , Boolean sub-algebras form upper/lower semi-modular lattices

atoms of such lattices form matroids independence definable there in all those forms they have significant relationships BUT independence of Boolean algebras turns out to

be a form of anti-matroidicity

Outline

Independence of Boolean sub-algebras

a number of sub-algebras {At} of a Boolean algebra B are independent (IB) if

example: collection of power sets of the partitions of a given finite set

application: subjective probability, different knowledge states

tA

Example: P4 example: set P4 of all partitions

(frames) of a set = {1,2,3,4}

it forms a lattice: each pair of elements admits inf and sup

0

coarsening

refinement

An analogy with projective geometry

let us then focus on the collection P() of disjoint partitions of a given set

similarity between independence of frames and ``independence” of vector subspaces

but vector subspaces are (modular) lattices

nniin VVVVspanVvvv ...),...,(0... 111

nniinn AAA ......0)(...)( 1111

Boolean sub-algebras of a finite set as semi-modular lattices

two order relations: 1 2 iff 1 coarsening of 2; 1 * 2 iff 1 refinement of 2;

L() =(P,) upper semi-modular lattice L*() = (P,*) lower semi-modular lattice upper semi-modularity:

for each pair x,y: x covers xy implies xy covers y

lower semi-modularity: for each pair, xy covers y implies x covers xy

Independence of atoms atoms (elements covering 0) of an upper

semi-modular lattice form a matroid

matroid (E, I2E) :1. I; 2. AI, A’A then A’I; 3. A1I, A2I, |A2|>|A1| then x A2 s.t. A1{x}I.

example: set E of columns of a matrix, endowed with usual linear independence

Three different relations the independence relation has 3 forms:

{l1,…, ln} I1 if lj ij li j=1,…,n; {l1,…, ln} I2 if lj i<j li = 0 j>1; {l1,…, ln} I3 if h(i li) = i h(li).

example: vectors of a vector space {v1,…, vn} I1 if vj span(li,ij) j=1,…,n; {v1,…, vn} I2 if vj span(li,i<j)= 0 j>1; {v1,…, vn} I3 if dim(span(li)) = n.

Their relations with IB

what is the relation of IB with I1, I2, I3

lower semi-modular case L*()

analogous results for the upper semi-modular case L()

IBI1 I2

(P,IB) is not a matroid! indeed, IB does not meet the

augmentation axiom 3. of matroids Proof: consider two independent frames

(Boolean subs of 2) A={1,2} pick another arbitrary frame A’ = {3}

trivially independent, 3 1,2

since |A|>|A’| we should form another indep set by adding 1 or 2 to 3

counterexample: 3 = 1 2

L as a geometric lattice a lattice is geometric if it is:

algebraic upper semi-modular each compact element is a join of atoms

classical example: projective geometries compact elements: finite-dimensional subspaces

for complete finite lattices each element is a join of a finite number of atoms: geometric = semi-modular

finite families of partitions are geometric lattices

Geometric lattices as lattices of flats

each geometric lattice is the lattice of flats of some matroid

flat: a set F which coincides with its closure F= Cl(F)

closure: Cl(X) = {xE : r(Xx)=r(X)}

rank r(X) = size of a basis (maximal independent set) of M|X

name comes from projective geometry, again

“Independence of flats” and IB

possible solution for the analogy between vectors and frames

vector subspaces are independent if their arbitrary representatives are, same for frames with respect to their events

formal definition: a collection of flats {F1,…,Fn} are FI if each selection {f1,…, fn} of representatives is independent in M: {f1,…, fn}I f1F1 ,…, fnFn

IB is FI for some matroid but this is the trivial matroid!

IB as opposed to matroidal independence we tried and reduce IB to some form of

matroidal independence in fact, independence of Boolean

algebras (at least in the finite case) is opposed to it

on the atoms of L*() IB collections are exactly those sets of frames which do not meet I3

as I3 is crucial for semi-modularity / matroidicity, Boolean independence works against both

Example: P4 example: partition lattice of a frame =

{1,2,3,4}

IB elements are those which do not meet semi-modularity

Conclusions

independence of finite Boolean sub-algebras is related to independence on lattices in both upper and lower semi-modular forms

cannot be explained as “independence of flats”

is indeed a form of “anti-matroidicity”

extension to general families of Boolean sub-algebras?