fabio cuzzolin inria rhone-alpes, grenoble, france 7/12/2007
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On the relationship between the notions of independence in matroids, lattices, and Boolean algebras. Fabio Cuzzolin INRIA Rhone-Alpes, Grenoble, France 7/12/2007. 21 st British Combinatorial Conference Reading, UK, July 9-13 2007. Outline. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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(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
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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
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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
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“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!
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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
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Example: P4 example: partition lattice of a frame =
{1,2,3,4}
IB elements are those which do not meet semi-modularity
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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?