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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

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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 {A t } 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

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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

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

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Independence of atoms atoms (elements covering 0) of an upper semi-modular lattice form a matroid matroid (E, I 2 E ) : 1. I; 2. A I, A A then A I; 3. A 1 I, A 2 I, |A 2 |>|A 1 | then x A 2 s.t. A 1 {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: {l 1,…, l n } I 1 if l j i j l i j=1,…,n; {l 1,…, l n } I 2 if l j i 1; {l 1,…, l n } I 3 if h( i l i ) = i h(l i ). example: vectors of a vector space {v 1,…, v n } I 1 if v j span(l i,i j) j=1,…,n; {v 1,…, v n } I 2 if v j span(l i,i 1; {v 1,…, v n } I 3 if dim(span(l i )) = n.

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Their relations with IB what is the relation of IB with I 1, I 2, I 3 lower semi-modular case L * ( ) analogous results for the upper semi-modular case L( ) IB I1I1 I2I2

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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) = {x E : r(X x)=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 {F 1,…,F n } are FI if each selection {f 1,…, f n } of representatives is independent in M: {f 1,…, f n } I f 1 F 1,…, f n F n 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 I 3 as I 3 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?

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