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Fabio Cuzzolin INRIA Rhone-Alpes, Grenoble, France 7/12/2007

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.

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Fabio Cuzzolin INRIA Rhone-Alpes, Grenoble, France 7/12/2007

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  1. 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

  2. Outline • 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

  3. 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  

  4. 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  

  5. 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

  6. 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 yimplies x covers xy

  7. Independence of atoms • atoms (elements covering 0) of an upper semi-modular lattice form a matroid • matroid (E, I2E) : • I; • AI, A’A then A’I; • A1I, A2I, |A2|>|A1| then x  A2 s.t. A1{x}I. • example: set E of columns of a matrix, endowed with usual linear independence

  8. Three different relations • the independence relation has 3 forms: • {l1,…, ln} I1 if lj  ij 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,ij) j=1,…,n; • {v1,…, vn} I2 if vj  span(li,i<j)= 0 j>1; • {v1,…, vn} I3 if dim(span(li)) = n.

  9. 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() I1 I2 IB

  10. (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

  11. 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

  12. 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

  13. “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  f1F1 ,…, fnFn • IB is FI for some matroid • but this is the trivial matroid!

  14. 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

  15. Example: P4 • example: partition lattice of a frame  = {1,2,3,4} • IB elements are those which do not meet semi-modularity

  16. 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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