The notion of independence in the theory of evidence: An algebraic study

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1 The notion of independence in the theory of evidence: An algebraic study Fabio Cuzzolin a a Perception group INRIA Rhône-Alpes 655, avenue de l Europe SAINT ISMIER CEDEX, France Abstract In this paper we discuss the nature of independence of sources in the theory of evidence from an algebraic point of view, starting from an analogy with projective geometries. Independence in Dempster s rule is equivalent to independence of frames as Boolean algebras. Collection of frames, in turn, can be given several algebraic interpretations in terms of semimodular lattices, matroids, and geometric lattices. Each of those structures are endowed with a particular notion of independence, which we prove to be distinct even though related to independence of frames. We show that the latter is in fact opposed to classical linear independence, giving collection of frames the structure of anti-matroids. Key words: Dempster s rule, family of frames, independence, matroid, semimodular lattice, geometric lattice. 1 Introduction The theory of evidence was born as a contribution to a mathematically rigorous description of the notion of subjective probability. In subjective probability, different observers (or experts ) of the same phenomenon possess in general different notions of what the decision space is. Mathematically, this translates into admitting the existence of several distinct representations of this decision space at different levels of refinement. Evidence will in general be address: Fabio.Cuzzolin@inrialpes.fr (Fabio Cuzzolin). URL: auteur=74 (Fabio Cuzzolin). Preprint submitted to Elsevier Science 13 September 2007

2 available on several of those domains or frames. In order for those experts to reach a consensus on the answer to the considered problem it is necessary for those frames to be mathematically related to each other. This idea is embodied in the theory of evidence by the notion of family of frames. The evidence gathered on distinct frames of the family (corresponding to different persons or sensors) can then be moved to a common frame or common refinement in order to be merged. In this context the notion of independence of frames IF [27] plays an important role. Evidence fusion is guaranteed to take place in all cases if and only if the involved frames are independent [10] in a very precise way which derives from Boolean theory. As Dempster s orthogonal sum assumes the conditional independence of the underlying probabilities generating belief functions through multi-valued mappings [12 14], it is not surprising to realize that combinability (in Dempster s approach) and independence of frames (in Shafer s formulation of the theory of evidence) are strictly intertwined. The study of independence of sources in the ToE can then be reduced to that of independence of frames. 1.1 Previous work The formal definition of evidence combination has been widely studied [41,40] in different mathematical frameworks [31,17]. An exhaustive review would be impossible here. In particular, some work has indeed been done on the issue of merging conflicting evidence [15,19,22,39], specially in critical situations in which the latter is derived from dependent sources [5]. Campos and de Souza [3] presented a method for fusing highly conflicting evidence which overcomes well known counterintuitive results. Liu [23] has recently formally defined when two basic belief assignments are in conflict by means of quantitative measures of both the mass of the combined belief assigned to the emptyset before normalization, and the distance between betting commitments of beliefs. Murphy [24], on her side, studied a related problem: the failure to balance multiple evidence. The notion of conflicting evidence has been widely used in the context of sensor fusion [4]: The matter has also been recently surveyed by Sentz and Ferson [26]. On the other side, though, not much work has been done on the properties of the families of compatible frames and the link with evidence combination. In [29] an analysis of the collections of partitions of a given frame in the context of the hierarchical representation of belief can nevertheless be found, while in [20] both the lattice-theoretical interpretation of families of frames and the meaning of the concept of independence are discussed. In [10] these themes were reconsidered: the structure of Birkhoff lattice of a family of frames was proven, and the crucial equivalence between independence of sources in Dempster s combination and independence of frames highlighted. 2

3 1.2 Contribution Here we build on the results obtained in [10] to complete the algebraic analysis of families of frames and conduct a comparative study of the notion of independence, so central in the theory of evidence, in an algebraic setup. The work is articulated in three steps (see the scheme of Figure 1). First we recall the fundamental result on the equivalence between independence of sources in Dempster s combination and independence of frames. In this incarnation independence of sources can indeed studied from an algebraic point of view, and compared with other classical forms of independence. In a second phase we prove that families of frames are endowed with three different algebraic structures, namely those of: 1. Boolean algebra, 2. semimodular lattice, and 3. geometric lattice. Each of those in turn admits its own particular notion of independence, in a wider sense. In a third step we study relationships and differences between all those different forms, and understand whether IF can be reduced to some major algebraic definition of independence. Our intuition comes from a striking resemblance between independence of frames and independence of vector subspaces. As we prove here this analogy is in fact a consequence of the fact that collections of vectors subspaces of a vector space V (or projective geometries [1]) share with families of frames the algebraic structures of semimodular and geometric lattice. The contribution of this work is then twofold: On one side, we complete the rich algebraic description of families of compatible frames by relating them to semimodular and geometric lattices, and matroids, extending some recent preliminary results [10]. On the other, we pose the notion of independence of frames in a wider context by highlighting its relation with classical independence in modern algebra. The overall picture is intriguing. Even though IF turns out not to be a cryptomorphic form of matroidal independence (being on the contrary related to anti-matroids), it possesses interesting relations with several forms of independence on lattices, pointing out the necessity of a more general, comprehensive definition of this very important notion. 1.3 Paper outline After recalling the fundamental assumption of independence of the bodies of evidence in Dempster s combination rule (Section 2), we show in Section 3 how to reduce it to independence of frames (as Boolean sub-algebras) by means of a recent result. The appealing formal similarity between independence of vectors and independence of frames (Section 4) strongly suggests that this could just be the reflection of a more basic similarity of the corresponding algebraic structures. 3

4 Sections 5, 6, 7, 8, are each devoted to one of those structures, and the associated independence relations. In Section 5 the classical notion of abstract linear independence on a matroid is exposed. Even though families of frames endowed with IF do not form a matroid, matroids are strictly related to other algebraic structures like semimodular and geometric lattices which do describe collections of compatible frames. In Section 6 we prove indeed that families of frames are both upper and lower semimodular lattices, according to the particular order relation we pick. On such structures matroidal independence can be extended, yielding several different relations whose meaning we thoroughly discuss and whose links with IF we highlight in Section 7. In Section 8 we complete the algebraic description of families of frames by proving that they are also geometric lattices. As a lattice is geometric iff is the lattice of flats of some matroid, compatible frames can then be seen as flats of a matroid. We therefore propose a new definition of independence of flats and discuss the possibility that it corresponds to evidential independence. We conclude (Section 9) by showing that the binary frames of a family are independent as Boolean algebras iff they are not independent as elements of the corresponding matroid. In a sense, then, we can claim that collections of independent frames are anti-matroids. Dempster's independence Pr 2 App Boolean sub-algebras IF = IB Cor 1-3, Th 3-13 families of frames Th 2 semimodular lattices independence on lattices Th 15 Pr 4 Th 14 Pr 4 Th 1,16,17 matroids geometric lattices independence of flats Pr 6 matroidal independence Fig. 1. A block diagram of the structure of the paper and its main contributions. 4

5 2 Independence of sources in Dempster s combination Independence of sources plays a central role in the theory of evidence. 2.1 Dempster s rule A basic probability assignment (b.p.a.) over a finite set or frame [27] Θ is a function m : 2 Θ [0, 1] on its power set 2 Θ = {A Θ} such that m( ) = 0, A Θ m(a) = 1, m(a) 0 A Θ. The belief function (b.f.) b : 2 Θ [0, 1] associated with a basic probability assignment m on Θ is defined as b(a) = B A m(b). Belief functions can be combined through an operator called Dempster s orthogonal sum. Definition 1. The orthogonal sum or Dempster s sum of two belief functions b 1, b 2 on Θ is a new belief function b 1 b 2 on Θ with b.p.a. m b1 b 2 (A) = B C=A m b1 (B)m b2 (C) B C m b1 (B)m b2 (C), (1) where m bi denotes the b.p.a. of b i. When the denominator of Equation (1) is zero the two functions are said to be non-combinable, and their orthogonal sum does not exists. 2.2 Multi-valued maps and independence of sources The notion of belief function derives originally from a series of Dempster s works on upper and lower probabilities induced by multi-valued mappings [12 14]. The following sketch of the nature of belief functions is abstracted from [28,32]. Let us consider a problem in which we have probabilities for a question Q 1 and we want to derive a degree of belief for a related question Q 2, Ω and Θ the sets of possible answers to Q 1 and Q 2 respectively. So, given a probability measure P on Ω we seek for the belief b(a) that A Θ contains the correct response to Q 2 (see Figure 2). If we call Γ(ω) the subset of answers to Q 2 compatible with ω Ω, each element ω tells us that the answer to Q 2 is somewhere in A whenever Γ(ω) A. The map Γ : Ω 2 Θ is called a multi-valued mapping from Ω to Θ. The degree of belief b(a) of an event A Θ is then the total probability of all answers ω that satisfy the above condition, namely b(a) = P ({ω Γ(ω) A}). Now let us consider two multi-valued mappings Γ 1, Γ 2 inducing two belief functions over a same frame Θ, Ω 1 and Ω 2 their domains and P 1, P 2 the probability measures over Ω 1 and Ω 2 respectively. If we suppose that the items of evidence 5

6 Ω ω P : Ω > [0,1] Γ Γ(ω) A Θ Θ b : 2 > [0,1] Fig. 2. A probability measure P on Ω induces a belief function b on Θ through a multi-valued mapping Γ. generating P 1 and P 2 independent, we are allowed to build the product space (Ω 1 Ω 2, P 1 P 2 ): detecting two outcomes ω 1 Ω 1 and ω 2 Ω 2 will then tell us that the answer to Q 2 is somewhere in Γ 1 (ω 1 ) Γ 2 (ω 2 ). However, if this intersection is empty the two pieces of evidence are in contradiction. We then need to condition the product measure P 1 P 2 over the set of pairs (ω 1, ω 2 ) whose images have non-empty intersection, namely Ω = {(ω 1, ω 2 ) Ω 1 Ω 2 Γ 1 (ω 1 ) Γ 2 (ω 2 ) }, P = P 1 P 2 Ω, Γ(ω 1, ω 2 ) = Γ 1 (ω 1 ) Γ 2 (ω 2 ). (2) The relation between the new belief function b induced by (2) and the pair of b.f.s being combined is exactly given by Dempster s rule (1). 3 Independence of sources and independence of frames Dempster s mechanism for evidence combination is then intimately connected to the assumption that the domains on which the evidence is present (in the form of a probability measure) are independent. This relationship is mirrored by the notion of independence of compatible frames [27]. 3.1 Families of frames Given two frames Θ and Θ, a map ρ : 2 Θ 2 Θ is a refining if it maps the elements of Θ to a disjoint partition of Θ : ρ({θ}) ρ({θ }) = θ, θ Θ; ρ({θ}) = Θ, θ Θ 6

7 with ρ(a) = θ A ρ({θ}) A Θ. Θ is called a refinement of Θ, Θ a coarsening of Θ. Shafer calls a structured collection of frames a family of compatible frames of discernment ([27], pages : see Appendix for the formal definition). In such a family, in particular, every pair of frames has a common refinement, i.e. a frame which is a refinement of both. Each finite collection of compatible frames has many common refinements. One of these is particularly simple [27]. Proposition 1. If Θ 1,..., Θ n are elements of a family of compatible frames F then there exists a unique common refinement Θ F of them such that θ Θ θ i Θ i for i = 1,..., n such that {θ} = ρ 1 ({θ 1 }) ρ n ({θ n }) (3) where ρ i denotes the refining between Θ i and Θ. This unique frame is called the minimal refinement Θ 1 Θ n of Θ 1,..., Θ n. In the example of Figure 3 we want to find out the position of a target point in an image. We can pose the problem on a frame Θ 1 = {c 1,..., c 5 } obtained c 1 c 2 c 3 c 4 c 5 c 1 c 2 c 3 c 4 c 5 c 1 Θ 1 Θ 3 r 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 r 1 r 2 r 3 r 4 r 5 r 6 r 2 r 3 e 11 e 21 ρ 1 c 11 c 12 Θ 2 r 4 e 31 ρ 3 r 5 r 6 e 41 e 51 e 60 ρ 2 e 1 e 11 e 21 e 31 e 60 Θ c 11 c 12 e 51 e 41 Fig. 3. An example of family of compatible frames. Different discrete quantizations of row and column ranges of an image have as common refinement the set of cells on the left. The refinings ρ 1, ρ 2, ρ 3 between all those frames are shown on the right. by partitioning the column range of the image into 5 intervals, or partition it into 10 intervals, yielding Θ 2 = {c 11, c 12,..., c 51, c 52 }. The row range can also be divided in, say, 6 intervals Θ 3 = {r 1,..., r 6 }. All those frames belong to a family of compatible frames, with the collection of cells Θ = {e 1,..., e 60 } depicted in Figure 3-left as common refinement and refinings shown in Figure 3-right. It is easy to verify that Θ meets condition (3) for the frames Θ 2, Θ 3 as, for example, {e 41 } = ρ 2 (c 11 ) ρ 3 (r 4 ) i.e. Θ is the minimal refinement of Θ 2, Θ 3. 7

8 3.2 Independence of frames Compatible frames admit the notion of independence [27]. Definition 2. Let Θ 1,..., Θ n be elements of a family of compatible frames, and ρ i : Θ i 2 Θ 1 Θ n the corresponding refinings to their minimal refinement. Θ 1,..., Θ n are independent if ρ 1 (A 1 ) ρ n (A n ) (4) whenever = A i Θ i for i = 1,..., n (see Figure 4). Θ 2 Θ 1 Α 2 Θ n Α 1 ρ 2 Α n ρ n ρ 1 Θ 1... Θn Fig. 4. Independence of frames. In particular, it is easy to see that if j [1,.., n] s.t. Θ j is a coarsening of some other frame Θ i, Θ j > 1, Θ 1,..., Θ n are not independent. Mathematically, families of compatible frames are collections of Boolean subalgebras of their common refinement ([30], see Appendix): Equation (4) is nothing but the independence condition for the associated sub-algebras. We denote in the following by IF the independence relation defined by Definition 2. It can be proven that an equivalent condition is [10] Θ 1 Θ n = Θ 1 Θ n (5) i.e. their minimal refinement is their Cartesian product. 3.3 Independence of frames and Dempster s rule Now, independence of frames and Dempster s rule are strictly related [10]. Proposition 2. Let Θ 1,..., Θ n be a set of compatible frames. Then they are independent iff all the possible collections of b.f.s b 1,..., b n defined on Θ 1,..., Θ n, respectively, are combinable over their minimal refinement Θ 1 Θ n. According to Proposition 2, independence of frames and independence of sources (which is at the root of Dempster s combination) are in fact equivalent. 8

9 This is not surprising when we compare the condition under which Dempster s sum is well defined (Equation 2) with independence of frames which reduces to Γ 1 (ω 1 ) Γ 2 (ω 2 ), (ω 1, ω 2 ) Ω 1 Ω 2 ρ 1 (A 1 ) ρ 2 (A 2 ), A 1 Θ 1, A 2 Θ 2 ρ 1 (θ 1 ) ρ 2 (θ 2 ), (θ 1, θ 2 ) Θ 1 Θ 2 (see [10] for a formal proof of Proposition 2). This result is the foundation on which an algebraic study of independence as introduced in the theory of evidence can be erected. 4 An algebraic study of independence 4.1 An analogy with vector subspaces The goal of this paper is to analyze the notion of independence of frames (as Boolean-theoretic incarnation of that of independence of sources) from an algebraic point of view. A powerful source of intuition is an intriguing similarity between independence of frames and independence of vector subspaces, summarized in the following diagram which makes use of Equation (5) v v n 0, v i V i span{v 1,..., V n } = V 1 V n ρ 1 (A 1 ) ρ n (A n ), A i Θ i Θ 1 Θ n = Θ 1 Θ n. (6) While a number of compatible frames Θ 1,..., Θ n are IF iff each selection of their representatives A i 2 Θ i has non-empty intersection, a collection of vectors subspaces V 1,..., V n is independent iff for each choice of vectors v i V i their sum is non-zero. In both cases this condition can be expressed by requiring that a certain composition of those frames/subspaces is equal to their Cartesian product. These two independence notions defined in apparently very different contexts can be formally obtained from each other under the following correspondence of quantities and operators: v i A i, V i 2 Θ i, +, 0, span. As we will see, both families of frames and collections of subspaces of a vector space or projective geometries share the algebraic structures of semimodular 9

10 lattice and geometric lattice. Each of those, in turn, admits its own notion of independence. It is natural to wonder how IF is related to those different forms of independence. 4.2 Outline of the paper The paradigm of abstract independence in modern algebra is the notion of matroid (Section 5). Matroidal independence, though, induces similar relations in other algebraic structures: in particular in those of semimodular and geometric lattice [33]. Even though families of frames are not matroids (5.2), they form semi-modular lattices (Section 6) so that IF inherits interesting relations with some forms of lattice-theoretic independence (Section 7). But families of frames also form geometric lattices (8.1), whose associated independence relation also shows similarities with IF (Section 8). Eventually, we will point out that IF is in fact opposed to matroidal independence (Section 9), a fact which has wider implications for this important notion in both Boolean algebra and subjective probability. 5 Matroids Matroids were introduced by Whitney in the Thirties [38]. He and other authors, among which van der Waerden [36], Mac Lane [21], and Teichmuller [35] recognized that several different concepts of dependence [18,1] in algebra (circuits in graphs, flats in affine geometries) have many properties in common with linear dependence of vectors. In particular, matroids have been an important source for semimodular lattices [2]. Let us start briefly introduce the basic notions of matroid theory [25]. 5.1 Matroids Definition 3. A matroid M = (E, I) is a pair formed by a ground set E, and a collection of independent sets I 2 E, such that: (1) I; (2) if I I and I I then I I; (3) if I 1 and I 2 are in I, and I 1 < I 2, then there is an element e of I 2 I 1 such that I 1 e I. Condition 3) is called augmentation axiom, and is the foundation of the notion of abstract independence in matroid theory. 10

11 The name matroid was coined by Whitney [38] because of a fundamental class of matroids which arise form matrices. The collection of columns of a matrix together with the collection of linearly independent (in the ordinary sense) sets of columns form indeed a matroid, called vector matroid. Consider as an example the matrix with column labels E = {1, 2, 3, 4, 5}. Obviously the collection of independent sets in E is I = {, {1}, {2}, {4}, {5}, {1, 2}, {1, 5}, {2, 4}, {2, 5}, {4, 5}}. It is interesting to see that linearly independent vectors in a vector space actually satisfy the augmentation axiom 3) of Definition 3. Let I 1 and I 2 be linearly independent subsets such that I 1 < I 2. Let W be the subspace spanned by I 1 I 2. Then dim W is at least I 2 (as I 2, the largest of the two collections, is linearly independent). Now suppose that I 1 e is linearly dependent for all e I 2 \ I 1. Then W is contained in the span of I 1, thus I 2 dim W I 1 < I 2 which is a contradiction. 5.2 Families of frames are not matroids It is natural to wonder whether IF could in fact be some form of matroidal independence, i.e. whether for each family F of compatible frames, (F, IF) is a matroid. Unfortunately this is not the case. Theorem 1. A family of compatible frames F endowed with Shafer s independence IF is not a matroid. Proof. In fact, IF does not meet the augmentation axiom 3) of Definition 3. Consider two independent frames I = {Θ 1, Θ 2 }. If we pick another arbitrary frame Θ 3 of the family, the collection I = {Θ 3 } is trivially IF. Suppose Θ 3 Θ 1, Θ 2. Then, since I > I, by augmentation we can form a new pair of independent frames by adding any of Θ 1, Θ 2 to Θ 3. But it is easy to find a counterexample, for instance by picking Θ 3 = Θ 1 Θ 2 (remember the remark after Definition 2). Independence of Boolean sub-algebras is then not independence in matroidal sense. Nevertheless independence exists in other forms defined on other algebraic structures. Two of them, in particular, inherit their own particular notion of independence from that of matroids. 11

12 5.3 Families of frames are related to matroids Both those structures are particular classes of lattice Lattices A partially ordered set or poset is a set P together with a binary relation such that, for all x, y, z in P the following conditions hold: (1) x x; (2) if x y and y x then x = y; (3) if x y and y z then x z. In a poset we say that x covers y (x y) if x y and there is no intermediate element in the chain linking them. A classical example is the power set 2 Θ of a set Θ together with the settheoretic inclusion relation. A poset has finite length if the length of all its chains (collections of consecutive elements) is bounded. Given two elements x, y P of a poset P their least upper bound sup P (x, y) = x y is the smallest element of P that is bigger than both x and y, while their greatest lower bound inf P (x, y) = x y is the biggest element of P that is smaller than both x and y. In the case of L = (2 Θ, ) sup is the usual set-theoretic union, A B = A B, while inf is the usual intersection A B = A B. By induction sup and inf can be defined for arbitrary finite collections too. However, not any pair of elements of a poset, in general, is guaranteed to admit inf and/or sup. Definition 4. A lattice L is a poset in which each pair of elements admits both inf and sup. When each arbitrary (even not finite) collection of elements of L admits both inf and sup, L is said complete. In this case there exist 0 L, 1 L called respectively initial and final element of L. 2 Θ is complete, with 0 = and 1 = {Θ}. The height h(x) of an element x in L is the length of the maximal chain from 0 to x. For the power set 2 Θ, the height of a subset A 2 Θ is simply its cardinality A Independence on semimodular and geometric lattices It is possible to define a matroidal independence relation on the atoms of a lattice, i.e. the elements of the lattice covering 0 (think of one-dimensional subspaces of a vector space V, Figure 5). In particular the lattice has to be semi-modular [33], as we will see in Section 6. For each upper semimodular 12

13 atoms 0 Fig. 5. A lattice can be represented as a (Hasse) diagram in which covering relations are drawn as undirected edges. The atoms of a lattice which initial element 0 (bounded below) are the elements covering 0. lattice L there exists a collection I of sets of atoms such that (A, I) is a matroid. On the other side, matroids are strictly related to geometric lattices, on which they induce a different definition of independence. As we are going to show in the following, families of frames are indeed both semimodular (Section 6) and geometric (Section 8.1) lattices. 6 The lattice of frames Collections of compatible frames (see Appendix) are collections of Boolean sub-algebras of (the power set of) their minimal refinement. In addition they possess the structure of lattice. Two different order relations between frames can be defined. According to the chosen ordering, the resulting lattice will be either upper or lower semimodular. This has two consequences: On one side, as vector subspaces also form a semimodular lattice it explains the similarity we pointed out in Section 4 and stimulated our curiosity (Section 6.4). On the other, it allows to introduce among frames several (pseudo) independence relations derived from matroidal independence, as we will see in Section Families of frames as lattices In a family of compatible frames one can define two distinct order relations on pairs of frames, both associated with the idea of refining (Section 3.1): Θ 1 Θ 2 ρ : 2 Θ 1 2 Θ 2 refining (7) 13

14 (Θ 1 is a coarsening of Θ 2 ), or Θ 1 Θ 2 ρ : 2 Θ 2 2 Θ 1 refining (8) i.e. Θ 1 is a refinement of Θ 2. (8) is clearly the inverse of (7). It makes sense to distinguish them explicitly as they generate two distinct algebraic structures, in turn associated with different extensions of the notion of matroidal independence, as we will see in Section 7. Now, as we have seen in [10], a family of frames F is a poset with respect to both (7) and (8). More precisely, after introducing the notion of maximal coarsening as the largest cardinality common coarsening Θ 1 Θ n of a given collection of frames Θ 1,, Θ n, we can prove that [10] Proposition 3. Both (F, ) and (F, ) where F is a family of compatible frames of discernment are lattices, with respectively i Θ i = i Θ i, i Θ i = i Θ i and i Θ i = i Θ i, i Θ i = i Θ i. 6.2 Upper and lower semimodularity A special class of lattices (modular lattices 3 ) arises from projective geometries, i.e. collections L(V ) of all subspaces of a vector space V. Modular lattices, as many authors have shown, are also related to abstract independence. This quality is retained by a wider class of lattices called semimodular lattices. Definition 5. A lattice L is upper semimodular if for each pair x, y of elements of L, x x y implies x y y. A lattice L is lower semimodular if for each pair x, y of elements of L, x y y implies x x y. Clearly if L is upper semimodular with respect to an order relation, than the corresponding dual lattice with order relation is lower semimodular, as x x y x y y x y x y x y. (9) For lattices of finite length, upper and lower semimodularity imply modularity. In this sense semimodularity is indeed one half of modularity. 6.3 Upper and lower semimodular lattices of frames Families of frames possess indeed the structure of semimodular lattice. Theorem 2. (F, ) is an upper semimodular lattice; (F, ) is a lower semimodular lattice. 3 A lattice L is modular iff if y x, x z = y z, x z = y z then x = y. 14

15 Proof. We just need to prove the upper semimodularity with respect to. Consider two compatible frames Θ, Θ, and suppose that Θ covers their minimal refinement Θ Θ (their inf with respect to ). The proof articulates into the following steps (see Figure 6): as Θ covers Θ Θ we have that Θ = Θ Θ + 1; this means that there exists a single element p Θ which is refined into two elements p 1, p 2 of Θ Θ, while all other elements of Θ are left unchanged: {p 1, p 2 } = ρ(p); this in turn implies that p 1, p 2 each belong to the image of a different element of Θ (otherwise Θ would itself be a refinement of Θ, and we would have Θ Θ = Θ): p 1 ρ (p 1), p 2 ρ (p 2); now, if we merge p 1, p 2 we obviously have a coarsening Θ of Θ : {p 1, p 2} = ρ (p ); but Θ is a coarsening of Θ, too, as we can build the refining σ : Θ 2 Θ : σ(q) = ρ (ρ (q)) where ρ (ρ (q)) is a subset of Θ q Θ : if q = p, σ(q) is {p} (ρ (p 1) \ {p 1 }) (ρ (p 2) \ {p 2 }); if q p, ρ (ρ (q)) is also a set of elements of Θ, as all elements of Θ but p are left unchanged by ρ. as Θ = Θ 1 Θ is the maximal coarsening of Θ, Θ : Θ = Θ Θ ; hence Θ covers Θ Θ, which is the sup of Θ, Θ in (F, ). Θ '' = Θ Θ' p'' Θ σ Q p ρ ρ'(p ') 1 ρ'' p' 1 ρ' p' 2 Θ' p p 1 2 ρ'(p ') 2 Θ Θ' Fig. 6. Proof of the upper semimodularity of (F, ). The lower semimodularity with respect to comes immediately from (9). 15

16 Theorem 2 strengthens the main result of [10], where we proved that finite families of frames are Birkhoff. A lattice is Birkhoff if x y x, y implies x, y x y. (Upper) semimodularity implies the Birkhoff property, but not vice-versa. We will here focus on finite families of frames. Given a set of compatible frames Θ 1,..., Θ n we can consider the set P (Θ) of all partitions of their minimal refinement Θ = Θ 1 Θ n. As the independence condition (Definition 2) involves only partitions of Θ 1 Θ n, we can conduct our analysis there. We denote by L(Θ) =. (P (Θ), ), L (Θ) =. (P (Θ), ) the two lattices associated with the set P (Θ) of partitions of Θ, with order relations (7), (8) respectively Example: the partition lattice P 4 Consider for example the partition lattice associated with a frame of size 4: Θ = {1, 2, 3, 4}, depicted in Figure 7, with order relation. Each edge Θ = 1,2,3,4 1/2,3,4 1,4/2,3 y 1,3,4/2 x 1,2,4/3 1,2,3/4 1,2/3,4 1,3/2,4 y' 1/2/3,4 1/2,4/3 1/2,3/4 x' 1,4/2/3 1,3/2/4 1,2/3/4 1/2/3/4 Fig. 7. The partition (lower) semimodular lattice L (Θ) for a frame Θ of size 4. Partitions A 1,..., A k of Θ are denoted by A 1 /.../A k. Partitions with the same number of elements are arranged on the same level. An edge between two nodes indicates that the bottom partition covers the top one. indicates here that the bottom partition covers the top one. To understand how inf and sup work in the frame lattice, pick the partitions x = {1/2, 3, 4}, x = {1/2, 3/4}. According to the diagram the partition which refines both and has smallest size (x x ) is Θ. Their sup x x is instead x, as x is a refinement of x. If we pick instead the pair of partitions y = {1, 2/3/4} and y = {1, 3/2/4}, we can notice that both y, y cover their inf y y = {1, 2, 3, 4} but in turn 16

17 their sup y y = {1/2/3/4} does not cover them, so (P (Θ), ) is not upper semimodular but lower semimodular. 6.4 Vector subspaces and families of frames We can now reinterpret the analogy we introduced in Section 4 between subspaces of a vector space V and frames of a family of compatible frames of discernment. Both are lattices: according to the chosen order relation we get an upper L(Θ) or lower L (Θ) semimodular lattice (see table) lattice L(V ) L (Θ) L(Θ) initial element 0 {0} 0 F Θ sup l 1 l 2 span(v 1, V 2 ) Θ 1 Θ 2 Θ 1 Θ 2 inf l 1 l 2 V 1 V 2 Θ 1 Θ 2 Θ 1 Θ 2 order relation l 1 l 2 V 1 V 2 Θ 1 coars. of Θ 2 Θ 1 refin. of Θ 2 height h(l 1 ) dim(v 1 ) Θ 1 1 Θ Θ 1 where 0 F is the unique set of a family F with cardinality 1 (whose existence and uniqueness are proved in [11]). As we mentioned in Section 5.3, matroidal independence can actually be introduced on semimodular lattices, possibly providing a rigorous explanation of the formal analogy between independence of vectors and independence of frames IF. In Section 7 we will then extensively explore the relations between IF and several extensions of matroidal independence on both L(Θ) and L (Θ). 7 Independence on lattices and independence of frames 7.1 Atom matroid of a semimodular lattice Consider again the classical example of linear independence of vectors. By definition v 1,..., v n are linearly independent iff α i v i = 0 α i = 0 i. i 17

18 V V This classical definition can though be given several equivalent formulations: I 1 : v j span(v i, i j) j = 1,..., n; I 2 : v j span(v 1,..., v j 1 ) = 0 j = 2,..., n; I 3 : dim(span(v 1,..., v n )) = n. (10) After recalling that one-dimensional subspaces are the atoms of a lattice (L(V )) for which span =, =, dim = h and 0 = 0 (see Section 6.4) we can express the relations (10) for collections of non-zero elements of an arbitrary semimodular lattice with initial element. Definition 6. The following relations on the elements of a semimodular lattice with initial element 0 can be defined: (1) {l 1,..., l n } are I 1 if l j l i l j l i l j j = 1,..., n; i j i j (2) {l 1,..., l n } are I 2 if l j l i = 0 j = 2,..., n; i<j (3) {l 1,..., l n } are I 3 if h( i l i ) = i h(l i ). Graphical interpretations of those relations in terms of Hasse diagrams are given in Figure 8. In the general case I 1, I 2, I 3 are distinct, and none of them originates a V li i=j l 3 l 1 l V 1 l 2 l2 h(l ) 2 l V 1 l 2 h(l ) 1 l 2 l j l i l i l i 1 2 n-1 0 = l 1 l = 2 (l V 1 l ) 2 l 1 h(l ) 1 l 3 0 h(l ) 2 I 1 I 2 I 3 Fig. 8. Graphical interpretation of the relations introduced in Definition 6. matroid (see Appendix). However, when defined on the atoms of an upper semimodular lattice with initial element they do coincide, and form a matroid [34]. 18

19 Proposition 4. The restrictions of the above relations to the set of the atoms A of an upper semimodular lattice L with initial element are equivalent, namely I 1 = I 2 = I 3 = I on A, and (A, I) is a matroid. As the partition lattice has both an upper L(Θ) and lower L (Θ) semimodular form, we can introduce two forms I 1, I 2, I 3 and I 1, I 2, I 3 of the above relations associated with L(Θ) and L (Θ), respectively. These can be seen as different forms of independence in lattice theory. As compatible frames form a semimodular lattice it is natural to suppose that some of those may indeed coincide with Shafer s independence of frames, or at least have some relation with it. 7.2 Boolean and lattice independence in the upper semimodular lattice L(Θ) Form of the relations In L(Θ) the relations introduced in Definition 6 become Θ 1,..., Θ n I 1 Θ j i j Θ i Θ j j = 1,..., n (11) Θ 1,..., Θ n I 2 Θ j i<j Θ i = Θ j = 2,..., n (12) n n Θ 1,..., Θ n I 3 Θ Θ i = ( Θ Θ i ) (13) i=1 i=1 as in the lattice L(Θ) we have Θ i Θ j = Θ i Θ j, Θ i Θ j = Θ i Θ j, h(θ i ) = Θ Θ i, and 0 = Θ. They read as follows: Θ 1,..., Θ n are I 1 iff no frame Θ j is a refinement of the maximal coarsening of all the others. They are I 2 iff j = 2,..., n Θ j does not have a non-trivial common refinement with the maximal coarsening of all its predecessors. The interpretation of I 3 is perhaps more interesting. The latter is equivalent to say that the coarsening that generates n i=1 Θ i can be broken up into n steps of the same length of the coarsenings that generates each of the frames Θ i starting from Θ: First Θ 1 is obtained from Θ by merging Θ Θ 1 elements, then Θ Θ 2 elements of this new frame are merged, and so on until we get n i=1 Θ i. We will return on this when discussing the dual relation on the lower semimodular lattice L (Θ). To study the logical implications between these lattice-theoretic relations and independence of frames, and between themselves, we first need an interesting lemma. Lemma 1. Θ 1,..., Θ n IF, n > 1 n i=1 Θ i = 0 F. 19

20 Proof. We prove Lemma 1 by induction. For n = 2, let us suppose Θ 1, Θ 2 IF. Then ρ 1 (A 1 ) ρ 2 (A 2 ) A 1 Θ 1, A 2 Θ 2, A 1, A 2 (ρ i denotes as usual the refining from Θ i to Θ 1 Θ 2 ). Suppose by absurd that their common coarsening has more than a single element, Θ 1 Θ 2 = {a, b}. But then ρ 1 (ρ 1 (a)) ρ 2 (ρ 2 (b)) = where ρ i denotes the refining between Θ 1 Θ 2 and Θ i, which goes against the hypothesis. Induction step. Suppose that the thesis is true for n 1. Then, since Θ 1,..., Θ n IF implies {Θ i, i j} IF, this implies by inductive hypothesis that Θ i = 0 F j = 1,..., n. i j Of course then, since 0 F is a coarsening of Θ j j, Θ j i j Θ i = Θ j 0 F = 0 F. Let us work our way up by considering increasingly general situations Atomic case The case of the atoms of L(Θ) (the frames of cardinality Θ 1) is in fact trivial. By Proposition 4, I 1 = I 2 = I 3 = I. On the other side, Theorem 3. Collections of atoms of L(Θ) are never IF. Proof. It comes from the fact that their minimal refinement can only be Θ, whose cardinality is way smaller than (n 1)...(n 1) (equivalent condition for IF, Equation (5)). More, as we will prove in Theorem 15, all atoms of L(Θ) are lattice-theoretic independent! For atoms of L(Θ), Boolean independence and lattice-theoretic independence are exactly the opposite, and both trivial relations Pairs of frames Let us next consider the simplified case of collections of just two frames. For n = 2 I 1, I 2, I 3 read respectively as Θ 1 Θ 2 Θ 1, Θ 2, Θ 1 Θ 2 = Θ, Θ + Θ 1 Θ 2 = Θ 1 + Θ 2. (14) 20

21 It is interesting to remark that Θ 1, Θ 2 I 1 Θ 1, Θ 2 Θ. We can prove the following logical implications. Theorem 4. (1) IF I 1 if Θ 1, Θ 2 0 F. (2) I 1 IF. (3) I 2 I 1 iff Θ 1, Θ 2 Θ. (4) I 3 I 1 iff Θ 1, Θ 2 Θ. (5) IF I 3 iff Θ i = 0 F and Θ j = Θ. Proof. (1) If Θ 1, Θ 2 are IF then Θ 1 is not a refinement of Θ 2, and vice-versa, unless one of them is 0 F. But then they are I 1 (Θ 1 Θ 2 Θ 1, Θ 2 ). (2) We can give a counterexample as in Figure 9 in which Θ 1, Θ 2 are I 1 (as Θ 1 Θ 2 Θ Θ 1 2 Fig. 9. Counterexample for the conjecture I 1 IF of Theorem 4. none of them is refinement of the other one) but their minimal refinement Θ 1 Θ 2 has cardinality 4 Θ 1 Θ 2 = 6 (hence they are not IF). (3) Obvious. (4) I 3 I 1 is equivalent to I 1 I 3. But Θ 1, Θ 2 I 1 reads as Θ 1 Θ 2 = Θ i which is equivalent to Θ 1 Θ 2 = Θ j. I.E. Θ 1, Θ 2 I 3 Θ + Θ j = Θ i + Θ j Θ = Θ i. But then Θ 1, Θ 2 I 3 Θ 1, Θ 2 I 1 iff Θ 1, Θ 2 Θ. (5) As Θ 1, Θ 2 are IF, Θ 1 Θ 2 = 1 so that I 3 Θ + 1 = Θ 1 + Θ 2. Now, by definition Θ Θ 1 Θ 2 = Θ 1 Θ 2 (last passage holding as those frames are IF) so that IF and I 3 imply Θ 1 + Θ 2 Θ 1 Θ Θ 1 1 Θ 1 Θ 2 Θ 2 = Θ 2 ( Θ 1 1) Θ 2 1 which holds iff the equality holds i.e. Θ 2 = 0 F, which implies Θ = Θ 1 Θ 2 = Θ 1. 21

22 In the singular case Θ 1 = 0 F, Θ 2 = Θ by Definitions (14) the pair {0 F, Θ} is I 2 and I 3, but not I 1. Besides, two frames can be both I 2 and IF without being singular in the above sense. y, y in Figure 7 provide such an example, as y y = Θ (I 2 ) and they are IF. As it well known that [34] on an upper-semimodular lattice (like L(Θ)) Proposition 5. I 3 I 2. the overall picture formed by the different independence relations for pairs of frames is as in Figure 10. Independence of frames and the most demanding I 1 IF I 2 I 3 Fig. 10. Relations between independence of frames IF and different forms of semimodular independence for pairs of frames in the upper semimodular lattice L(Θ). form of lattice-theoretic independence I 3 are mutual exclusive relations, both stronger than the weakest form I 1. Some of those features are retained by the general case too General case, n > 2 The situation is somehow different in the general case of a collection of n frames. IF and I 1, in particular, turn out to be incompatible. Theorem 5. Θ 1,..., Θ n IF, n > 2 then Θ 1,..., Θ n I 1. Proof. If Θ 1,..., Θ n are IF then any subset of those frames is IF (otherwise we could find empty intersections in Θ 1 Θ n ). But then by Lemma 1 i L {1,...,n} Θ i = 0 F for all subsets L of {1,..., n} with at least 2 elements: L > 1. But then, as L = {i j, i {1,..., n}} has cardinality n 1 > 1 (as n > 2) i j Θ i = 0 F for all j {1,..., n} so that Θ j i j Θ i = Θ j 0 F = Θ j j {1,..., n} and Θ 1,..., Θ n are not I 1. 22

23 Theorem 6. Θ 1,..., Θ n IF, n > 2 then Θ 1,..., Θ n I 2. Proof. If Θ 1,..., Θ n IF then Θ 1,..., Θ k 1 IF for all k = 3,..., n. But by Lemma 1 this implies i<k Θ i = 0 F so that Θ k i<k Θ i = Θ k k > 2. But now, Θ 1,..., Θ n IF with n > 2 implies Θ k Θ k. That holds because, as n > 2 there is at least one frame Θ i in the collection Θ 1,..., Θ n different from 0 F, and clearly {Θ i, Θ} are not IF (as Θ i is a non-trivial coarsening of Θ). Hence Θ k i<k Θ i Θ k > 2 which is in fact a much stronger condition than I 2. A singular case is that in which one of the frames is Θ itself. By Definitions (11) and (12) of I 1 and I 2, if j : Θ j = Θ then Θ 1,..., Θ n I 2 Θ 1,..., Θ n I 1. From Proposition 5 it follows that Corollary 1. Θ 1,..., Θ n IF, n > 2 then Θ 1,..., Θ n I 3. so that putting together the results of Theorems 3,4 and 6 and Corollary 1 we get that IF and I 3 are incompatible in all significant cases. Corollary 2. If Θ 1,..., Θ n are IF then they are not I 3, unless n = 2, Θ 1 = 0 F and Θ 2 = Θ. We will comment on those results after having discussed the lower semimodular case. 7.3 Boolean and lattice independence in the lower semimodular lattice L (Θ) Analogously, the candidate independence relations associated with the lower semimodular lattice L (Θ) read as Θ 1,..., Θ n I 1 Θ j i j Θ i Θ j j = 1,..., n (15) Θ 1,..., Θ n I2 j 1 Θ j Θ i = 0 F j = 2,..., n (16) i=1 Θ 1,..., Θ n I3 n n Θ i 1 = ( Θ i 1) (17) i=1 i=1 as Θ i Θ j = Θ i Θ j, Θ i Θ j = Θ i Θ j, h (Θ i ) = Θ i 1, and 0 = 0 F. The frames Θ 1,..., Θ n are I1 iff none of them is a coarsening of the minimal refinement of all the others: In other words, there is no proper subset of 23

24 {Θ 1,..., Θ n } which has still Θ 1... Θ n as common refinement. Θ 1,..., Θ n are I2 iff j > 1 Θ j does not have a non-trivial common coarsening with the minimal refinement of its predecessors. Again, the third form I3 is the more remarkable. On its side, it has a very interesting semantics in terms of probability spaces: As the dimension of the polytope of probability measures definable on a domain of size k is k 1, Θ 1,..., Θ n are I3 iff the dimension of the probability polytope for the minimal refinement is the sum of the dimensions of the polytopes associated with the individual frames. Θ 1,..., Θ n I 3 dimp n i=1 Θ i = i dim P Θi. (18) Accordingly, as I 3 can be written as Θ n n Θ i = ( Θ Θ i ) dim P Θ dim P n i=1 Θ = i i=1 i=1 i (dim P Θ dim P Θi ) its twin relation can also be interpreted by saying that the difference of the dimensions of the probability simplices on Θ and n i=1 Θ i is the sum of the individual differences. On the other side, it is very interesting to point out a stunning analogy between independence of frames and I 3 : While condition (5) for IF Θ 1 Θ n = Θ 1 Θ n says that the minimal refinement is the Cartesian product of the individual frames, Equation (18) for I3 basically states that the probability simplex of the minimal refinement is a Cartesian product of the individual ones. We will consider their relationship in more detail in the last part of the paper General case In this case it is easier to describe first the general framework, and later prove stronger statements holding in specific situations. Theorem 7. Θ 1,..., Θ n IF and Θ j 0 F j then Θ 1,..., Θ n I 1. Proof. Let us suppose that Θ 1,..., Θ n are IF but not I 1, i.e. j : Θ j coarsening of i j Θ i. We need to prove that A 1 Θ 1,..., A n Θ n s.t. ρ 1 (A 1 ) ρ n (A n ) = where ρ i denotes the refining from Θ i to Θ 1 Θ n. Since Θ j is a coarsening of i j Θ i then there exists a partition Π j of i j Θ i associated with Θ j, and a refining ρ from Θ j to i j Θ i. 24

25 As {Θ i, i j} are IF, for all θ i j Θ i there exist θ i Θ i, i j s.t. {θ} = i j ρ i(θ i ) where ρ i is the refining to i j Θ i. Now, θ belongs to a certain element A of the partition Π j. By hypothesis (Θ j 0 F j) Π j contains at least two elements. But then we can choose θ j = ρ 1 (B) with B another element of Π j. In that case we obviously get ρ j (θ j ) i j ρ i (θ i ) = which implies that {Θ i, i = 1,..., n} IF against the hypothesis. Does IF imply I 1 even when Θ i = 0 F? The answer is negative. Θ 1,..., Θ n I 1 means that i s.t. Θ j is a coarsening of i j Θ i : But if Θ i = 0 F then Θ i is a coarsening of i j Θ i. The reverse implication dose not hold: IF and I 1 remain distinct. Theorem 8. Θ 1,..., Θ n I 1 Θ 1,..., Θ n IF. Proof. We need a simple counterexample. Consider two frames Θ 1 and Θ 2 in which Θ 1 is not a coarsening of Θ 2 (Θ 1, Θ 2 are I 1). Then Θ 1, Θ 2 Θ 1 Θ 2 but it easy to find an example (see Figure 11) in which Θ 1, Θ 2 are not IF. Θ 2 Θ 1 ρ 2 ρ 1 Θ 1 Θ 2 Fig. 11. A counterexample to I 1 IF. Besides, like in the upper semimodular case I 2 does not imply I 1. Theorem 9. Θ 1,..., Θ n I 2 Θ 1,..., Θ n I 1. Proof. Figure 12 shows a counterexample to the conjecture I2 I1. Given Θ 1 Θ j 1 and Θ j, one possible choice of Θ j+1 s.t. Θ 1,..., Θ j+1 are I2 but not I1 is shown. IF is a stronger condition than I 2 also. Theorem 10. Θ 1,..., Θ n IF Θ 1,..., Θ n I 2. 25

26 Θ Θ j Θ j Θ j Θ Θ j Θ Θ Θ j j Fig. 12. A counterexample to I 2 I 1. Proof. We first need to show that Θ 1,..., Θ n are IF iff j = 1,..., n the pair {Θ j, i j Θ i } is IF. As a matter of fact (5) can be written as Θ j Θ i = Θ j ( i j Θ i ) {Θ j, Θ i } IF. i j i j But then by Lemma 1 we get as desired. It follows from Theorems 7 and 10 that, unless some frame is unitary, Corollary 3. IF I 1 I 2. i.e. independence of frames is a more demanding requirement than both the first two forms of lattice-theoretic independence. The converse is however false. Think of a pair of frames (n = 2), for which Θ 1 Θ 2 Θ 1, Θ 2 (I 1), Θ 1 Θ 2 = 0 F (I 2). Now, those conditions are met by the counterexample of Figure 11, in which the two frames are not IF Pairs of frames We can indeed maintain something stronger when considering only pairs of frames. For n = 2 I 1, I 2, I 3 read respectively as Θ 1 Θ 2 Θ 1, Θ 2, Θ 1 Θ 2 = 0 F, Θ 1 Θ 2 = Θ 1 + Θ 2 1. (19) Unlike the general case, for pairs I 2 does not imply I 1. 26

27 Theorem 11. If Θ 1, Θ 2 0 F then Θ 1, Θ 2 I 2 implies Θ 1, Θ 2 I 1. Proof. Obvious by Equation (19). If one of the frames is unitary, all independence conditions hold but I 1. Theorem 12. If Θ j = 0 F j {1, 2} then Θ 1, Θ 2 I 2, I 3, IF, I 1. Proof. If for instance Θ 2 = 0 F then by (19) Θ 1 0 F = 0 F = Θ 2 and Θ 1, Θ 2 are not I 1 while they are I 2. As Θ 1 Θ 2 = 0 F Θ 1 = Θ 1, Θ 2 = 1 they are I 3 again by (19). Finally, their are IF as Θ 1 Θ 2 = Θ 1 = 1 Θ 1 = Θ 2 Θ 1 (according to Equation (5)). Can we have non-unitary pairs of frames which are both I 3 and I 1? Figure 9 provides such an example: as Θ 1 Θ 2 = 4 = Θ 1 + Θ 2 1 = the two frames are I 3, none of them is refinement of the other, and they are both non-unitary Atomic case The atoms of the lattice L (Θ) are nothing but the binary partitions of Θ, i.e. {Ω L (Θ) : Ω = 2}. For them, both I 1, I 2 are trivial. Theorem 13. If Θ 1,..., Θ n A then Θ 1,..., Θ n are both I 1 and I 2. Proof. If Θ j A j then Θ j i j Θ i = 0 F Θ j j and Θ 1,..., Θ n are I 1. But then by Definition 16 Θ 1,..., Θ n are also I Comments Figure 13 illustrates what we have learned about the relations between independence of frames and the various forms of lattice-theoretic independence in both the upper (left) and lower (right) semimodular lattice of frames. Only the general case of a collection of more than two non-atomic frames is shown for sake of simplicity: special cases (Θ i = 0 F for L (Θ), Θ i = Θ for L(Θ)) are also neglected. In the upper semimodular case, minding the special case in which one of the frames is Θ itself, independence of frames IF is mutually exclusive with all lattice-theoretic relations I 1, I 2, I 3 (Theorems 5, 6 and Corollary 1) unless we consider two non-atomic frames, for which IF I 1 (Theorem 4.1). In fact they are the negation of each other in the case of atoms of L(Θ) (frames of size n 1), when I = I 1 = I 2 = I 3 is trivially true for all frames, while 27

28 I 1 I 3 * IF * I 2 I I IF 1 3 I 2 * Fig. 13. Left: Relations between independence of frames IF and all different forms of semimodular independence on the upper semimodular lattice L(Θ). Right: Relations on the lower semimodular lattice L (Θ). IF is never met (Theorem 3). The relation between I 1 and I 2, I 3 is not yet understood, but we know that the latter imply the former when dealing with pairs. In the lower semimodular case IF is a stronger condition than both I 1 and I 2 (Theorems 7, 10) which are indeed trivial for binary partitions of Θ (Theorem 13). On the other side, as we will prove in Section 5, and notwithstanding the analogy coming from Equation (18), IF is mutually exclusive with the third independence relation even in its lower semimodular incarnation. Some common features emerge: the first two forms of lattice independence are always trivially met by atoms of the related lattice. More, independence of frames and the third form of lattice independence are mutually exclusive in either case. The lower semimodular case is clearly more interesting, though. On L(Θ) independence of frames and lattice-theoretic independence are basically unrelated (see Figure 13-left). Their lower-semimodular counterparts, instead, have meaningful links with IF even though distinct from it. The knowledge of which collections of frames are I 1, I 2 and I 3 tells us much about IF frames, as collections of Boolean independent frames are in I 1 I 2 I 3. We know that IF is strictly included in I1 I2 (Section 7.3.1), but the possibility that independence of frames may indeed coincide with I1 I2 I3 is still to be explored. 8 The lattice of frames as a geometric lattice of flats Shafer s independence of frames is then distinct from but related to latticetheoretic independence. This, though, does not fully explain the analogy we discussed in Section 4 between the concepts of independence in the theory of 28

29 evidence and linear algebra. However, a different explanation of this formal similarity is possible in terms of a different algebraic structure. As we anticipated in Section 5.3, matroids are strictly related to another class of lattices which comes from projective geometric: geometric lattices. As we show here, families of frames are indeed geometric lattices. This provides an explanation of the similarity between projective geometries and families of frames, both in terms of their algebraic structures and the associated independence relation, which is alternative to that in terms of semi-modular lattices we thoroughly analyzed in Section Families of frames as geometric lattices Geometric lattices are based on an abstract definition of the notion of compactness in terms of joins of a lattice. Definition 7. An element p of a lattice L is called compact iff if there exists a subset S L which admits a join ( S) which covers p (p S), then there exists a finite subset F S such that p F. A lattice L is called algebraic if: L is complete (admits 0 = L and 1 = L); each element p of L is a join of compact elements. L is called geometric if: it is algebraic; it is upper semimodular; each compact element of L is a join of atoms: p L a 1,, a m A such that p = i a i. The name geometric lattice comes in fact from the familiar example of projective geometries. Projective geometries are complete lattices L(V ). Compact elements in L(V ) are exactly the finite-dimensional subspaces of V. Each finite dimensional subspace is a span of one-dimensional subspaces or vectors (the atoms of L(V )). When a complete lattice L is finite, each of its elements are joins of a finite number of atoms. In this case, geometricity reduces to semimodularity. But then it follows immediately that Theorem 14. L(Θ) is a geometric lattice. L (Θ) is not a geometric lattice. Theorem 14 gives in fact an alternative formal explanation of the parallelism between L(V ) and L(Θ). As their semi-modularity does, their geometricity also allows to define a different form of independence inherited but different from matroidal independence. 29