Families of compatible frames of discernment as semimodular lattices

Shafer’s mathematical theory of evidence is one of the most remarkable attempts to introduce a complete generalized probability theory. One of its features is the central role of the notion of different level of knowledge of a given phenomenon, embodied into the idea of compatible frames of discernment . In this work we are going to analyze the algebraic basement of the concept of family of frames F = {S,R}. The double structure composed by frames S = {Θ} and refinings R = {ω : 21 → 22 , Θi ∈ S} is discussed and the correspondence between them analyzed. We will distinguish among complete, finite generated and general families of frames. The monoidal and modular structure of these collections is proved by discussing the properties of the internal operation of minimal refinement ⊗. By recalling the equivalence lattice of all the partitions of a set, we will introduce the lattice structure by defining a dual operation called maximal coarsening ⊕ and proving its properties. New axioms are introduced in order to reflect the principle of duality and give a constructive form to the theory. The equivalence among these different versions is easily proved. In addiction, the assumption of finite knowledge is shown to be the counterpart of finite generated families. The analogy between the projective space of the linear subspaces of a finite dimensional vector space V and the notion of family of frames is discussed, with a particular attention to the idea of independence. Starting from the analysis of the linear independence relation among atoms of a semimodular lattice we will introduce a relation {p1, ..., pn} LI ⇔ h(p1 ∨ · · · ∨ pn) = h(p1) + ... + h(pn), p1, ..., pn ∈ L (where h(x) is the rank of x ∈ L) that generalizes to arbitrary elements of a Birkhoff lattice bounded below L. The equivalence between internal independence of a collection of frames Θ1, ..., Θn as Boolean subalgebras of their minimal refinement Θ1⊗ · · · ⊗Θn and their external independence as elements of a Birkhoff lattice bounded below is proved.