The introduction of belief functions as the most suitable mathematical descriptions of empirical evidence
or subjective states of belief, and the related mechanisms for their combination in a belief revision
process is the most important contribution of the theory of evidence. Another major pillar of evidential
reasoning is the formalization of the idea of structured collection of representations of the external world,
encoded by the notion of 'family of frames'.
We lay the foundations for a rigorous algebraic analysis of the conflict problem, by studying the algebraic
structure of the families of compatible frames as mathematical objects obeying a small number of axioms, originally proposed by Shafer.
We distinguish finite from general
families of frames, describe the monoidal properties of compatible collections of both frames and refinings,
and introduce the internal operation of 'maximal coarsening', which in turns induces in a family of
frames the structures of Birkhoff, upper semimodular and lower semimodular lattice.
We outline a proposal for dealing with possibly conflicting belief functions defined on different compatible frames in an algebraic setting, based
on building a new collection of combinable b.f.s via a 'pseudo Gram-Schmidt'
algorithm. To investigate this possibility, we analyse
the algebraic structure of families of frames and show that they form
upper and lower semimodular lattices, depending on which order relation we pick
We then take a step forward and investigate the relation between Shafer's definition of independence of frames and various extensions of
independence to compatible frames as elements of a semimodular lattice, in order to draw some conclusions on a conjectured algebraic solution to
the conflict problem. We study relationships and differences between the different forms of lattice-theoretical
independence, and understand whether classical independence of frames can be reduced to one of them.
As a result, we pose the notion of independence of frames in a wider context by highlighting its relation with classical independence in modern
algebra. Although independence of frames turns out not to be a cryptomorphic form of matroidal independence, it does exhibit correlations with
several extensions of matroidal independence to lattices, stressing the need for a more general, comprehensive definition of
this widespread and important notion.