One of the major ideas of Shafer's mathematical theory of evidence is the introduction of uncertainty descriptions on different representation domains of phenomena, called families of compatible frames of discernment. Here we are going to analyze these families of frames from an algebraic point of view, study the properties of minimal refinements of collections of domains and introduce the internal operation of maximal coarsening to establish the structure of semimodular lattice. Motivated by the search for a solution of the conflict problem that arises in sensor fusion applications, we will show the connection between classical independence of frames as Boolean subalgebras and independence of frames as elements of a locally finite Birkhoff lattice. This will eventually suggest a potential algebraic solution of the conflict problem.