Algebraic Structures Related to the Combination of Belief Functions

Based on a new demand — the commutativity of belief functions combination with refinement/coarsening of the frame of discernment — the role of the disjunctive rule of combination has increased. To compare the nature of this rule with a more frequent but also more controversional one, i.e. with Dempster’s rule, an algebraic analysis was used. The basic necessary definitions both from the Dempster-Shafer theory and from algebra are recalled. An algebraic investigation of the Dempster’s semigroup — the algebraic structure of binary belief functions with the Dempster’s rule of combination is briefly recalled as well. After this, a new algebraic structure of binary belief functions with the disjunctive rule of combination is defined. The structure is studied, and the results are discussed in a comparison with those ones of the classical Dempster’s rule. In the end, an impact of new algebraic results to the field of decision making and some ideas for future research are presented.