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.