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.