Construction of systems of sets related to the plurality functions

In the present paper we provide a way of construction of all m-elements consistent systems. These kinds of families of sets are the parameters determining the solutions of some functional equation, which express the consistency condition appearing in characterizing the plurality functions. First, we formulate the idea of extending p-elements family to such m-tuples families which are m-elements consistent systems. Then we study some of their properties and we use them in the constructing of extending p-elements families to m-elements consistent systems and all such systems. Finally, we deal with m-elements consistent systems which satisfy an additional condition.