Arrow-type results under fuzzy preferences based on filter and ultrafilter

In this paper, we determine, for a given t-norm T, the algebraic structures of the set of decisive coalitions of a Fuzzy Aggregation Rule (FAR) defined from the set of profiles of fuzzy T-pre-orders to the set of fuzzy T-pre-orders. More precisely, we show that the set of decisive coalitions is a pre-filter on the finite set of voters when the FAR satisfies Pareto Condition (PC). When the FAR satisfies PC and Independence of Irrelevant Alternatives (IIA), we determine additional conditions on social preferences (range) under which the set of decisive coalitions is respectively a filter and an ultrafilter on the set of voters. From these two results, we outline two fuzzy Arrow-type results: the generator of the filter is an oligarchy (that is a decisive coalition where every member has a veto) for the FAR and the generator of the ultrafilter is a dictator for the FAR. We deduce that the usual FAR, defined in Dutta (1987) [7] and analyzed in Fono and Andjiga (2005) [8], is an oligarchy by showing that its set of decisive coalitions is the filter having the set of voters as unique element.