Aggregation operators preserving quasiconvexity

Quasiconvexity of a fuzzy set is the necessary and sufficient condition for its cuts to be convex. We study the class of those two variable aggregation operators that preserve quasiconvexity on a bounded lattice, i.e. A(μ, ν) is quasiconvex for quasiconvex lattice valued fuzzy sets μ, ν. The class of all such aggregation operators is characterized by a lattice identity that they have to fulfill. In case of a unit interval we show the construction of aggregation operators preserving quasiconvexity from a pair of real valued functions on the unit interval. As a consequence we get that the intersection of quasiconvex fuzzy sets is quasiconvex if and only if the intersection is based on the minimum triangular norm.