Weak and graded properties of fuzzy relations in the context of aggregation process

We study graded properties ( α -properties) of fuzzy relations, which are parameterized versions of properties of a fuzzy relation defined by L.A. Zadeh. Namely, we take into account fuzzy relations which are: α -reflexive, α -irreflexive, α -symmetric, α -antisymmetric, α -asymmetric, α -connected, α -transitive, where α ∈ [ 0 , 1 ] . We also pay our attention to the composed versions of these basic properties, e.g. an α -equivalence, α -orders. We also consider the so-called “weak” properties of fuzzy relations which are the weakest versions of the standard properties of fuzzy relations. We take into account the same types of properties as in the case of the graded ones. Using functions of n variables we consider an aggregated fuzzy relation of given fuzzy relations. We give conditions for functions to preserve graded and weak properties of fuzzy relations.