B-properties of fuzzy relations in aggregation process — the “converse problem”

In this paper the problem of connections between input fuzzy relations Ri, …, R„ on a set X and the output fuzzy relation Rf = F (Ri, …, R„) on X is studied, where F is a function of the type F : [0,1]ⁿ → [0,1] and RF is an aggregated fuzzy relation. Namely, fuzzy relation RF = F(R1, …, Rn) is assumed to have a given property and the properties of fuzzy relations Ri, …, R„ are examined. This approach to checking connections between input fuzzy relations and the output fuzzy relation is a new one. In the literature the problem of preservation by an aggregation function F diverse types of properties of fuzzy relations Ri, …, R„ is examined. The properties, which are examined in this paper, depend on their notions on binary operations B : [0,1]² → [0,1], i.e. they are generalized versions of known properties of fuzzy relations.