Properties of fuzzy relations and aggregation process in decision making

Abstract

In this contribution connections between input fuzzy relations R₁, . . . ,R_n on a set <em>X</em> and the output fuzzy relation<br />R_F = F(R₁, . . . ,R_n) are studied. F is a function of the form F : [0, 1]ⁿ → [0, 1] and R_F is called an aggregated fuzzy<br />relation. In the literature the problem of preservation, by a function <em>F</em>, diverse types of properties of fuzzy relations<br />R₁, . . . ,R_n is examined. Here, it is considered the converse approach. Namely, fuzzy relation R_F = F(R₁, . . . ,R_n) is<br />assumed to have a given property and then it is checked if fuzzy relations R₁, . . . ,R_n have this property. Moreover, a<br />discussion on the mentioned two approaches is provided. The properties, which are examined in this paper, depend on<br />their notions on binary operations B : [0, 1]² → [0, 1]. By incorporating operation B these properties are generalized<br />versions of known properties of fuzzy relations.