Uninorms, as binary operations on the unit interval, have been widely applied in information aggregation. The class of almost equitable uninorms appears when the contradictory information is aggregated. It is proved that among various uninorms of which either underlying t-norm or t-conorm is continuous, only the representable uninorms belong to the class of almost equitable uninorms. As a byproduct, a characterization for the class of representable uninorms is obtained.
Diverse classes of fuzzy relations such as reflexive, irreflexive, symmetric, asymmetric, antisymmetric, connected, and transitive fuzzy relations are studied. Moreover, intersections of basic relation classes such as tolerances, tournaments, equivalences, and orders are regarded and the problem of preservation of these properties by n-ary operations is considered. Namely, with the use of fuzzy relations R1,…,Rn and n-argument operation F on the interval [0,1], a new fuzzy relation RF=F(R1,…,Rn) is created. Characterization theorems concerning the problem of preservation of fuzzy relations properties are given. Some conditions on aggregation functions are weakened in comparison to those previously given by other authors.
Homogeneity, as one type of invariantness, means that an aggregation function is invariant with respect to multiplication by a constant, and quasi-homogeneity, as a relaxed version, reflects the original output as well as the constant. In this paper, we characterize all homogeneous/quasi-homogeneous n-ary aggregation functions and present several methods to generate new homogeneous/quasi-homogeneous n-ary aggregation functions by aggregation of given ones.
Fuzzy logic has been used for flexible database querying for more than 30 years. This paper examines some of the issues of flexible querying which seem to have potential for further research and development from theoretical and practical points of view. More precisely, defining appropriate fuzzy sets for queries, calculating matching degrees for commutative and non-commutative query conditions, preferences, merging constraints and wishes, empty and overabundant answers, and views on practical realizations are discussed in this paper. Suggestions how to solve them and integrate into one compact solution are also outlined in this paper.