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Idempotent uninorms on finite ordinal scales. (English) Zbl 1178.03070
Normally in fuzzy set theory the unit interval \([0,1]\) serves as an evaluation set for the degrees of membership, and triangular norms and conorms are used to represent intersections respectively unions of such fuzzy sets. In practical applications, however, it happens frequently that linguistic variables occur with a finite number of totally ordered values, such as very low, low, medium, high and very high. In this contribution such a finite ordinal scale is considered. Triangular norms and conorms are special instances of uninorms, introduced by J. Dombi, where the neutral element can be any number in the unit interval. So this paper is interesting both from a theoretical as well as a practical point of view. More particularly, the authors give a characterization of a special class of uninorms on a finite ordinal scale, namely the idempotent ones. Remarkable in this characterization is the prominent role of the neutral element of the uninorm. As a corollary of this characterization it is shown that the total number of idempotent uninorms on a set of \(n+1\) elements equals 2 to the power \(n\).
Reviewer: E. Kerre (Gent)

MSC:
03E72 Theory of fuzzy sets, etc.
68T37 Reasoning under uncertainty in the context of artificial intelligence
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