On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory by Susanne Saminger-Platz & Radko Mesiar

Author:Susanne Saminger-Platz & Radko Mesiar
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(ii) symmetric if if and only if .

The graph of a multi-function p will be denoted by G(p), i.e., if and only if .

A symmetric multi-function is surjective, i.e., for all there exists an such that if and only if we have for all . The graph of a symmetric, surjective, non-increasing multi-function is a connected line. For any uninorm with continuous underlying functions we denote and let be such that for some if and only if . Then either or or . Note that if then U is non-continuous in (B, 1), if then U is non-continuous in (0, A), and if then U is non-continuous in (0, 1). Further, we have . Now we can introduce the result describing the set of points of discontinuity of a uninorm with continuous underlying functions via a multi-function. Note that in the results on idempotent uninorms and uninorms locally internal on A(e) introduced in the previous section we can see examples of such a multi-function (see also Fig. 1).

Theorem 8

Let be a uninorm with continuous underlying t-norm and t-conorm. Then there exists a symmetric, surjective, non-increasing multi-function r on such that U is continuous on where .

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