Convex linear \(T\)-\(S\) functions: a generalization of Frank’s equation. (English) Zbl 1284.39019
Summary: The main purpose of this paper is to solve the functional equation \(T+\lambda S=\mathrm{Min}+\lambda\mathrm{Max}\), for \(0\leq\lambda\leq\infty\) and a pair \((T,S)\) of a t-norm \(T\) and a t-conorm \(S\). This equation arises when we consider a convex linear combination of a t-norm and a t-conorm, and set out the problem of finding the intersection of the segments determined by the pairs \((\mathrm{Min},\mathrm{Max})\) and \((T,S)\).
MSC:
| 39B22 | Functional equations for real functions |
| 03E72 | Theory of fuzzy sets, etc. |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 68T37 | Reasoning under uncertainty in the context of artificial intelligence |
Keywords:
aggregation function; associativity; convex linear combination; Frank’s equation; copula; dual copula; triangular norm; triangular conormReferences:
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