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Extended triangular norms. (English) Zbl 1165.03010
A triangular norm \(T\) (a non-decreasing commutative associative binary operation on \([0,1]\) with neutral element \(e= 1\)) is extended to act on fuzzy truth values (special fuzzy subsets of \([0,1]\)) based on a (possibly different) t-norm \(T_*\). Analytical forms of extended t-norms \(T\) based on \(T_*\) are discussed for several special cases. First of all, the case \(T=\min\) is discussed. Next, the extended continuous \(T\) based on \(T_*= \min\) is considered. For \(T= T_L\) (Łukasiewicz t-norm given by \(T_L(x,y)= \max(0,x+ y-1)\)), the case of \(T_*\) being a continuous Archimedean t-norm is studied. Also, the case \(T= T_*= \pi\) (product t-norm) is involved. Finally, extensions of continuous triangular norms based on the drastic product t-norm \(T_D\) (\(T_D(x,y)= 0\) whenever \((x,y)\in [0,1[^2\)) is considered. The introduced results may serve to develop new families of type-2 fuzzy logic systems.

03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
Full Text: DOI
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