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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.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03E72 Theory of fuzzy sets, etc.
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