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On the distributivity of uninorms over nullnorms. (English) Zbl 1279.03047

A uninorm \(U\) is an associative, commutative and increasing binary operation on the interval \([0,1]\) with a neutral element. Similarly, a nullnorm \(V\) is an associative, commutative and increasing binary operation on the interval \([0,1]\) with an absorbing element \(k\in[0,1]\) such that \(V(0,x)=x\) for all \(x\leq k\) and \(V(1,x)=x\) for all \(x\geq k\). In this paper, the authors investigate the distributivity of uninorms over nullnorms, i.e., the equation \(U(x,V(y,z))=V(U(x,y),U(x,z))\). They consider the case when the uninorm is continuous in \((0,1)^2\) or is representable. It is proved that the absorbing element of the nullnorm is an idempotent element of the uninorm if the distributivity equation holds. Some interesting results are obtained when the nullnorm is continuous.

MSC:

03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
26B99 Functions of several variables
39B99 Functional equations and inequalities
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