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On locally internal monotonic operations. (English) Zbl 1022.03038

Summary: This paper deals with monotonic binary operations \(F: [0,1]^2 \to[0,1]\) with the property (called locally internal property) that the value of any point \((x,y)\) is always one of its arguments \(x,y\). After stating a theorem that characterizes this kind of operations, some special cases are studied in detail by considering additional properties of the operations: commutativity, existence of a neutral element and associativity. In case of locally internal, associative monotonic operations with neutral element, a characterization theorem gives an improvement of a well-known theorem of Czogala and Drewniak on idempotent, associative and increasing operations with neutral element, as well as an improvement of a characterization theorem for left (and right) continuous, idempotent uninorms.

MSC:

03E72 Theory of fuzzy sets, etc.
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References:

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