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.


03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[1] Czogala, E.; Drewniak, J., Associative monotonic operations in fuzzy set theory, Fuzzy Sets and Systems, 12, 249-269 (1984) · Zbl 0555.94027
[2] De Baets, B., Idempotent uninorms, European J. Oper. Res., 118, 3, 631-642 (1999) · Zbl 0933.03071
[3] Fodor, J.; Yager, R.; Rybalov, A., Structure of uninorms, Internat. J. Uncertainty, Fuzziness Knowledge-Based Systems, 5, 411-427 (1997) · Zbl 1232.03015
[4] Martín, J.; Mayor, G., On locally internal aggregation functions, Internat. J. Uncertainty, Fuzziness Knowledge-based Systems, 69, 235-241 (1999) · Zbl 1087.68667
[5] Mayor, G.; Torrens, J., On some classes of idempotent operators, Internat. J. Uncertainty, Fuzziness Knowledge-based Systems, 5, 401-410 (1997) · Zbl 1232.03042
[6] Moser, B.; Tsiporkova, E.; Klement, P., Convex combinations in terms of triangular normsa characterization of idempotent, bisymmetrical and self-dual compensatory operators, Fuzzy Sets and Systems, 104, 97-108 (1999) · Zbl 0928.03063
[7] Yager, R.; Rybalov, A., Uninorm aggregation operators, Fuzzy Sets and Systems, 80, 111-120 (1996) · Zbl 0871.04007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.