Idempotent uninorms. (English) Zbl 0933.03071

De Baets, Bernard

Eur. J. Oper. Res. 118, No. 3, 631-642 (1999).

Summary: Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. Two broad classes of idempotent uninorms are fully characterized: the class of left-continuous ones and the class of right-continuous ones. In particular, the important subclasses of conjunctive left-continuous idempotent uninorms and of disjunctive right-continuous idempotent uninorms are characterized by means of super-involutive and sub-involutive decreasing unary operators. As a consequence, it is shown that any involutive negator gives rise to a conjunctive left-continuous idempotent uninorm and to a disjunctive right-continuous idempotent uninorm.

MSC:

03E72

Theory of fuzzy sets, etc.

Keywords:

fuzzy sets; t-norms; t-conorms; idempotent uninorms; involutive negator

References:

[2]	Czogała, E.; Drewniak, J., Associative monotonic operations in fuzzy set theory, Fuzzy Sets and Systems, 12, 249-269 (1984) · Zbl 0555.94027
[4]	Dombi, J., Basic concepts for the theory of evaluation: The aggregative operator, European Journal of Operational Research, 10, 282-293 (1982) · Zbl 0488.90003
[5]	Dombi, J., A general class of fuzzy operators, the de Morgan classs of fuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets and Systems, 8, 149-163 (1982) · Zbl 0494.04005
[6]	Esteva, F.; Domingo, X., Sobre funciones de negación en [0,1], Stochastica, 4, 141-166 (1980)
[8]	Fodor, J.; Yager, R.; Rybalov, A., Structure of uninorms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5, 411-427 (1997) · Zbl 1232.03015
[10]	Klement, E.-P.; Mesiar, R.; Pap, E., On the relationship of associative compensatory operators to triangular norms and conorms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 4, 129-144 (1996) · Zbl 1232.03041
[13]	Ling, C., Representation of associative functions, Publications Mathematicae Debrecen, 12, 189-212 (1965) · Zbl 0137.26401
[14]	Mayor, G.; Torrens, J., On some classes of idempotent operators, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5, 401-410 (1997) · Zbl 1232.03042
[15]	Mesiar, R., Choquet-like integrals, Journal of Mathematical Analysis and Applications, 194, 477-488 (1995) · Zbl 0845.28010
[17]	Mesiar, R.; Rybárik, J., Pan-operations structure, Fuzzy Sets and Systems, 74, 365-369 (1995) · Zbl 0859.28010
[20]	Trillas, E., Sobre funciones de negación en la teoriá de conjuntos diffusos, Stochastica, 3, 47-59 (1979)
[21]	Wangming, W., Commutative implications on complete lattices, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2, 333-341 (1994) · Zbl 1232.03022
[22]	Yager, R.; Rybalov, A., Uninorm aggregation operators, Fuzzy Sets and Systems, 80, 111-120 (1996) · Zbl 0871.04007