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Logical operators on complete lattices. (English) Zbl 0741.03010
A concept of consistency among logical operators is given with a criterion to decide whether a group of logical operators is suitable for fuzzy reasoning. Also, a necessary and sufficient condition for a kind of logical operators on $$[0,1]$$ is obtained.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03G10 Logical aspects of lattices and related structures 06B23 Complete lattices, completions
##### Keywords:
consistency; logical operators; fuzzy reasoning
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##### References:
 [1] Alsina, C.; Trillas, E.; Valverde, L., On some logical connectives for fuzzy sets theory, J. math. anal. appl., 93, 15-26, (1983) · Zbl 0522.03012 [2] Balbes, Raymond; Dwinger, Philip, Distributive lattices, (1974), University of Missouri Press Rolla · Zbl 0321.06012 [3] Baldwin, J.F., A new approach to approximate reasoning using fuzzy logic, Fuzzy sets syst., 2, 309-325, (1979) · Zbl 0413.03017 [4] Bellman, R.; Giertz, M., On the analytic formalism of the theory of fuzzy set, Inform. sci., 5, 149-156, (1973) · Zbl 0251.02059 [5] Dubois, D.; Prade, H., New results about properties and semantics of fuzzy set-theoretic operators, () · Zbl 0593.04004 [6] Dubois, D.; Prade, H., Fuzzy sets and systems—theory and application, (1980), Academic Press New York [7] Ling, C.H., Representation of associative functions, Publ. math. debrecen, 12, 182-212, (1965) · Zbl 0137.26401 [8] Lowen, R., On fuzzy complements, Inform. sci., 14, 107-113, (1978) · Zbl 0416.03047 [9] Miyakoshi, M.; Shimbo, M., Solutions of composite fuzzy relational equations, Fuzzy sets syst., 16, 53-63, (1985) · Zbl 0582.94031 [10] Mizumoto, M.; Zimmermann, H.J., Comparison of fuzzy reasoning methods, Fuzzy sets syst., 8, 253-283, (1982) · Zbl 0501.03013 [11] Rosser, J.B.; Turquette, A.R., Many-valued logics, (1952), North-Holland Amsterdam · Zbl 0063.06587 [12] Schweizer, B.; Sklar, A., Associative functions and abstract semi-groups, Publ. math. debrecen, 10, 69-81, (1963) · Zbl 0119.14001 [13] Trillas, E., Sobre functions de negation en la teoria de conjuntos difusos, Stochastics, 111, 1, 47-60, (1979) [14] Trillas, E.; Valverde, L., On implication and indistinguishability in the setting of fuzzy logic, () · Zbl 0564.03027 [15] Trillas, E.; Valverde, L., On mode and implication in approximate reasoning, () · Zbl 0546.03015 [16] Wu, Wangming, The sup-T compositions of fuzzy relations, J. Shanghai teachers’ coll., 3, 57-65, (1983), (in Chinese). · Zbl 0599.54007 [17] Wu, Wangming, Triangular norms, triangular co-norms, and pseudocomplements, J. Shanghai normal univ., 4, 1-10, (1984), (in Chinese). · Zbl 0597.04003 [18] Wu, Wangming, The inf-α compositions of fuzzy relations, J. Shanghai normal univ., 2, 1-7, (1985), (in Chinese). · Zbl 0633.54005 [19] Wu, Wangming, Fuzzy reasoning and fuzzy relational equations, Fuzzy sets syst., 20, 67-78, (1986) · Zbl 0629.94031 [20] Zadeh, L.A., A fuzzy-algorithmic approach to the definition of complex or imprecise concepts, Int. J. man-Mach. stud., 8, 249-291, (1976) · Zbl 0332.68068
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