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Pseudo-t-norms and implication operators on a complete Brouwerian lattice. (English) Zbl 1013.03020
Fuzzy Sets Syst. 132, No. 1, 113-124 (2002); corrigendum ibid. 153, No. 2, 295-296 (2005).
Given a complete Brouwerian lattice $$L$$, a pseudo-t-norm is a binary operation $$T$$ on $$L$$ such that $$T(1,a) = a$$, $$T(0,a) = 0$$ and $$T(a,b) \leq T(a,c)$$ whenever $$b \leq c$$. Motivated by problems of fuzzy logic, the authors extend previous works on relations between t-norms and implication functions. More precisely, they study in detail the relations between the set of all infinitely $$\vee$$-distributive pseudo-t-norms and the set of all infinitely $$\wedge$$-distributive implications on $$L$$. Some examples and particular cases are presented. [See also the notes on this article in the same journal 153, No. 2, 289-294 (2005; Zbl 1086.03019).]

##### MSC:
 03B52 Fuzzy logic; logic of vagueness
##### Keywords:
t-norms; weak t-norm; pseudo-t-norm; implication
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##### References:
 [1] De Baets, B.; Mesiar, R., Triangular norms on product lattices, Fuzzy sets and systems, 104, 61-75, (1999) · Zbl 0935.03060 [2] Dubois, D.; Prade, H., Fuzzy sets in approximate reasoning, part 1: inference with possibility distributions, Fuzzy sets and systems, 40, 143-202, (1991) · Zbl 0722.03017 [3] Fodor, J.C., On fuzzy implication operators, Fuzzy sets and systems, 42, 293-300, (1991) · Zbl 0736.03006 [4] Fodor, J.C., Strict preference relations based on weak t-norms, Fuzzy sets and systems, 43, 327-336, (1991) · Zbl 0756.90006 [5] Fodor, J.C., A new look at fuzzy connectives, Fuzzy sets and systems, 57, 141-148, (1993) · Zbl 0795.04008 [6] Fodor, J.C., Contrapositive symmetry of fuzzy implications, Fuzzy sets and systems, 69, 141-156, (1995) · Zbl 0845.03007 [7] Fodor, J.; Jenei, S., On reversible triangular norms, Fuzzy sets and systems, 104, 43-51, (1999) · Zbl 0965.03068 [8] Fodor, J.C.; Keresztfalvi, T., A characterization of the hamacher family of t-norms, Fuzzy sets and systems, 65, 51-58, (1994) · Zbl 0845.03008 [9] Klement, E.P.; Mesiar, R.; Pap, E., Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms, Fuzzy sets and systems, 104, 3-13, (1999) · Zbl 0953.26008 [10] Kundu, S.; Chen, J.H., Fuzzy logic or lukasiewicz logic: a clarification, Fuzzy sets and systems, 95, 369-379, (1998) · Zbl 0924.03031 [11] Ma, Z.; Wu, W.-M., Logical operators on complete lattices, Inform. sci., 55, 77-97, (1991) · Zbl 0741.03010 [12] Mesiar, R.; Novak, V., Operations Fitting triangular-norm-based biresiduation, Fuzzy sets and systems, 104, 77-84, (1999) · Zbl 0936.03025 [13] Miyakosn, M.; Shimbo, M., Solutions of composite fuzzy relational equations with triangular norms, Fuzzy sets and systems, 16, 53-63, (1985) · Zbl 0582.94031 [14] Pavelka, J., On fuzzy logic II, Zeitschr. f. math. logik und grundlagen d. math., 25, 119-134, (1979) · Zbl 0446.03015 [15] Smets, P.; Margrez, P., Implication in fuzzy logic, Internat. J. approx. reason., 1, 327-347, (1987) · Zbl 0643.03018 [16] Tüksen, I.B.; Kreinovich, V.; Yager, R., A new class of fuzzy implications. axioms of fuzzy implication revisited, Fuzzy sets and systems, 100, 267-272, (1999) · Zbl 0939.03030 [17] Turunen, E., Algebraic structures in fuzzy logic, Fuzzy sets and systems, 52, 181-188, (1992) · Zbl 0791.03010 [18] G.-J. Wang, Theory of L-Fuzzy Topological Spaces, Shaanxi Normal University Press, Xi’an, China, 1988 (in Chinese). [19] Wang, Z.-D., On L-subsets and TL-subalgebras, Fuzzy sets and systems, 65, 59-69, (1994) · Zbl 0843.08002 [20] Wang, Z.-D., Primary TL-submodules and P-primary TL-submodules, Fuzzy sets and systems, 88, 237-254, (1997) · Zbl 0919.16036 [21] Weber, S., A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy sets and systems, 11, 115-134, (1983) · Zbl 0543.03013 [22] Wu, W.-M., Triangular norms, triangular co-norms, and pseudocomplements, J. Shanghai normal univ., 4, 1-10, (1984), (in Chineses) [23] Wu, W.-M., A multivalued logic system with respect to t-norms, (), 101-118 [24] Yager, R., An approach to inference in approximate reasoning, Int. J. man – mach. stud., 13, 323-338, (1980) [25] Yu, Y.-D., L-sets and L-universal algebras, (), 91-114 [26] Yu, Y.-D.; Wang, Z.-D., TL-subrings and TL-ideals, part 1: basic concepts, Fuzzy sets and systems, 68, 93-103, (1994) · Zbl 0847.16034
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