# zbMATH — the first resource for mathematics

On a class of left-continuous $$\text t$$-norms. (English) Zbl 1012.03032
In the paper mutual relations between t-norms and logical weak negations are investigated. The attention is focused on so-called nilpotent minimum t-norms, whose construction is generalized in the paper. In the main results the weak negations compatible with a given t-norm are characterized and, vice versa, also a characterization of continuous t-norms compatible with a given weak negation function is given.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness
##### Keywords:
fuzzy connectives; left-continuous t-norms; weak negations
Full Text:
##### References:
 [1] Cignoli, R.; Esteva, F.; Godo, L.; Torrens, A., Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft computing, 4, 106-112, (2000) [2] Climescu, A.C., Sur l’équation fonctionelle de l’associativé, Bull. école polytechns iassy, 1, 1-16, (1946) [3] Esteva, F.; Domingo, X., Sobre negociaciones fuertes y debiles en [0,1], Stochastica, IV, 2, 141-166, (1980) [4] Esteva, F.; Godo, L., Monoidal t-norm based logictowards a logic for left-continuous t-norms, Fuzzy sets and systems, 124, 271-288, (2001) · Zbl 0994.03017 [5] Esteva, F.; Godo, L.; Hájek, P.; Navara, M., Residuated fuzzy logics with an involutive negation, Arch. math. logic, 39, 103-124, (2000) · Zbl 0965.03035 [6] J. Fodor, Nilpotent minimum and related connectives for fuzzy logic, Proceedings of the FUZZ-IEEE’95, 1995, pp. 2077-2082. [7] Hájek, P., Methamatematics of fuzzy logic, (1998), Kluwer Dordrecht [8] Hájek, P., Basic fuzzy logic and BL-algebras, Soft computing, 2, 124-128, (1998) [9] Höhle, U., Commutative, residuated l-monoids, (), 53-106 · Zbl 0838.06012 [10] Jenei, S., New family of triangular norms via contrapositive symmetrization of residuated implications, Fuzzy sets and systems, 110, 157-174, (2000) · Zbl 0941.03059 [11] S. Jenei, Structure of Girard monoids on [0, 1], in: S. Rodabaugh, E.P. Klement (Eds.), Topological and Algebraic Structures in Fuzzy Sets, Kluwer, Dordrecht, in press. · Zbl 1036.03035 [12] S. Jenei, F. Montagna, A proof of standard completeness for Esteva and Godo’s logic MTL, Studia Logica, in press. · Zbl 0997.03027 [13] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0972.03002 [14] Mostert, P.S.; Shields, A.L., On the structure of semigroups on a compact manifold with boundary, Ann. of math., 65, 117-143, (1957) · Zbl 0096.01203 [15] E. Trillas, Sobre funciones de negación en la teoría de conjuntos difusos. Stochastica I(1) (1979) 47-60. (English translation in: S. Barro, A. Bugarin, A. Sobrino (Eds.), Advances in Fuzzy logic, Public. Univ. Santiago de Compostela, Spain, 1998, pp. 31-45).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.