×

zbMATH — the first resource for mathematics

Triangular norms on product lattices. (English) Zbl 0935.03060
The original concept of a triangular norm (t-norm) has been introduced by Schweizer and Sklar as associative, commutative, monotone \([0,1]^2- [0,1]\) mappings satisfying the boundary condition \((\forall x\in[0,1])\) \((T(x,1)= x)\). Many authors have extended this notion to arbitrary bounded partially ordered sets. The present paper continues the research of t-norms on bounded posets. Firstly, the authors compare their approach to several existing generalizations of t-norms such as t-norms on a finite chain and interval-valued t-norms as studied by Jenei and Nguyen-Walker.
Secondly they introduce, illustrate and study from a lattice-theoretical point of view, three classes associated with an arbitrary t-norm on a bounded poset: the class of idempotent elements, of zero divisors and of nilpotent elements. Furtheron, a stronger Archimedean notion than the diagonal inequality as introduced by De Cooman and Kerre is launched. The rest of the paper discusses the concept of direct product of two t-norms on the product poset of the underlying bounded posets.
More particularly, the class of idempotent elements, of zero divisors and of nilpotent elements of such a direct product are determined and the cancellation law is studied. Finally, the t-norms on a product lattice that are a direct product of two t-norms are characterized and in particular on the unit square. The paper contains many examples and appropriate counterexamples.
Reviewer: E.Kerre (Gent)

MSC:
03E72 Theory of fuzzy sets, etc.
Software:
OEIS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agustí, J.; Esteva, F.; Garcia, P.; Godo, L.; Lopez de Mantaras, R.; Sierra, C., Local multi-valued logics in modular expert systems, J. expt. theor. artificial intell., 6, 303-321, (1994) · Zbl 0815.68108
[2] Birkhoff, G., ()
[3] De Baets, B., Oplossen Van vaagrelationele vergelijkingen: een ordetheoretisch benadering, (), 389
[4] De Baets, B., An order-theoretic approach to solving sup-\(T\) equations, (), 67-87 · Zbl 0874.04005
[5] De Baets, B., Database entry A030453, ()
[6] De Cooman, G.; Kerre, E., Order norms on bounded partially ordered sets, J. fuzzy math., 2, 281-310, (1994) · Zbl 0814.04005
[7] Drossos, C.; Navara, M., Generalized t-conorms and closure operators, (), 22-26
[8] Godo, L.; Sierra, C., A new approach to connective generation in the framework of expert systems using fuzzy logic, (), 157-162
[9] Goguen, J., L-fuzzy sets, J. math. anal. appl., 18, 145-174, (1967) · Zbl 0145.24404
[10] Jenei, S., A more efficient method for defining fuzzy connectives, Fuzzy sets and systems, 90, 25-35, (1997) · Zbl 0922.03074
[11] E.-P. Klement, R. Mesiar, E. Pap, Triangular Norms, in preparation. · Zbl 0972.03002
[12] Ling, C., Representation of associative functions, Publ. math. debrecen, 12, 189-212, (1965) · Zbl 0137.26401
[13] Mayor, G.; Torrens, J., On a class of operators for expert systems, Internat. J. intell. systems, 8, 771-778, (1993) · Zbl 0785.68087
[14] Menger, K., Statistical metrics, (), 535-537 · Zbl 0063.03886
[15] Nguyen, H.; Walker, E., A first course in fuzzy logic, (1997), CRC Press Boca Raton · Zbl 0856.03019
[16] Ray, S., Representation of a Boolean algebra by its triangular norms, Mathware soft comput., 4, 63-68, (1997) · Zbl 0873.06009
[17] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), Elsevier Amsterdam · Zbl 0546.60010
[18] Smutná, D., On a peculiar t-norm, Busefal, 75, (1998), in press
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.