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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)

03E72 Theory of fuzzy sets, etc.
Full Text: DOI
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