Triangular norms on product lattices.

*(English)*Zbl 0935.03060The 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.

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. |

##### Keywords:

direct product of t-norms; triangular norm; t-norms on bounded posets; idempotent elements; zero divisors; nilpotent elements; product poset; cancellation law; product lattice##### Software:

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\textit{B. De Baets} and \textit{R. Mesiar}, Fuzzy Sets Syst. 104, No. 1, 61--75 (1999; Zbl 0935.03060)

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##### References:

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