Triangular norms on product lattices

In this paper, triangular norms (t-norms) are studied in the general setting of bounded partially ordered Sets, with emphasis on finite chains, product lattices and the real unit square. The sets of idempotent elements, zero divisors and nilpotent elements associated to a t-norm are introduced and related to each other. The Archimedean property of t-norms is discussed, in particular its relationship to the diagonal inequality. The main subject of the paper is the direct product sf t-norms on product posets. It is shown that the direct product of t-norms without zero divisors is again a t-norm without zero divisors. A weaker version of the Archimedean property is presented and it is shown that the direct product of such pseudo-Archimedean t-norms is again pseudo-Archimedean, A generalization of the cancellation law is presented, in the same spirit as the definition of the set of zero divisors. It is shown that the direct product of cancellative t-norms is again cancellative. Direct products of t-norms on a product lattice are characterized as t-norms with partial mappings that show some particular morphism behaviour. Finally, it is shown that in the case of the real unit square, transformations by means of an automorphism preserve the direct product structure. (C) 1999 Elsevier Science B.V. All rights reserved.