Triangular norms on product lattices

In this paper, triangular norms (t-norms) are studied in the general settin g of bounded partially ordered Sets, with emphasis on finite chains, produc t lattices and the real unit square. The sets of idempotent elements, zero divisors and nilpotent elements associated to a t-norm are introduced and r elated 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 pres ented 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 divi sors. It is shown that the direct product of cancellative t-norms is again cancellative. Direct products of t-norms on a product lattice are character ized as t-norms with partial mappings that show some particular morphism be haviour. Finally, it is shown that in the case of the real unit square, tra nsformations by means of an automorphism preserve the direct product struct ure. (C) 1999 Elsevier Science B.V. All rights reserved.