Direct product and algebraic properties of tringular norms onbounded partially ordered sets

In this study, triangular norms are studied in the general setting of bounded partially ordered sets, with accent 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 intercourse to the diagonal inequality. The main subject of the study is the direct product of 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 ineffective 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 put forward, in the same logic 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 conduct. Finally, it is shown that in the case of the real unit square, transformations by means of an automorphism safe guard the direct product formation.