The greatest common divisor and other triangular norms on the set of natural numbers

This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, ...,+∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characterization of those t-norms on N which are a direct product of tnorms on {0, 1, 2, ...,+∞} is given and, after introducing the concept of T -prime (prime with respect to a tnorm T ), a theorem about the existence of a T -prime decomposition is obtained. This result generalizes the fundamental theorem of arithmetic.