Mayor, Gaspar; Monreal, Jaume The greatest common divisor and other triangular norms on the extended set of natural numbers. (English) Zbl 1167.11004 Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 17, No. 1, 35-45 (2009). A triangular norm \(T\) (triangular conorm \(S\)) on a bounded poset \(P\) with top element \(1\) and bottom element \(0\) is a non-decreasing commutative associative operation on \(P\) with neutral element \(1\) (neutral element \(0\)). The paper is focused on the study of triangular norms and conorms on the set \(N= \{1,2,\dots,\infty\}\) of extended natural numbers ordered by the divisibility and with top element \(\infty\). After recalling some results on (divisible) triangular norms and conorms on the chain \(L= \{0,1,\dots,\infty\}\), triangular norms on \(N\) which are direct products of triangular norms on \(L\) are introduced and characterized. Recall that for a triangular norm \(T\) on \(N\), a number \(p\in N\setminus\{0\}\) is called \(T\)-prime if \(T(m,p)= 1\) for all \(m< p\). As a main result of this paper, for a triangular norm \(T\) and a divisible triangular conorm \(S\) on \(N\), an \(S\)-decomposition of any \(n\geq 2\) into \(T\)-primes \(p_1,\dots, p_r\) (and possibly one not \(T\)-prime number \(n_r\)) is shown, thus generalizing the fundamental theorem of arithmetic. Reviewer: Radko Mesiar (Bratislava) MSC: 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 06A06 Partial orders, general Keywords:natural number; greatest common divisor; primes; triangular norm; triangular conorm; divisible triangular norm PDFBibTeX XMLCite \textit{G. Mayor} and \textit{J. Monreal}, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 17, No. 1, 35--45 (2009; Zbl 1167.11004) Full Text: DOI References: [1] DOI: 10.1002/int.4550080703 · Zbl 0785.68087 · doi:10.1002/int.4550080703 [2] DOI: 10.1016/B978-044451814-9/50007-0 · doi:10.1016/B978-044451814-9/50007-0 [3] DOI: 10.1016/S0165-0114(98)00259-0 · Zbl 0935.03060 · doi:10.1016/S0165-0114(98)00259-0 [4] DOI: 10.1007/978-94-011-0215-5_5 · doi:10.1007/978-94-011-0215-5_5 [5] Klement E. P., Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms (2005) · Zbl 1063.03003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.