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The greatest common divisor and other triangular norms on the extended set of natural numbers. (English) Zbl 1167.11004

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.

MSC:

11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
06A06 Partial orders, general
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[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
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