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Zbl 1087.20041
Klement, Erich Peter; Mesiar, Radko; Pap, Endre
Archimedean components of triangular norms. (English)
[J] J. Aust. Math. Soc. 78, No. 2, 239-255 (2005). ISSN 1446-7887; ISSN 1446-8107/e

A triangular norm (shortly t-norm) $T$ is a binary operation on the real closed interval $[0,1]$ compatible with the natural order ``$\leq$" of $[0,1]$ such that $([0,1],T)$ is an Abelian, totally ordered semigroup having 1 as neutral element. Continuous t-norms are just ordinal sums of continuous Archimedean t-norms (turning $[0,1]$ into a topological semigroup).\par In this paper the Archimedean components of triangular norms are studied. The main results are the following: A characterization of a function $T\colon [0,1]^2\to[0,1]$ which is a continuous Archimedean t-norm. The result states that continuous Archimedean t-norms are characterized by having continuous additive generators. A characterization of a function $T\colon[0,1]^2\to[0,1]$ which is continuous t-norm in terms of continuous Archimedean t-norms and ordinal sums. Each continuous t-norm is uniquely determined by its non-trivial Archimedean components. If $T$ is a t-norm, then $([0,1],T)$ is an ordinal sum of semigroups if and only if $T$ is an ordinal sum of t-subnorms. If $T$ is a t-norm, then the ordinal sum of its Archimedean components is the strongest t-norm which has the same Archimedean components as $T$. Extensions of Archimedean components to triangular norms and a characterization of the class of triangular norms which are uniquely determined by their Archimedean components are given. Some construction methods for Archimedean components are given at the end.
[Niovi Kehayopulu (Athens)]

MSC 2000:: *20M10 General structure theory of semigroups
06F05 Ordered semigroups
03E72 Fuzzy sets (logic)
20M05 Free semigroups

Keywords: ordered semigroups; triangular norms; Archimedean components; ordinal sums

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