Elsevier

Fuzzy Sets and Systems

Volume 121, Issue 2, 16 July 2001, Pages 183-190
Fuzzy Sets and Systems

Continuous Archimedean t-norms and their bounds

https://doi.org/10.1016/S0165-0114(00)00032-4Get rights and content

Abstract

Upper and lower bounds in the class of continuous Archimedean t-norms are studied. The existence of strict (nilpotent) bounds of finite families of strict (nilpotent) t-norms is shown in a constructive way, based on the additive generators of the corresponding t-norms. In general, a nilpotent lower bound and a strict upper bound of a finite system of any continuous Archimedean t-norms can be found. Some examples and some applications are given. By the duality, similar results for bounds of continuous Archimedean t-conorms can be derived.

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    In other words, an algebraic structure is said to be Archimedean whenever any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other. In the context of aggregation functions, the concept of Archimedean t-norms was first studied by Schweizer and Sklar [6], and, since then, Archimedean t-norms have been largely explored in the various works, e.g., by Klement et al. [7], Saminger-Platz [8], Jenei [9,10], Marko and Mesiar [11]. For the context of interval fuzzy logic [12–14], see, e.g., the work by Deschrijver [15].

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Supported by grants GAČR 402/99/0032 and VEGA 1/7146/20, 2/6087/99.

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