Smooth Convex t-Norms Do Not Exist

descriptionPublicationkeyboard_double_arrow_right Article 01 Feb 1988Publisher:JSTORJournal:Proceedings of the American Mathematical Society, volume 102, page 317 (issn: 0002-9939,

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Authors: Alsina, Claudi; Thomás, Maria S.;

doi: 10.2307/2045882

Smooth Convex t-Norms Do Not Exist

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Abstract

A binary operation on [0,1] which is associative, commutative, nondecreasing in each place, and has 1 as a unit element is said to be a t-norm. An example of continuous convex t-norm is \(W(x,y)=Max(x+y-1,0),\) x,y\(\in [0,1]\). The authors prove that smooth convex t-norms do not exist.

Keywords

convexity, Systems of functional equations and inequalities, binary operation, Convexity of real functions of several variables, generalizations, continuous convex t-norm

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