On the distributivity of fuzzy implications over representable uninorms
type: Item
instance of: scholarly article
title: On the distributivity of fuzzy implications over representable uninorms
zbMATH Open document ID: 1201.03013
author: Michał Baczyński
published in: Fuzzy Sets and Systems
publication date: 2010-09-07T00:00:00Z
review text: The classical tautology \((p\land q)\to r\equiv (p\to r) \lor (q\to r)\) rewritten in the language of fuzzy connectives can be understand as a functional equation \[ I(U_1(x,y),z)= U_2(I(x,z), I(y,z)), \] where \(U_1\), \(U_2\) are representable uninorms and \(I\) is a fuzzy implication. The author shows that this functional equation has no solution. On the other hand, the functional equation \[ I(x,U_1(y,z))= U_2(I(x,y), I(x,z)) \] (generalizing the tautologies \(x\to (y\land z)\equiv (x\to y)\land(x\to z)\) and \(x\to (y\lor z)\equiv (x\to y)\lor (x\to z)\)) admits several solutions. The author shows that then the fuzzy implication \(I\) cannot be continuous, and he characterizes the solutions where the vertical sections of \(I\) are continuous. As a by-product, some new results concerning the Cauchy functional equation \(f(x+ y)= f(x)+ f(y)\) on the extended real line \([-\infty,+\infty]\) are shown.
reviewed by: Radko Mesiar
Mathematics Subject Classification ID: 03B52
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 5780076
zbMATH Keywords: fuzzy implication
zbMATH Keywords: representable uninorm
zbMATH Keywords: Cauchy functional equation
MaRDI profile type: Publication
full work available at URL: j.fss.2010.03.005
OpenAlex ID: W2062098912
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DOI: 10.1016/J.FSS.2010.03.005
description: scientific article; zbMATH DE number 5780076