Solution to an open problem: a characterization of conditionally cancellative t-subnorms
type: Item
instance of: scholarly article
title: Solution to an open problem: a characterization of conditionally cancellative t-subnorms
zbMATH Open document ID: 1273.26037
author: Balasubramaniam Jayaram
published in: Aequationes Mathematicae
publication date: 2012-12-19T00:00:00Z
review text: The author proves that for any \(t\)-subnorm \(M\!:[0,1]^2\rightarrow [0,1]\), having the residual operation \(I(x,y)=\sup\{t\in [0,1] \mid M(x,t)\leq y\}\), \(x,y\in [0,1]\), the following conditions are equivalent: (i) \(I(x,M(x,y))=\max(n(x),y)\), \(x,y\in [0,1]\), with \(n(x)=I(x,0)\) for all \(x\in [0,1]\) and (ii) \(M\) is a conditionally cancelative \(t\)-subnorm. This is the solution to problem 11 of U. Höhle [\textit{E. P. Klement, R. Mesiar} and \textit{E. Pap}, Fuzzy Sets Syst. 145, No. 3, 471--479 (2004; Zbl 1050.03019)]. Finally, this result leads to the conclusion of the non-existence of a conditionally cancelative \(t\)-subnorm whose natural negation is involutive.
reviewed by: Gabriela Cristescu
Mathematics Subject Classification ID: 26E50
Mathematics Subject Classification ID: 26D07
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 54E70
zbMATH DE Number: 6116949
zbMATH Keywords: triangular norms
zbMATH Keywords: conditional cancellativity
zbMATH Keywords: t-subnorm
zbMATH Keywords: fuzzy negation
zbMATH Keywords: natural negation
zbMATH Keywords: residuation
zbMATH Keywords: open problem
MaRDI profile type: Publication
full work available at URL: s00010-012-0143-0
OpenAlex ID: W2030543064
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DOI: 10.1007/S00010-012-0143-0
description: scientific article; zbMATH DE number 6116949