Solution of an open problem on pseudo-Archimedean t-norms
type: Item
instance of: scholarly article
title: Solution of an open problem on pseudo-Archimedean t-norms
zbMATH Open document ID: 1162.03031
author: Kunlun Zhang
author: Donghai Li
author: Li-Xia Song
published in: Information Sciences
publication date: 2009-03-24T00:00:00Z
review text: Let \((P,\wedge,\vee,\bot,\top)\) be a bounded lattice. A t-norm \(T\) on the lattice is any commutative and associative binary operation on \(P\) which is non-decreasing with respect to the lattice order and has the unit element \(\top\). For the special case when the lattice is the chain \([0,1]\) it is known that every t-norm with the pseudo-Archimedean property which is not cancellative has a nilpotent element. In the paper under review the authors show that this is no longer true for t-norms on general bounded lattices. Namely, they construct two examples of a bounded lattice and for each one of them a pseudo-Archimedean t-norm which is not cancellative and has no nilpotent elements. This work was motivated by an open question posed by \textit{B. De Baets} and \textit{R. Mesiar} [Fuzzy Sets Syst. 104, No.~1, 61--75 (1999; Zbl 0935.03060)].
reviewed by: Peter Sarkoci
Mathematics Subject Classification ID: 03E72
zbMATH DE Number: 5531313
zbMATH Keywords: bounded lattice
zbMATH Keywords: triangular norm
zbMATH Keywords: t-norms on bounded lattices
zbMATH Keywords: idempotent elements
zbMATH Keywords: zero divisors
zbMATH Keywords: nilpotent elements
zbMATH Keywords: cancellation law
MaRDI profile type: Publication
full work available at URL: j.ins.2008.08.001
OpenAlex ID: W2016707814
cites work: Triangular norms on product lattices
cites work: Metamathematics of fuzzy logic
cites work: Triangular norms. Position paper I: Basic analytical and algebraic properties.
DOI: 10.1016/J.INS.2008.08.001
description: scientific article; zbMATH DE number 5531313