Residuation in fuzzy algebra and some applications
type: Item
instance of: scholarly article
title: Residuation in fuzzy algebra and some applications
zbMATH Open document ID: 0845.04007
DOI: 10.1016/0165-0114(94)00252-3
published in: Fuzzy Sets and Systems
publication date: 1996-06-18T00:00:00Z
review text: Applications of systems of linear equations over structures different from the classical field were studied in the 1960s and in recent years have again received increasing attention as a tool for modelling discrete event systems or fuzzy relations. Several authors have observed that a system of the form \(R \otimes x = b\) always has a `principal' solution \(x^* \), i.e. such that the system is solvable if and only if \(x^*\) is a solution; and in this case \(x^*\) is the maximum solution. The connection with residuation theory was pointed out by the first author [Minimax algebra (Lect. Notes Econ. Math. Syst. 166 (1979; Zbl 0399.90052)] and \textit{A. Di Nola} [Fuzzy Sets Syst. 34, No. 3, 365-376 (1990; Zbl 0701.04003)]. The aim of this paper is to show further implications of residuation theory for fuzzy algebra, including a method for testing linear dependence of sets of vectors. Next we address `the inverse problem' studied by \textit{W. Pedrycz} [Fuzzy Sets Syst. 49, No. 3, 339-355 (1992; Zbl 0805.04005)] and propose an efficient algorithm for its solution. Analogies of this approach with methods used in a similar algebra, where multiplication is a group operation, are stressed.
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 06F99
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 15A06
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 891004
zbMATH Keywords: inversion of linear relations
zbMATH Keywords: Chebyshev approximation
zbMATH Keywords: systems of linear equations
zbMATH Keywords: fuzzy relations
zbMATH Keywords: residuation
zbMATH Keywords: fuzzy algebra
zbMATH Keywords: linear dependence of sets of vectors
author: R. A. Cuninghame-Green
author: Katarina Cechlárová
MaRDI profile type: Publication
full work available at URL: 0165-0114(94)00252-3
OpenAlex ID: W2051106936
cites work: Q4056047
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description: scientific article; zbMATH DE number 891004