Abstract
We introduce a notion of an idempotent semilinear space and consider two systems of linear-like equations. These systems are equivalent to systems of fuzzy relation equations with sup-*and \(\inf\)-→ compositions. We show that the theory of Galois connections can be successfully used in characterizing whether these systems are solvable and, if so, finding their solutions sets. Moreover, because the two types of systems of linear-like equations are dual according to this theory, it is sufficient to investigate only one system.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Di Nola, A., Sessa, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer, Dordrecht (1989)
De Baets, B.: Analytical solution methods for fuzzy relation equations. In: Dubois, D., Prade, H. (eds.) The Handbooks of Fuzzy Sets Series, vol. 1, pp. 291–340. Kluwer, Dordrecht (2000)
Nosková, L., Perfilieva, I.: System of fuzzy relation equations with sup− * composition in semi-linear spaces: minimal solutions. In: Proc. FUZZ-IEEE Conf. on Fuzzy Systems, London, pp. 1520–1525 (2007)
Perfilieva, I., Nosková, L.: System of fuzzy relation equations with inf → composition: complete set of solutions. Fuzzy Sets and Systems 159, 2256–2271 (2008)
Raikov, D.A.: Vector spaces. FizMatLit, Moscow (1962)
Kurosh, A.G.: Lectures on General Algebra. FizMatLit, Moscow (1962)
Golan, J.S.: Semirings and their Applications. Kluwer Academic Publishers, Dordrecht (1999)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning I, II, III. Information Sciences 8-9, 199–257, 301–357, 43–80 (1975)
Bandler, W., Kohout, L.: Semantics of implication operators and fuzzy relational products. Int. J. Man-Machine Studies 12, 89–116 (1980)
Gottwald, S.: On the existence of solutions of systems of fuzzy equations. Fuzzy Sets and Systems 12, 301–302 (1984)
Klawonn, F.: Fuzzy points, fuzzy relations and fuzzy functions. In: Novák, V., Perfilieva, I. (eds.) Discovering the World with Fuzzy Logic, pp. 431–453. Springer, Berlin (2000)
Perfilieva, I., Tonis, A.: Compatibility of systems of fuzzy relation equations. Int. J. of General Systems 29, 511–528 (2000)
Perfilieva, I., Gottwald, S.: Solvability and approximate solvability of fuzzy relation equations. Int. J. of General Systems 32, 361–372 (2003)
Sanchez, E.: Resolution of composite fuzzy relation equations. Information and Control 30, 38–48 (1976)
Wang, X.: Infinite fuzzy relational equations on a complete Brouwerian lattice. Fuzzy Sets and Systems 138, 657–666 (2003)
Štěpnička, M., Jayaram, B.: On the suitability of the bandler-kohout subproduct as an inference mechanism. IEEE Trans. on Fuzzy Syst. (2010) (to appear)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Perfilieva, I. (2010). Fuzzy Relation Equations in Semilinear Spaces. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods. IPMU 2010. Communications in Computer and Information Science, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14055-6_57
Download citation
DOI: https://doi.org/10.1007/978-3-642-14055-6_57
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14054-9
Online ISBN: 978-3-642-14055-6
eBook Packages: Computer ScienceComputer Science (R0)