Abstract
There has been considerable interest in the nature of fuzzy points and in their relationship to crisp points. This paper describes a very general approach to the characterization of points which treats them as generalized maps from one object to another in a 2-categorical framework. It will be shown that fuzzy points are also amenable to this treatment, and that, if the category (of locales) is assumed to be equipped with a KZ monad, the locales of fuzzy sets exhibit behavior which allows for such useful results as the justification of the Zadeh Extension Principle and the unification of change of base and change of set in fuzzy set theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Mac Lane, S. and Birkhoff, G.: Algebra (Third Edition). AMS Chelsea Publishing, Providence, Rhode Island (1999)
Mac Lane, S. and Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer-Verlag, New York (1992)
Warner, M.W.: Frame-fuzzy points and membership. Fuzzy Sets and Systems 42 (1991) 335–344
Vickers, S.: Locales are not pointless. In: Hankin, C. et. al. (eds): Theory and Formal Methods 1994. Imperial College Press, London (1995) 199–216
Rodabaugh, S.: Powerset Operator Foundations For Poslat Fuzzy Set Theories And Topologies. In: Hohle, U. and Rodabaugh, S. (eds.): Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory. Kluwer Academic Publishers, Boston (1999) 91–116
Kock, A.: Monads for which structures are adjoint to units. Journal of Pure and Applied Algebra 104 (1995) 41–59
Vickers, S.: Information Systems for Continuous Posets. Theoretical Computer Science 114 (1993) 201–229
Vickers, S.: Localic Completion of Quasimetric Spaces. Research Report DoC97/2, Dept. of Computing, Imperial College, London
Johnstone, P.: Stone Spaces. Cambridge University Press, Cambridge (1982)
Vickers, S.: Topology via Logic. Cambridge University Press, Cambridge (1989)
Rodabaugh, S.: Categorical Foundations Of Variable-Basis Fuzzy Topology. In: Hohle, U. nd Rodabaugh, S. (eds.): Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory. Kluwer Academic Publishers, Boston (1999) 273–388
Koslowski, J.: Monads and Interpolads in Bicategories. Theory and Application of Categories 8 (1997) 182–212
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Barone, J.M. (2002). Fuzzy Points and Fuzzy Prototypes. In: Pal, N.R., Sugeno, M. (eds) Advances in Soft Computing — AFSS 2002. AFSS 2002. Lecture Notes in Computer Science(), vol 2275. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45631-7_63
Download citation
DOI: https://doi.org/10.1007/3-540-45631-7_63
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43150-3
Online ISBN: 978-3-540-45631-5
eBook Packages: Springer Book Archive