Abstract
Traditionally, rough sets build upon relations based on ordinary sets, i.e. relations on X as subsets of X×X. A starting point of this paper is the equivalent view on relations as mappings from X to the (ordinary) power set PX. Categorically, P is a set functor, and even more so, it can in fact be extended to a monad (P,η,μ). This is still not enough and we need to consider the partial order (PX,≤). Given this partial order, the ordinary power set monad can be extended to a partially ordered monad. The partially ordered ordinary power set monad turns out to contain sufficient structure in order to provide rough set operations. However, the motivation of this paper goes far beyond ordinary relations as we show how more general power sets, i.e. partially ordered monads built upon a wide range of set functors, can be used to provide what we call rough monads.
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Eklund, P., Galán, M.A., Gähler, W., Medina, J., Ojeda Aciego, M., Valverde, A.: A note on partially ordered generalized terms. In: Proc. of Fourth Conference of the European Society for Fuzzy Logic and Technology and Rencontres Francophones sur la Logique Floue et ses applications (Joint EUSFLAT-LFA 2005), pp. 793–796 (2005)
Eklund, P., Gähler, W.: Partially ordered monads and powerset Kleene algebras. In: Proc. 10th Information Processing and Management of Uncertainty in Knowledge Based Systems Conference (IPMU 2004) (2004)
Gähler, W.: General Topology – The monadic case, examples, applications. Acta Math. Hungar. 88, 279–290 (2000)
Gähler, W., Eklund, P.: Extension structures and compactifications. In: Categorical Methods in Algebra and Topology (CatMAT 2000), pp. 181–205 (2000)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)
Järvinen, J.: On the structure of rough approximations. Fundamenta Informaticae 53, 135–153 (2002)
Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, pp. 3–41. Princeton University Press, Princeton (1956)
Kortelainen, J.: A Topological Approach to Fuzzy Sets, Ph.D. Dissertation, Lappeenranta University of Technology, Acta Universitatis Lappeenrantaensis 90 (1999)
Manes, E.G.: Algebraic Theories. Springer, Heidelberg (1976)
Pawlak, Z.: Rough sets. Int. J. Computer and Information Sciences 5, 341–356 (1982)
Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)
Tarski, A.: On the calculus of relations. J. Symbolic Logic 6, 65–106 (1941)
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Eklund, P., Galán, M.A. (2006). Monads Can Be Rough. In: Greco, S., et al. Rough Sets and Current Trends in Computing. RSCTC 2006. Lecture Notes in Computer Science(), vol 4259. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11908029_9
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DOI: https://doi.org/10.1007/11908029_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-47693-1
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