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M-valued Measure of Roughness for Approximation of L-fuzzy Sets and Its Topological Interpretation

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Computational Intelligence (IJCCI 2014)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 620))

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Abstract

We develop a scheme allowing to measure the “quality” of rough approximation of fuzzy sets. This scheme is based on what we call “an approximation quadruple” \((L,M,\varphi ,\psi )\) where L and M are cl-monoids (in particular, \(L=M=[0,1]\)) and \(\psi : L \rightarrow M\) and \(\varphi : M \rightarrow L\) are satisfying certain conditions mappings (in particular, they can be the identity mappings). In the result of realization of this scheme we get measures of upper and lower rough approximation for L-fuzzy subsets of a set equipped with a reflexive transitive M-fuzzy relation R. In case the relation R is also symmetric, these measures coincide and we call their value by the measure of roughness of rough approximation. Basic properties of such measures are studied. A realization of measures of rough approximation in terms of L-fuzzy topologies is presented.

The support of the ESF project 2013/0024/1DP/1.1.1.2.0/13/APIA/VIAA/045 is kindly announced.

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Notes

  1. 1.

    The subscripts \(_L\) and \(_M\) will be usually omitted as soon as it is clear from the context in which monoid we are working.

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Authors and Affiliations

  1. Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju, Jeonbuk, 561-756, Republic of Korea

    Sang-Eon Han

  2. Institute of Mathematics and Computer Science, University of Latvia, Raina Boulevard 29, Riga, 1459, Latvia

    Alexander Šostak

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  1. Sang-Eon Han
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  2. Alexander Šostak
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Correspondence to Alexander Šostak .

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Editors and Affiliations

  1. Ingenería Informática, Escuela Técnica Superior de, Granada, Spain

    Juan Julian Merelo

  2. aSEEB-ISR-IST, Technical University of Lisbon (IST), Lisbon, Portugal

    Agostinho Rosa

  3. Facultad de Informática, University of Murcia, Murcia, Spain

    José M. Cadenas

  4. University of Coimbra, Coimbra, Portugal

    António Dourado

  5. Images, Signals and Intelligence, University PARIS-EST Créteil (UPEC), Créteil, France

    Kurosh Madani

  6. Instituto Politécnico de Setúbal (IPS), Setúbal, Portugal

    Joaquim Filipe

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Han, SE., Šostak, A. (2016). M-valued Measure of Roughness for Approximation of L-fuzzy Sets and Its Topological Interpretation. In: Merelo, J.J., Rosa, A., Cadenas, J.M., Dourado, A., Madani, K., Filipe, J. (eds) Computational Intelligence. IJCCI 2014. Studies in Computational Intelligence, vol 620. Springer, Cham. https://doi.org/10.1007/978-3-319-26393-9_15

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  • DOI: https://doi.org/10.1007/978-3-319-26393-9_15

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