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On the measure of M-rough approximation of L-fuzzy sets

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Abstract

We develop an approach allowing to measure the “quality” of rough approximation of fuzzy sets. It is based on what we call “an approximative quadruple” \(Q=(L,M,\varphi ,\psi )\) where L and M are complete lattice commutative monoids and \(\varphi : L \rightarrow M\), \(\psi : M \rightarrow L\) are mappings satisfying certain conditions. By realization of this scheme, we get measures of upper and lower rough approximation for L-fuzzy subsets of a set equipped with an M-preoder \(R: X\times X \rightarrow M\). In case R is symmetric, these measures coincide. Basic properties of such measures are studied. Besides, we present an interpretation of measures of rough approximation in terms of LM-fuzzy topologies.

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Notes

  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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Acknowledgements

The first named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2016R1D1A3A03918403). The second named author expresses gratefulness to Chonbuk National University and KIAS (Korea Institute for Advanced Study) for the financial support of his visits to Chonbuk National university in years 2012, 2014 and 2016 during which an essential part of this research was done. Both authors are grateful to the anonymous referee for reading the paper carefully and making valuable comments which allowed to eliminate some mistakes and to improve the exposition of the material.

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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 CS, University of Latvia, Riga, 1459, Latvia

    Alexander Šostak

  3. Faculty of Physics and Mathematics, University of Latvia, Riga, 1002, 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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Communicated by A. Di Nola.

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Han, SE., Šostak, A. On the measure of M-rough approximation of L-fuzzy sets. Soft Comput 22, 3843–3855 (2018). https://doi.org/10.1007/s00500-017-2841-y

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