On the measure of M-rough approximation of L-fuzzy sets
- 133 Downloads
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.
KeywordsL-fuzzy set Upper M-rough approximation operator Lower M-rough approximation operator Measure of inclusion Measure of M-rough approximation of an L-fuzzy set Ditopology LM-ditopology
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.
Compliance with ethical standards
Conflict of interest
The authors confirm that they do not have conflict of interests.
- Eļkins A, Šostak A, Uļjane I (2016) On a category of extensional fuzzy rough approximation \(L\)-valued spaces. In: Information processing and management of uncertainty in knowledge-based systems, IPMU2016, Einhofen, The Netherlands, June 20–24, 2016, Proceedings, Part II, pp 48–60Google Scholar
- Han S-E, Kim IS, Šostak A (2014) On approximate-type systems generated by L-relations. Inf Sci 281:8–20Google Scholar
- Kehagias A, Konstantinidou M (2003) \(L\)-valued inclusion measure, \(L\)-fuzzy similarity, and \(L\)-fuzzy distance. Fuzzy Sets Syst 136:313–332 L-fuzzy set; upper M-rough approximation operator; lower M-rough approximation operator; measure of inclusion; measure of M-rough approximation of an L-fuzzy set; ditopology, LM-ditopologyMathSciNetCrossRefMATHGoogle Scholar
- Klawonn F (2000) Fuzzy points, fuzzy relations and fuzzy functions. In: Novák V, Perfilieva I (eds) Discovering the world with fuzzy logic. Springer, Berlin, pp 431–453Google Scholar
- Rosenthal KI (1990) Quantales and their applications, Pitman research notes in mathematics, vol 234. Longman Scientific and Technical, HarlowGoogle Scholar
- Yao YY (1998b) On generalizing Pawlak approximation operators. In: Proceedings of the first international conference on rough sets and current trends in computing, pp 298–307Google Scholar
- Zadeh (1965) Fuzzy sets, information and control 8:338–353Google Scholar