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

- 99 Downloads

## 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.

## Keywords

*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*-ditopology

## Notes

### 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.

### Compliance with ethical standards

### Conflict of interest

The authors confirm that they do not have conflict of interests.

## References

- Brown LM, Ertürk R, Dost Ş (2000) Ditopological texture spaces and fuzzy topology, I. Basic concepts. Fuzzy Sets Syst 110:227–236CrossRefMATHGoogle Scholar
- Bustinice H (2000) Indicator of inclusion grade for interval-valued fuzzy sets. Application for approximate reasoning based on interval-valued fuzzy sets. Int J Approx Reason 23:137–209MathSciNetCrossRefGoogle Scholar
- Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209CrossRefMATHGoogle Scholar
- 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
- Fang J (2007) I-fuzzy Alexandrov topologies and specialization orders. Fuzzy Sets Syst 158:2359–2374MathSciNetCrossRefMATHGoogle Scholar
- Goguen JA (1967) \(L\)-fuzzy sets. J Math Anal Appl 18:145–174MathSciNetCrossRefMATHGoogle Scholar
- Han S-E, Šostak A (2016) \(M\)-valued measure of roughness of \(L\)-fuzzy sets and its topological interpretation. In: Merelo JJ, Rosa A, Cadenas JM, Duarado A, Madani K, Filipe J (eds) Studies in computational intelligence, vol 620. Springer, Berlin, pp 251–266CrossRefGoogle Scholar
- Han S-E, Kim IS, Šostak A (2014) On approximate-type systems generated by L-relations. Inf Sci 281:8–20Google Scholar
- Hao J, Li Q (2011) The relation between \(L\)-fuzzy rough sets and \(L\)-topology. Fuzzy Sets Syst 178:74–83CrossRefMATHGoogle Scholar
- Höhle U (1992) \(M\)-valued sets and sheaves over integral commutative CL-monoids. In: Rodabaugh SE, Höhle U, Klement EP (eds) Applications of category theory to fuzzy subsets. Kluwer Academic Publishers, Dordrecht, pp 33–72CrossRefGoogle Scholar
- Höhle U (1995) Commutative residuated \(l\)-monoids. In: Höhle U, Klement EP (eds) Nonclassical logics and their applications to fuzzy subsets. Kluwer Academic Publishers, Docrecht, Boston, pp 53–106CrossRefGoogle Scholar
- Höhle U (1998) Many-valued equalities, singletons and fuzzy partitions. Soft Comput 2:134–140CrossRefGoogle Scholar
- Järvinen J (2002) On the structure of rough approximations. Fundam Inform 53:135–153MathSciNetMATHGoogle Scholar
- Järvinen J, Kortelainen J (2007) A unified study between modal-like operators, topologies and fuzzy sets. Fuzzy Sets Syst 158:1217–1225CrossRefMATHGoogle 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
- Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, DordrechtCrossRefMATHGoogle Scholar
- Kortelainen J (1994) On relationship between modified sets, topological spaces and rough sets. Fuzzy Sets Syst 61:91–95MathSciNetCrossRefMATHGoogle Scholar
- Lai H, Zhang D (2006) Fuzzy preoder and fuzzy topology. Fuzzy Sets Syst 157:1865–1885CrossRefMATHGoogle Scholar
- Menger K (1951) Probabilistic geometry. Proc NAS 27:226–229MathSciNetCrossRefMATHGoogle Scholar
- Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356CrossRefMATHGoogle Scholar
- Pu P, Liu Y (1980) Fuzzy topology I: neighborhood structure of a fuzzy point. J Math Anal Appl 76:571–599MathSciNetCrossRefMATHGoogle Scholar
- Qin K, Pei Z (2005) On the topological properties of fuzzy rough sets. Fuzzy Sets Syst 151:601–613MathSciNetCrossRefMATHGoogle Scholar
- Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155MathSciNetCrossRefMATHGoogle Scholar
- Rosenthal KI (1990) Quantales and their applications, Pitman research notes in mathematics, vol 234. Longman Scientific and Technical, HarlowGoogle Scholar
- Schweitzer B, Sklar A (1983) Probabilistic metric spaces. North Holland, New YorkMATHGoogle Scholar
- Skowron A (1988) On the topology in information systems. Bull Pol Acad Sci Math 36:477–480MathSciNetMATHGoogle Scholar
- Šostak A (1985) On a fuzzy topological structure. Suppl Rend Circ Matem Palermo Ser II 11:125–186MATHGoogle Scholar
- Šostak A (2010) Towards the theory of M-approximate systems: fundamentals and examples. Fuzzy Sets Syst 161:2440–2461MathSciNetCrossRefMATHGoogle Scholar
- Tiwari SP, Srivastava AK (2013) Fuzzy rough sets. Fuzzy preoders and fuzzy topoloiges. Fuzzy Sets Syst 210:63–68CrossRefMATHGoogle Scholar
- Valverde L (1985) On the structure of F-indistinguishability operators. Fuzzy Sets Syst 17:313–328MathSciNetCrossRefMATHGoogle Scholar
- Wiweger A (1988) On topological rough sets. Bull Pol Acad Sci Math 37:51–62MathSciNetGoogle Scholar
- Yao YY (1998a) A comparative study of fuzzy sets and rough sets. Inf Sci 109:227–242MathSciNetCrossRefMATHGoogle 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 L (1971) Similarity relations and fuzzy orderings. Inf Sci 3:177–200MathSciNetCrossRefMATHGoogle Scholar
- Zadeh (1965) Fuzzy sets, information and control 8:338–353Google Scholar
- Zeng W, Li H (2006) Inclusion measures, similarity measures and the fuzziness of fuzzy sets and their relations. Int J Intell Syst 21:639–653CrossRefMATHGoogle Scholar