# 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.

## 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.

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