The axiomatic characterizations on L-fuzzy covering-based approximation operators

Abstract

Axiomatic characterization is the foundation of L-fuzzy rough set theory: the axiom sets of approximation operators guarantee the existence of L-fuzzy relations or L-fuzzy coverings that reproduce the approximation operators. Axiomatic characterizations of approximation operators based on L-fuzzy coverings have not been fully explored, although those based on L-fuzzy relations have been studied thoroughly. Focusing on three pairs of widely used L-fuzzy covering-based approximation operators, we establish an axiom set for each of them, and their independence is examined. It should be noted that the axiom set of each L-fuzzy covering-based approximation operator is different from its crisp counterpart, with an either new or stronger axiom included in the L-fuzzy version.

Keywords:

• L-fuzzy rough set
• L-fuzzy covering
• axiom set
• residuated lattice

Acknowledgements

The authors thank the reviewers and the editor for their valuable comments and suggestions. This work is also dedicated to the first and third authors’ master’s supervisor, Professor Guangwu Meng on the occasion of his 60th birthday.

Notes

No potential conflict of interest was reported by the authors.

1 If we let $\mathcal{C}=\{A=f(A)|A\in L^X\}$ as that in Belohlavek (2001), then we have not $1\in \mathcal{C}$ by (U1) generally. This means that $\mathcal{C}$ may not be an L-fuzzy covering.

2 This fact is pointed out by one anonymous referee.

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant number 11501278], [grant number 11471152]; Shandong Provincial Natural Science Foundation, China [grant number ZR2013AQ011], [grant number ZR2014AQ011] and the Ke Yan Foundation of Liaocheng University  [grant number 318011505].