Most of the research in fuzzy rough sets and fuzzy topological structures have been studied on the basis of fuzzy partially ordered sets. Instead of fuzzy partially ordered sets, the concept of distance functions in complete co-residuated lattices is introduced. Using distance functions, we define Alexandrov pretopology, Alexandrov precotopology and fuzzy interior (fuzzy closure) operators in complete co-residuated lattices, and we investigate their properties. Moreover, we prove that there exist isomorphic categories and Galois correspondence between topological categories.
HájekP., Metamathematices of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
8.
HerrlichH. and HušekM., Galois connections categorically, J Pure Appl Algebra68 (1990), 165–180.
9.
HöhleU., KlementE.P., Non-classical logic and their applications to fuzzy subsets, Kluwer Academic Publishers, Boston, 1995.
10.
HöhleU. and RodabaughS.E., Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht, 1999.
11.
JunshengQ. and Bao QingHu., On (⊙, &) -fuzzy rough sets based on residuated and co-residuated lattices, Fuzzy Sets Syst336 (2018), 54–86.
12.
KimY.C., Join-meet preserving maps and Alexandrov fuzzy topologies, Journal of Intelligent and Fuzzy Systems28 (2015), 457–467.
13.
KimY.C., Join-meet preserving maps and fuzzy preorders, Journal of Intelligent and Fuzzy Systems28 (2015), 1089–1097.
14.
KoJ.M. and KimY.C., Fuzzy complete lattices, Alexandrov L-fuzzy topologies and fuzzy rough sets, Journal of Intelligent and Fuzzy Systems38 (2020), 3253–3266.
15.
KoJ.M. and KimY.C., Preserving maps and approximation operators in complete co-residuated lattices, J Korean Inst Intell Syst30(5) (2020), 389–398.
16.
LaiH. and ZhangD., Fuzzy preorder and fuzzy topology, Fuzzy Sets Syst157 (2006), 1865–1885.
17.
MaZ.M. and HuB.Q., Topological and lattice structures of L-fuzzy rough set determined by lower and upper sets, Information Sciences218 (2013), 194–204.
PawlakZ., Rough sets: Theoretical Aspects of Reasoning about Data, System Theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, The Netherlands, (1991).
20.
PeiZ., PeiD. and ZhengL., Topology vs generalized rough sets, Int J Approx Reason52 (2011), 231–239.
21.
QiaoJ. and HuB.Q., A short note on L-fuzzy approximation spaces and L-fuzzy pretopological spaces, Fuzzy Sets Syst312 (2017), 126–134.
22.
RadzikowskaA.M. and KerreE.E., A comparative study of fuzy rough sets, Fuzzy Sets Syst126 (2002), 137–155.
23.
RamadanA.A., ElkordyE.H. and El-DarderyM., L-fuzzy approximition spaces and L-fuzzy toplogical spaces, Ir J Fuzzy Syst13(1) (2016), 115–129.
24.
RodabaughS.E. and KlementE.P., Topological and Algebraic Structures In Fuzzy Sets, The Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003.
25.
SheY.H. and WangG.J., An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and Mathematics with Applications58 (2009), 189–201.
26.
TurunenE., Mathematics Behind Fuzzy Logic, A Springer-Verlag Co., 1999.
27.
WangC.Y. and HuB.Q., Granular variable precision fuzzy rough sets with general fuzzy relations, Fuzzy Sets Syst275 (2015), 39–57.
28.
WangC.Y., Topological structures of L-fuzzy rough sets and similarity sets of L-fuzzy relations, Int J Approx Reason83 (2017), 160–175.
