We introduce the concepts of fuzzy join complete lattices and Alexandrov L-pretopologies in complete residuated lattices. We show that fuzzy join complete lattices, Alexandrov L-pretopologies, fuzzy meet complete lattices and Alexandrov L-precotopologies are equivalent. Moreover, we define L-preinterior operators (resp. L-preclosure operators) as a viewpoint of fuzzy joins (resp. fuzzy meet) and fuzzy rough sets. Furthermore their properties and examples are investigated.
BělohlávekR., Fuzzy Relational Systems, Kluwer Academic Publishers, New York, 2002.
2.
FanL., A new approach to quantitative domain theory, Electronic Notes in Theoretic Computer Science45 (2011).
3.
HájekP., Metamathematices of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, 1998.
4.
HaoJ. and LiQ., The relationship between L-fuzzy rough set and L-topology, Fuzzy Sets and Systems178 (2011), 74–83.
5.
HöhleU. and KlementE.P., Non-classical logic and their applications to fuzzy subsets, Kluwer Academic Publishers, Boston, 1995.
6.
HöhleU. and RodabaughS.E., Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht, 1999.
7.
KimY.C., Join-meet preserving maps and Alexandrov fuzzy topologies, Journal of Intelligent and Fuzzy Systems28 (2015), 457–467.
8.
KimY.C., Join-meet preserving maps and fuzzy preorders, Journal of Intelligent and Fuzzy Systems28 (2015), 1089–1097.
9.
KimY.C., Categories of fuzzy preorders, approximation operators and Alexandrov topologies, Journal of Intelligent and Fuzzy Systems31 (2016), 1787–1793.
10.
KimY.C. and KoJ.M., Images and preimages of filterbases, Fuzzy Sets and Systems157 (2006), 1913–1927.
11.
LaiH. and ZhangD., Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems157 (2006), 1865–1885.
12.
MaZ.M. and HuB.Q., Topological and lattice structures of Lfuzzy 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).
15.
QiaoJ. and HuB.Q., Granular variable precision L-fuzzy rough sets based on residuated lattices, Fuzzy Sets and Systems336 (2018), 148–166.
16.
RadzikowskaA.M. and KerreE.E., A comparative study of fuzy rough sets, Fuzzy Sets and Systems126 (2002), 137–155.
17.
RamadanA. and LiL., Categories of lattice-valued closure (interior) operators and Alexandroff L-fuzzy topologies, Iranian Journal of Fuzzy Systems16(3) (2019), 73–84.
18.
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.
19.
SheY.H. and WangG.J., An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and Mathematics with Applications58 (2009), 189–201.
20.
TiwariS.P. and SrivastavaA.K., Fuzzy rough sets, fuzzy preorders and fuzzy topologies, Fuzzy Sets and Systems210 (2013), 63–68.
21.
WangC.Y., Topological characterizations of generalized fuzzy rough sets, Fuzzy Sets and Systems312 (2017), 109–125.
22.
WangC.Y., Topological structures of L-fuzzy rough sets and similarity sets of L-fuzzy relations, International Journal of Approximate Reasoning83 (2017), 160–175.
23.
WangC.Y. and HuB.Q., Granular variable precision fuzzy rough sets with general fuzzy relations, Fuzzy Sets and Systems275 (2015), 39–57.
24.
WardM. and DilworthR.P., Residuated lattices, Trans Amer Math Soc45 (1939), 335–354.
25.
ZhangD., An enriched category approach to many valued topology, Fuzzy Sets and Systems158 (2007), 349–366.
26.
XieW.X., ZhangQ.Y. and FanL., Fuzzy complete lattices, Fuzzy Sets and Systems160 (2009), 2275–2291.
27.
ZhangQ.Y. and FanL., Continuity in quantitive domains, Fuzzy Sets and Systems154 (2005), 118–131.
28.
ZhangQ.Y., XieW.X. and FanL., The Dedekind-MacNeille completion for fuzzy posets, Fuzzy Sets and Systems160 (2009), 2292–2316.
29.
ZhaoF.F., LiL.Q., SunS.B. and JinQ., Rough approximation operators based on quantale-valued fuzzy generalized neighborhood systems, Iranian Journal of Fuzzy Systems16(6) (2019), 53–63.
