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Abstract

In this paper, we introduce the concepts of Alexandrov L-neighborhood filters, Alexandrov L-topologies and Alexandrov L-convergence structures in complete residuated lattices. We investigate the Galois correspondences among Alexandrov L-neighborhood filters, Alexandrov L-topologies and Alexandrov L-convergence structures. Moreover, we investigate their topological properties and give their examples.

Keywords

Complete residuated lattices

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References

Bĕlohlávek , Fuzzy Relational Systems, Kluwer Academic Publishers, New York, 2002.

Fang , Stratified L-order convergence structures, Fuzzy Sets and Systems 161 (2010), 2130–2149.

Fang , Relationships between L-ordered convergence structures and strong L-tologies, Fuzzy Sets and Systems 161 (2010), 2923–2944.

Hohle and

S.E.

Rodabaugh , Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht, 1999.

Jager , A category of L-fuzzy convergence spaces, Quaes-tiones Math 24 (2001), 501–517.

Jager , Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems 156 (2005), 1–24.

Jager , Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets and Systems 158 (2007), 424–435.

Jin and

L.Q.

Li , One-axiom characterizations on lattice-valued closure(interior) operators, Journal of Intelligent and Fuzzy Systems 31 (2016), 1679–1688.

Jin ,

L.Q.

Li and

G.W.

Meng , On the relationships between types of L-convergence spaces, Iranian Journal of Fuzzy Systems 13(1) (2016), 93–103.

10.

Y.C.

Kim , Join-meet preserving maps and Alexandrov fuzzy topologies, Journal of Intelligent and Fuzzy Systems 28 (2015), 457–467.

11.

Y.C.

Kim , Categories of fuzzy preorders, approximation operators and Alexandrov topologies, Journal of Intelligent and Fuzzy Systems 31 (2016), 1787–1793.

12.

Y.C.

Kim and

J.M.

Ko , Images and preimages of L-filter bases, Fuzzy Sets and Systems 173 (2005), 93–113.

13.

J.M.

Ko and

Y.C.

Kim , Alexandrov L-filters and Alexan-drovL-convergence spaces, Journal of Intelligent and Fuzzy Systems. 10.3233/JIFS-171723

14.

Lai and

Zhang , Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems 157 (2006), 1865–1885.

15.

Li ,

Jin and

Hu , Lattice-valued convergence associated with CNS spaces, Fuzzy Sets and Systems. 10.1016Zj.fss.2018.05.023

16.

L.Q.

Li and

Jin , On adjunctions between Lim, L-Top, and SL-Lim, Fuzzy Sets and Systems 182 (2011), 66–78.

17.

Z.M.

Ma and

B.Q.

Hu , Topological and lattice structures of Lfuzzy rough set determined by lower and upper sets, Information Sciences 218 (2013), 194–204.

18.

Ma ,

Zhan ,

M.I.

Ali and

Mehmood , A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review 49(4) (2018), 511–529.

19.

Orpen and

Jager , Lattice-valued convergence spaces, Fuzzy Sets and Systems 162 (2011), 1–24.

20.

Pawlak , Rough sets, Internat J Comput Inform Sci 11 (1982), 341–356.

21.

Pang , Degrees of separation properties in stratified L-generalized convergence spaces using residuated implication, Filomat 31(20) (2017), 6293–6305.

22.

Pang , Stratified L-ordered filter spaces, Quaestiones Mathematicae 40(5) (2017), 661–678.

23.

Pang and

Yi , Several types of enriched (LM)-fuzzy convergence spaces, Fuzzy Sets and Systems 321 (2017), 55–72.

24.

A.M.

Radzikowska and

E.E.

Kerre , A comparative study of fuzy rough sets, Fuzzy Sets and Systems 126 (2002), 137–155.

25.

S.E.

Rodabaugh and

E.P.

Klement , 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.

26.

Y.H.

She and

G.J.

Wang , An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and Mathematics with Applications 58 (2009), 189–201.

27.

S.P.

Tiwari and

A.K.

Srivastava , Fuzzy rough sets fuzzy preorders and fuzzy topologies, Fuzzy Sets and Systems 210 (2013), 63–68.

28.

Ward and

R.P.

Dilworth , Residuated lattices, Trans Amer Math Soc 45 (1939), 335–354.

29.

W.X.

Xie ,

Q.Y.

Zhang and

Fan , Fuzzy complete lattices, Fuzzy Sets and Systems 160 (2009), 2275–2291.

30.

Yao , On many-valued L-fuzzy convergence spaces, Fuzzy Sets and Systems 159 (2008), 2503–2519.

31.

Zhan and

Xu , Two types of coverings based multigranulation rough fuzzy sets and applications to decision making, Artificial Intelligence Review, 2018. 10.1007/s10462-018-9649-8

32.

Zhang , An enriched category approach to many-valued topology, Fuzzy Sets and Systems 158 (2007), 349–366.

33.

Zhang and

Zhan , Novel classes of fuzzy soft ß-coverings based fuzzy rough sets with applications to multicriteria fuzzy group decision making, Soft Computing (2018). 10.1007/s00500-018-3470-9

34.

Q.Y.

Zhang and

Fan , Continuity in quantitive domains, Fuzzy Sets and Systems 154 (2005), 118–131.

35.

Q.Y.

Zhang ,

W.X.

Xie and

Fan , The Dedekind-MacNeille completion for fuzzy posets, Fuzzy Sets and Systems 160 (2009), 2292–2316.

Alexandrov L -topologies and Alexandrov L -convergence structures 1

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