This paper studies T-similarity of L-fuzzy relations and its algebraic structures where L is a complete residuated lattice. L-fuzzy topologies induced by L-fuzzy rough approximation operators are investigated and T-similarity of L-fuzzy relations is first introduced. Next, a characteristic condition for L-fuzzy relations to be T-transitive is established. Finally, algebraic structures on T-similarity of L-fuzzy relations are obtained and an example is given in order to illustrate the practical significance and possible applications of the proposed concepts.
BlountK. and TsinakisC., The structure of residuated lattices, International Journal of Algebra and Computation13(4) (2003), 437–461.
2.
ChangC., Fuzzy topological spaces, Journal of Mathematical Analysis and Applications24 (1968), 182–190.
3.
ChenP., LaiH. and ZhangD., Core ective hull of nite strong L-topological spaces, Fuzzy Sets and Systems182 (2011), 79–92.
4.
ChenP. and ZhangD., Alexandro L-co-topological spaces, Fuzzy Sets and Systems161 (2010), 2505–2514.
5.
DuboisD. and PradeH., Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems17 (1990), 191–208.
6.
DuboisD. and PradeH., Two foldfuzzy sets and rough sets-some issues in knowledge representation, Fuzzy Sets and Systems23 (1987), 3–18.
7.
GoguenJ.A., L-fuzzy sets, Journal of Mathematical Analysis and Applications18 (1967), 145–174.
8.
HaoJ. and LiQ., The relationshipbetween L-fuzzy rough set and L-topology, Fuzzy Sets and Systems178 (2011), 74–83.
9.
HájekP., Mathematics of fuzzy logics, Kluwer Academic Publishers, Boston, 1998.
10.
LaiH. and ZhangD., Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems157 (2006), 1865–1885.
11.
KortelainenJ., On the relationship between modified sets, topological spaces and rough sets, Fuzzy Sets and Systems61 (1994), 91–95.
12.
KunchevaL.I., Fuzzy rough sets: Application to feature selection, Fuzzy Sets and Systems51 (1992), 147–153.
13.
LowenR., Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis and Applications56 (1976), 621–633.
14.
LashinE.F., KozaeA.M., Abo KhadraA.A. and
MedhatT., Rough set theory for topological spaces, International Journal of Approximate Reasoning40 (2005), 35–43.
15.
LiuY. and LuoM., Fuzzy topology, World Scientific Publishing, Singapore, 1998.
16.
LiZ. and CuiR., Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators, Information Sciences305 (2015), 219–233.
17.
LiZ. and CuiR., T-similarity of fuzzy relations and related algebraicstructures, Fuzzy Sets and Systems275 (2015), 130–143.
18.
LiZ., XieT. and LiQ., Topological structure of generalized rough sets, Computers and Mathematics with Applications63 (2012), 1066–1071.
19.
LiuG. and ZhuW., The algebraic structures of generalized rough set theory, Information Sciences178 (2008), 4105–4113.
20.
LaiH. and ZhangD., Fuzzy preorder and fuzzy topology, Fuzzy Sets and Systems157 (2006), 1865–1885.
21.
MorsiN.N. and YakoutM.M., Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems100 (1998), 327–342.
22.
MiJ., WuW. and ZhangW., Constructive and axiomatic approaches for the study of the theory of rough sets, Pattern Recognition Artificial Intelligence15 (2002), 280–284.
23.
NandaS., Fuzzy rough sets, Fuzzy Sets and Systems45 (1992), 157–160.
24.
PalS.K., Soft data mining computational theory of perceptions and rough-fuzzy approach, Information Sciences163 (2004), 5–12.
25.
PawlakZ., Rough sets, International Journal of Computer and Information Science11 (1982), 341–356.
26.
PawlakZ., Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991.
27.
PuB. and LiuY., Fuzzy Topology I. Neighborhood strucure of a fuzzy point and Moore-Smith convergence, Journal of Mathematical Analysis and Applications79 (1980), 571–599.
28.
PalS.K. and MitraP., Case generation using rough sets with fuzzy representations, Transactions on Knowledge and Data Engineering16(3) (2004), 292–300.
29.
PeiZ., PeiD. and ZhengL., Topology vs generalized rough sets, International Journal of Approximate Reasoning52 (2011), 231–239.
30.
PawlakZ. and SkowronA., Rudiments of rough sets, Information Sciences177 (2007), 3–27.
31.
PawlakZ. and SkowronA., Rough sets: Some extensions, Information Sciences177 (2007), 28–40.
32.
PawlakZ. and SkowronA., Rough sets and Boolean reasoning, Information Sciences177 (2007), 41–73.
33.
QinK. and PeiZ., On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems151 (2005), 601–613.
34.
RadzikowskaA.M. and KerreE.E., A comparative study of fuzzy rough sets, Fuzzy Sets and Systems126 (2002), 137–155.
35.
RadzikowskaA.M. and KerreE.E., Fuzzy rough sets based on residuated lattices, in: Transactions on Rough Sets II, in: LNCS3135 (2004), 278–296.
36.
SostakA., Towards the theory of M-approximate systems: Fundamentals and examples, Fuzzy Sets and Systems161 (2010), 2440–2461.
37.
SkowronA. and StepaniukJ., Tolerance approximation spaces, Fundamenta Informaticae27 (1996), 245–253.
38.
SheY. and WangG., An axiomatic approach of fuzzy rough sets based on residuated lattices, Computers and Mathematics with Applications58 (2009), 189–201.
39.
SlowinskiR. and VanderpootenD., Similarity relation as a basis for rough approximations, ICS Research Report53 (1995), 249–250.
40.
TiwariS.P. and SrivastavaK., Fuzzy rough sets preoders and fuzzy topologies, Fuzzy Sets and Systems210 (2013), 63–68.
WangG., Theory of L-fuzzy topological spaces, Shaanxi Normal University Press, Xi’an, 1988.
43.
WardM. and DilworthR.P., Residuated lattices, Transactions of the American Mathematical Society45 (1939), 335–354.
44.
WuW. and ZhouL., On intuitionistic fuzzy topologies based on intuitionistic fuzzy reflexive and transitive relations, Soft Computing15 (2011), 1183–1194.
45.
WuW., MiJ. and ZhangW., Generalized fuzzy rough sets, Information Sciences151 (2003), 263–282.
46.
WuQ., WangT., HuangY., LiJ., Topology theory on rough sets, IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics38 (2008), 68–77.
47.
YaoY.Y., Constructive and algebraic methods of the theory of rough sets, Information Sciences109 (1998), 21–47.
48.
YaoY.Y., Two views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning15 (1996), 291–317.
49.
YeungD.S., ChenD., TsangE.C.C., LeeJ.W.T. and WangX., On the generalization of fuzzy rough sets, IEEE Transactions on Fuzzy Systems13 (2005), 343–361.
50.
YangL. and XuL., Topological properties of generalized approximation spaces, Information Sciences181 (2011), 3570–3580.
51.
ZhangD., An enriched category approach to many valued topology, Fuzzy Set sand Systems158 (2007), 349–366.
52.
ZhangX., DaiJ. and YuY., On the union and intersection operations of rough sets based on various approximation spaces, Information Sciences292 (2015), 214–229.
53.
ZhangX., MiaoD., LiuC. and LeM., Constructive methods of rough approximation operators and multigranulation rough sets, Knowledge-Based Systems91 (2016), 114–125.
54.
ZhangW., WuW., LiangJ. and LiD., Rough set theory and methods, Chinese Scientific Publishers, Beijing, 2001.
55.
ZhouL., WuW. and ZhangW., On intuitionistic fuzzy rough sets and their topological structures, International Journal of General Systems38 (2009), 589–616.
56.
ZhangW., WangJ., LiuW. and FangJ., An introduction to fuzzy mathematics, Xi’an Jiaotong University Press, Xi’an, 1991.
57.
ZhangX., ZhouB. and LiP., A general frame for intuitionistic fuzzy rough sets, Information Sciences216 (2012), 34–49.
