In this paper, the concepts of (L, M)-fuzzy internal relations and (L, M)-fuzzy enclosed relations between two L-fuzzy sets are introduced. They are defined respectively to be M-fuzzy subsets of LX × LX satisfying a set of axioms. A categorical approach is provided to present these relations. It is proved that the category of (L, M)-fuzzy internal relation spaces, the category of (L, M)-fuzzy enclosed relation spaces and the category of (L, M)-fuzzy topological spaces are isomorphic. In addition, some (L, M)-fuzzy internal relations and (L, M)-fuzzy enclosed relations are naturally constructed from (L, M)-fuzzy quasi-uniformities and (L, M)-fuzzy S-quasi-proximities.
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References
1.
C.L.Chang, Fuzzy topological spaces, J Math Anal Appl24 (1968), 182–190.
2.
J.Fang, Categories isomorphic to L-FTOP, Fuzzy Sets and Systems157 (2006), 820–831.
3.
G.Gierz, et al., A Compendium of Continuous Lattices, Springer, Berlin, 1980.
4.
U.Höhle, Upper semicontinuous fuzzy sets and applications, J Math Anal Appl78 (1980), 659–673.
5.
U.Höhle and S.E.Rodabaugh, (Eds), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Kluwer Academic Publishers (Boston/Dordrecht/ London), 1999.
6.
A.K.Katsaras, On fuzzy syntopogenous structures, J Math Anal Appl99 (1983), 219–236.
7.
T.Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz, Poznan, Poland, 1985.
E.S.Kim, S.H.Ahn and D.H.Park, The relationship between L-fuzzy proximities and L-fuzzy quasi-uniformities, Iranian Journal of Fuzzy Systems9(6) (2012), 101–111.
10.
Y.Liu and M.Luo, Fuzzy Topology, World Scientific Publishing Co. Pte. Ltd, Singapore, 1997.
11.
B.Pang, Degrees of separation properties in stratified L-generalized convergence spaces using residual implication, Filomat31(20) (2017), 6293–6305.
PangB., Categorical properties of L-fuzzifying convergence spaces, Filomat32(11) (2018), 4021–4036.
14.
B.Pang and F.-G.Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets Syst313 (2017), 61–74.
15.
B.Pang and F.-G.Shi, Strong inclusion orders between L-subsets and its applications in L-convex spaces, Quaest Math41(8) (2018), 1021–1043.
16.
B.Pang and F.-G.Shi, Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces, Fuzzy Sets Syst (2018). DOI: 10.1016/j.fss.2018.05.012
17.
B.Pang and Z.-Y.Xiu, Stratified L-prefilter convergence structures in stratified L-topological spaces, Soft Comput22 (2018), 7539–7551.
18.
B.Pang and Z.-Y.Xiu, An axiomatic approach to bases and sub-bases in L-convex spaces and their applications, Fuzzy Sets Syst (2018). DOI: 10.1016/j.fss.2018.08.002
19.
B.Pang and Y.Zhao, Characterizations of L-convex spaces, Iran J Fuzzy Syst13(4) (2016), 51–61.
20.
S.E.Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies, Chapter 2 in [5], pp. 91–116.
21.
A.P.Šostak, On a fuzzy topological structure, Rendiconti Ciecolo Matematico Palermo (Suppl Ser II)11 (1985), 89–103.
