Characterizations of $L$-concavities and $L$-convexities via derived relations

This paper is to characterize $L$-concavities and $L$-convexities via some derived forms of relations and operators. Specifically, notions of $L$-concave derived internal relation space and $L$-concave derived hull space are introduced. It is proved that the category of $L$-concave derived internal relation spaces and the category of $L$-concave derived hull spaces are isomorphic to the category of $L$-concave spaces. Also, notions of $L$-convex derived enclosed relation space and $L$-convex derived hull space are introduced. It is proved that the category of $L$-convex derived enclosed relation spaces and the category of $L$-convex derived hull spaces are isomorphic to the category of $L$-convex spaces.

Keywords:

L-concave derived internal relation space, L-convex derived enclosed relation space L-concave derived hull space, L-convex derived hull space,

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[1] S.Z. Bai, Q-convergence of ideas in fuzzy lattices and its applications, Fuzzy Sets Syst. 92 (3), 357-363, 1997.
[2] S.Z. Bai, Pre-semi-closed sets and PS-convergence in L-fuzzy topological spaces, J. Fuzzy Math. 9, 497-509, 2001.
[3] F.H. Chen, Y. Zhong and F.G. Shi, M-fuzzifying derived spaces, J. Intel. Fuzzy Syst. 36 (1), 79-89, 2019.
[4] J.L. Kelly, General topology, Van Nastrand, New York, 1955.
[5] E.Q. Li and F.G. Shi, Some properties of M-fuzzifying convexities induced by Morders, Fuzzy Sets Syst. 350 (1), 41-54, 2018.
[6] C.Y. Liao and X.Y. Wu, L-topological-convex spaces generated by L-convex bases, Open Math. 17 (1), 1547-1566, 2019.
[7] Y. Maruyama, Lattice-valued fuzzy convex geometry, RIMS Kakyuroku 1641, 22-37, 2009.
[8] B. Pang, L-fuzzifying convex structures as L-convex structures, J. Nonlinear Convex Anal. 21, 2831-2841, 2020.
[9] B. Pang, Convergence structures in M-fuzzifying convex spaces, Quaest. Math. 43, 1541-1561, 2020.
[10] B. Pang, Hull operators and interval operators in (L,M)-fuzzy convex spaces, Fuzzy Sets and Systems 45, 106-127, 2021.
[11] B. Pang, Fuzzy convexities via overlap functions, IEEE Trans. Fuzzy Syst. 2022, DOI: 10.1109/TFUZZ.2022.3194354.
[12] B. Pang and F.G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets and Systems 313, 61-74, 2017.
[13] B. Pang and F.G. Shi, Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces, Fuzzy Sets and Systems 369, 20-39, 2019.
[14] B. Pang and Z.Y. Xiu, An axiomatic approach to bases and subbases in L-convex spaces and their applications, Fuzzy Sets and Systems 369, 40-56, 2019.
[15] B.M. Pu and Y.M. Liu, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (2), 571-599, 1980.
[16] M.V. Rosa. On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets and Systems 62 (1), 97-100, 1994.
[17] C. Shen, F.H. Chen and F. G. Shi, Derived operators on M-fuzzifying convex spaces, J. Intell. Fuzzy Systems 37 (3), 2687-2696, 2019.
[18] F.G. Shi, The fuzzy derived operator induced by the derived operator of ordinary set and fuzzy topology induced by the fuzzy derived operators, Fuzzy Systems Math. 5, 32-37, 1991.
[19] F.G. Shi and Z.Y. Xiu, A new approach to the fuzzification of convex structures, J. Appl. Math. 1, 1-12, 2014.
[20] F. G. Shi and Z.Y. Xiu, (L,M)-fuzzy convex structures, J. Nonlinear Sci. Appl. 10, 3655-3669, 2017.
[21] M.L.J. van de Vel, Theory of convex structures, Noth-Holland, Amsterdam, 1993.
[22] K. Wang and F.G. Shi, M-fuzzifying topological convex spaces, Iran. J. Fuzzy Syst. 15 (6), 159-174, 2018.
[23] X.Y.Wu, B. Davvaz and S.Z. Bai, On M-fuzzifying convex matroids and M-fuzzifying independent structures, J. Intell. Fuzzy Syst. 33 (1), 269-280, 2017.
[24] X.Y. Wu and E.Q. Li, Category and subcategories of (L,M)-fuzzy convex spaces, Iran. J. Fuzzy Syst. 15 (1), 129-146, 2019.
[25] X.Y. Wu and C.Y. Liao, (L,M)-fuzzy topological-convex spaces, Filomat 33 (19), 6435-6451, 2019.
[26] X.Y. Wu, C.Y. Liao and Y.H. Zhao, L-topological derived neighborhood relations and L-topological derived remotehood relations, Filomat 36 (5), 1433-1450, 2022.
[27] X.Y. Wu, Q. Liu, C.Y. Liao and Y.H. Zhao, L-topological derived internal (resp. enclosed) relation spaces, Filomat 35, 2497-2516, 2021.
[28] X.Y. Wu and F.G. Shi, L-concave bases and L-topological-concave spaces, J. Intell. Fuzzy Systems 35 (1), 4731-4743, 2018.
[29] X. Xin, F.G. Shi and S.G. Li, M-fuzzifying derived operators and difference derived operators, Iran. J. Fuzzy Syst. 7 (2), 71-81, 2010.
[30] Z.Y. Xiu and B. Pang, Base axioms and subbase axioms in M-fuzzifying convex spaces, Iran. J. Fuzzy Syst. 15 (2), 75-87, 2018.
[31] Z.Y. Xiu, Q.H. Li and B. Pang, Fuzzy convergence structures in the framework of L-convex spaces, Iran. J. Fuzzy Syst. 17 (4), 139-150, 2020.
[32] H. Yang and B. Pang, Fuzzy points based betweenness relations in L-convex spaces, Filomat 35, 3521-3532, 2021.
[33] S.J. Yang and F.G. Shi, M-fuzzifying independent spaces, J. Intell. Fuzzy Syst. 34 (1), 11-21, 2018.
[34] W. Yao, Moore-Smith convergence in (L,M)-fuzzy topology, Fuzzy Sets and Systems 190 (1), 47-62, 2012.
[35] W. Yao and L. X. Lu, Moore-Smith convergence in M-fuzzifying topological spaces,J. Math. Res. Exposition 31 (5), 770-780, 2011.
[36] L. Zhang and B. Pang, Monoidal closedness of the category of $\top$-semiuniform convergence spaces, Hacet. J. Math. Stat. 51, 1348-1370, 2022.
[37] L. Zhang and B. Pang, A new approach to lattice-valued convergence groups via ⊤-filters, Fuzzy Sets and Systems 2022, https://doi.org/10.2989/16073606.2021.1973140.
[38] F.F. Zhao and B. Pang, Equivalence among L-closure (interior) operators, L-closure (interior) operators and L-enclosed (internal) relations, Filomat 36, 979-1003, 2022.
[39] Y. Zhong, Derived operators of M-fuzzifying matroids, J. Intell. Fuzzy Systems 35 (20), 4673-4683, 2018.