Document Type: Research Paper


1 College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, P.R.China

2 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P.R.China


Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $\mathscr{B}$ (resp. $\varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with  $\mathscr{B}$ (resp. $\varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbases can be used to characterize CP mappings and CC mappings between $M$-fuzzifying convex spaces.


[1] P. Dwinger, Characterizations of the complete homomorphic images of a completely distribu-
tive complete lattice I, Indagationes Mathematicae (Proceedings), 85 (1982), 403{414.
[2] W. Kubis, Abstract Convex Structures in Topology and Set Theory, PhD thesis, University
of Silesia Katowice, 1999.
[3] M. Lassak, On metric B-convexity for which diameters of any set and its hull are equal, Bull.
Acad. Polon. Sci., 25 (1977), 969{975.
[4] Y. Maruyama, Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 164 (2009), 22{37.
[5] J. V. Mill, Supercompactness and Wallman Spaces, Math. Centre Tracts 85, Amsterdam
[6] B. Pang and F. G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets Syst.,
313 (2017), 61{74.
[7] B. Pang and Y. Zhao, Characterizations of L-convex spaces, Iranian Journal of Fuzzy Sys-
tems, 13(4) (2016), 51{61.
[8] M. V. Rosa, On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets
Syst., 62 (1994), 97{100.
[9] M. V. Rosa, A Study of Fuzzy Convexity with Special Reference to Separation Properties,
PhD thesis, Cochin University of Science and Technology, 1994.
[10] F. G. Shi and E. Q. Li, The restricted hull operator of M-fuzzifying convex structures, J.
Intell. Fuzzy Syst., 30 (2015), 409{421.
[11] F. G. Shi and Z. Y. Xiu, A new approach to the fuzzi cation of convex structures, J. Appl.
Math., vol. 2014, Article ID 249183.
[12] V. P. Soltan, d-convexity in graphs, Soviet Math. Dokl., 28 (1983), 419{421.
[13] M. L. J. Van de Vel, Theory of Convex Structures, North-Holland, Amsterdam 1993.
[14] J. C. Varlet, Remarks on distributive lattices, Bull. Acad. Polon. Sci., 23 (1975), 1143{1147.
[15] X. Y. Wu and S. Z. Bai, On M-fuzzifying JHC convex structures and M-fuzzifying Peano
interval spaces, J. Intell. Fuzzy Syst., 30 (2016), 2447{2458.
[16] Z. Y. Xiu and F. G. Shi, M-fuzzifying interval spaces, Iranian Journal of Fuzzy Systems,
14(1) (2017), 145{162.
[17] M. S. Ying, A new approach for fuzzy topology (I), Fuzzy Sets Syst., 39 (1991), 303{321.
[18] Y. L. Yue and J. M. Fang, Bases and subbases in I-fuzzy topologiacl spaces, J. Math. Res.
Exposition, 26(1) (2006), 89{95.
[19] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 238{353.