Document Type: Research Paper


1 College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610000, P.R. China

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


In this paper,  we introduce  the notion of $M$-fuzzifying interval spaces, and discuss the relationship between $M$-fuzzifying interval spaces and $M$-fuzzifying convex structures.
It is proved that  the category  {\bf MYCSA2}  can be embedded in  the category  {\bf  MYIS}  as a reflective subcategory, where  {\bf MYCSA2} and   {\bf  MYIS} denote  the category of $M$-fuzzifying convex structures of  $M$-fuzzifying  arity $\leq 2$  and  the category of $M$-fuzzifying interval spaces, respectively.
 Under the framework of $M$-fuzzifying interval spaces,   subspaces and product spaces   are presented  and  some of their fundamental  properties are obtained.


[1] M. Berger, Convexity, American Mathematical Monthly, 97(8) (1990), 650{678.
[2] P. Dwinger, Characterizations of the complete homomorphic images of a completely distribu-
tive complete lattice I, Indagationes Mathematicae (Proceedings), 85 (1982), 403{414.
[3] J. Eckho , Helly, Radon, and Caratheodory type theorems, Handbook of convex geometrry,
Vol. A, B, North-Holland (1993), 389{448.
[4] J. M. Fang, Sums of L-fuzzy topological spaces, Fuzzy Sets and Systems, 157 (2005), 739{754.
[5] G. Gierz, et al., Continuous lattices and domains, Encyclopedia of Mathematics and its
Applications, 93, Cambridge University Press, Cambridge, 2003.
[6] T. E. Gantner, R. C. Steinlage and R. H. Warren, Compactness in fuzzy topological spaces,
J. Math. Anal. Appl., 62 (1978), 547{562.
[7] H. L. Huang and F. G. Shi, L-fuzzy numbers and their properties, Information Sciences, 178
(2008), 1141{1151.
[8] U. Hohle, Probabilistsche Metriken auf der Menge nicht negativen verteilungsfunktionen,
Aequationes Math., 18 (1978), 345{356.
[9] W. Kubis, Abstract convex structures in topology and set theory, PhD thesis, University of
Silesia Katowice, 1999.
[10] N. N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets and Systems, 27 (1988),
[11] Y. Maruyama, Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 1641 (2009), 22{37.
[12] C. V. Negoita and D. A. Ralescu, Applications of fuzzy sets to systems analysis, Interdisciplinary
Systems Research Series, vol. 11, Birkhauser, Basel, Stuttgart and Halsted Press,
New York, 1975.
[13] S. Philip, Studies on fuzzy matroids and related topics, PhD thesis, Cochin University of
Science and Technology, 2010.
[14] S. E. Rodabaugh, Separation axioms: representation theorems, compactness, and compacti-
cations, in: U.oHle, S.E. Rodabaugh (Eds.), Mathematics of fuzzy sets: logic, topology, and
measure theory, The Handbooks of Fuzzy Sets Series, vol. 3, Kluwer Academic Publishers,
Dordrecht, (1999), 481{552.

[15] M. V. Rosa, On fuzzy topology, fuzzy convexity spaces, and fuzzy local convexity, Fuzzy Sets
and Systems, 62 (1994), 97{100.
[16] M. V. Rosa, A study of fuzzy convexity with special reference to separation properties, PhD
thesis, Cochin University of Science and Technology, 1994.
[17] F. G. Shi, Theory of L -nested sets and L -nested sets and its applications, Fuzzy Systems
and Mathematics, in Chinese, 4 (1995), 65{72.
[18] F. G. Shi, L-fuzzy sets and prime element nested sets, Journal of Mathematical Research
and Exposition, in Chinese, 16 (1996), 398{402,
[19] F. G. Shi, Theory of molecular nested sets and its applications, Journal of Yantai Teachers
University (Natural Science), in Chinese, 1 (1996), 33{36.
[20] F. G. Shi, L-fuzzy relation and L-fuzzy subgroup, Journal of Fuzzy Mathematics, 8 (2000),
[21] F. G. Shi and Z. Y. Xiu, A new approach to the fuzzi cation of convex structures, Journal
of Applied Mathematics, Article ID 249183, (2014).
[22] F. G. Shi, (L;M)-fuzzy metric spaces, Indian Journal of mathematics, 39 (1991), 303{321.
[23] F. G. Shi and E. Q. Li, The restricted hull operator of M-fuzzifying convex structures, Journal
of Intelligent and Fuzzy Systems., 30(1) (2016), 409{421.
[24] M. L. J. Van de Vel, Theory of Convex Structures, North Holland, N. Y., 1993.
[25] G. J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47 (1992),