Characterizations of $L$-convex spaces

Document Type : Research Paper


Shenzhen Graduate School, Harbin Institute of Technology, 518055 Shen- zhen, P.R. China


In this paper, the concepts of $L$-concave structures, concave $L$-interior operators and concave $L$-neighborhood systems are introduced. It is shown that the category of $L$-concave spaces and the category of concave $L$-interior spaces are isomorphic, and they are both isomorphic to the category of concave $L$-neighborhood systems whenever $L$ is a completely distributive lattice. Also, it is proved that these categories are all isomorphic to the category of $L$-convex spaces whenever $L$ is a completely distributive lattice with an order-reversing involution operator.


[1] S. Abramsky and A. Jung, Domain theory, S. Abramsky, D. Gabbay, T.S.E. Mailbaum (Eds.),
Handbook of Logic in Computer Science, Oxford University Press, Oxford (1994), 1{168.
[2] V. Chepoi, Separation of two convex sets in convexity structures, J. Geom., 50 (1994), 30{51.
[3] E. Ellis, A general set-separation theorem, Duke Math. J., 19 (1952), 417{421.
[4] J. Eckho , Radon's theorem in convex product structures I, Monatsh. Math., 72 (1968),
[5] J. Eckho , Radon's theorem in convex product structures II, Monatsh. Math., 73 (1969),
[6] R. E. Jamison, A general theory of convexity, Dissertation, University ofWashington, Seattle,
Washington, 1974.
[7] D. C. Kay and E. W. Womble, Axiomatic convexity theory and the relationship between the
Caratheodory, Helly and Radon numbers, Paci c J. Math., 38 (1971), 471{485.
[8] 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.
[9] F. W. Levi, On Helly's theorem and the axioms of convexity, J. Indian Math. Soc., 15 Part
A (1951), 65{76.
[10] Y. Maruyama, Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 164 (2009), 22{37.
[11] K. Menger, Untersuchungen uber allgemeine Metrik, Math. Ann., 100(1928), 75{163.
[12] B. Pang and F. G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets Syst.,
[13] B. Pang and F. G. Shi, L-hull operators and L-interval operators in L-convex spaces, Sub-
[14] M. V. Rosa, On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets
Syst., 62 (1994), 97{100.
[15] F. G. Shi and Z. Y. Xiu, A new approach to the fuzzi cation of convex structures, J. Appl.
Math., 2014 (2014), 12 pages.
[16] G. Sierkama, Caratheodory and Helly-numbers of convex-product-structures, Paci c J. Math.,
61 (1975), 272{282.
[17] G. Sierkama, Relationships between Caratheodory, Helly, Radon and Exchange numbers of
convex spaces, Nieuw Archief Wisk, 25 (1977), 115{132.
[18] V. P. Soltan, Some questions in the abstract theory of convexity, Soviet Math. Dokl., 17
(1976), 730{733.
[19] V. P. Soltan, D-convexity in graphs, Soviet Math. Dokl., 28 (1983), 419{421.
[20] V. P. Soltan, Introduction to the axiomatic theory of convexity, (Russian) Shtiinca, Kishinev
[21] M. Van De Vel, Finite dimensional convex structures II: the invariants, Topology Appl., 16
(1983), 81{105.
[22] M. Van De Vel, Binary convexities and distributive lattices, Proc. London Math. Soc., 48
(1984), 1{33.
[23] M. Van De Vel, Theory of convex structures, North-Holland, Amsterdam 1993.
[24] J. C. Varlet, Remarks on distributive lattices, Bull. Acad. Polon. Sci., 23 (1975), 1143{1147.