## Subcategories of the category of $$L$$-convex spaces.(English)Zbl 1372.52001

This paper presents several subcategories of the category of $$L$$-convex spaces (namely, stratified $$L$$-convex spaces, convex-generated $$L$$-convex spaces, weakly induced $$L$$-convex spaces and induced $$L$$-convex spaces) and discusses the relationship between them.

### MSC:

 52A01 Axiomatic and generalized convexity 03E72 Theory of fuzzy sets, etc. 06A15 Galois correspondences, closure operators (in relation to ordered sets)
Full Text:

### References:

 [1] Adámek, J.; Herrlich, H.; Strecker, G. E., Abstract and concrete categories, (1990), Wiley New York · Zbl 0695.18001 [2] Chepoi, V., Separation of two convex sets in convexity structures, J. Geom., 50, 30-51, (1994) · Zbl 0807.52002 [3] Demirci, M., ($$\mathcal{Z}_1, \mathcal{Z}_2$$)-complete partially ordered sets and their representations by $$\mathcal{Q}$$-spaces, Appl. Categ. Struct., 21, 703-723, (2013) · Zbl 1317.06001 [4] Demirci, M., Fundamental duality of abstract categories and its applications, Fuzzy Sets Syst., 256, 73-94, (2014) · Zbl 1339.54008 [5] Demirci, M., Stratified categorical fixed-basis fuzzy topological spaces and their duality, Fuzzy Sets Syst., 267, 1-17, (2015) · Zbl 1392.54007 [6] Ellis, E., A general set-separation theorem, Duke Math. J., 19, 417-421, (1952) · Zbl 0047.28601 [7] Eckhoff, J., Radon’s theorem in convex product structures I, Monatshefte Math., 72, 303-314, (1968) · Zbl 0159.51701 [8] Eckhoff, J., Radon’s theorem in convex product structures II, Monatshefte Math., 73, 17-30, (1969) · Zbl 0174.53701 [9] Huang, H. L.; Shi, F. G., L-fuzzy numbers and their properties, Inf. Sci., 178, 1141-1151, (2008) · Zbl 1136.03326 [10] Jamison, R. E., A general theory of convexity, (1974), University of Washington Seattle, Washington, Dissertation · Zbl 0282.28001 [11] Kay, D. C.; Womble, E. W., Axiomatic convexity theory and the relationship between the Carathéodory, Helly and Radon numbers, Pac. J. Math., 38, 471-485, (1971) · Zbl 0235.52001 [12] Lassak, M., On metric B-convexity for which diameters of any set and its hull are equal, Bull. Acad. Pol. Sci., 25, 969-975, (1977) · Zbl 0379.52010 [13] Levi, F. W., On Helly’s theorem and the axioms of convexity, J. Indian Math. Soc., 15, 65-76, (1951), Part A · Zbl 0044.19101 [14] Maruyama, Y., Lattice-valued fuzzy convex geometry, RIMS Kokyuroku, 164, 22-37, (2009) [15] Menger, K., Untersuchungen über allgemeine metrik, Math. Ann., 100, 75-163, (1928) · JFM 54.0622.02 [16] Preuss, G., Foundations of topology-an approach to convenient topology, (2002), Kluwer Academic Publisher Dordrecht, Boston, London · Zbl 1058.54001 [17] Rosa, M. V., On fuzzy topology fuzzy convexity spaces and fuzzy local convexity, Fuzzy Sets Syst., 62, 97-100, (1994) · Zbl 0854.54010 [18] Shi, F. G., Theory of $$L_\beta$$-nested sets and $$L_\alpha$$-nested sets and its applications, Fuzzy Syst. Math., 4, 65-72, (1995), (in Chinese) · Zbl 1266.03063 [19] Shi, F. G.; Xiu, Z. Y., A new approach to the fuzzification of convex structures, J. Appl. Math., 2014, (2014), 12 pages [20] F.G. Shi, Z.Y. Xiu, $$(L, M)$$-fuzzy convex structures, submitted for publication. [21] Sierkama, G., Carathéodory and Helly-numbers of convex-product-structures, Pac. J. Math., 61, 272-282, (1975) [22] Sierkama, G., Relationships between Carathéodory, Helly, Radon and exchange numbers of convex spaces, Nieuw Arch. Wiskd., 25, 115-132, (1977) [23] Soltan, V. P., Some questions in the abstract theory of convexity, Sov. Math. Dokl., 17, 730-733, (1976) · Zbl 0348.52002 [24] Soltan, V. P., D-convexity in graphs, Sov. Math. Dokl., 28, 419-421, (1983) · Zbl 0553.05060 [25] Soltan, V. P., Introduction to the axiomatic theory of convexity, (1984), Shtiinca Kishinev, (Russian) · Zbl 0559.52001 [26] Van De Vel, M., Finite dimensional convex structures II: the invariants, Topol. Appl., 16, 81-105, (1983) · Zbl 0556.52001 [27] Van De Vel, M., Binary convexities and distributive lattices, Proc. Lond. Math. Soc., 48, 3, 1-33, (1984) · Zbl 0505.06005 [28] Van De Vel, M., Theory of convex structures, (1993), North-Holland Amsterdam · Zbl 0785.52001 [29] Van Mill, J., Supercompactness and wallman spaces, Math. Cent. Tracts, 85, (1977), Amsterdam · Zbl 0407.54001 [30] Varlet, J. C., Remarks on distributive lattices, Bull. Acad. Pol. Sci., 23, 1143-1147, (1975) · Zbl 0323.06010 [31] Wang, G. J., Theory of topological molecular lattices, Fuzzy Sets Syst., 47, 351-376, (1992) · Zbl 0783.54032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.