The characterization for the sobriety of $L$-convex spaces

[1] J. Ad´amek, H. Herrlich, G. E. Strecker, Abstract and concrete categories, Wiley, New York, 1990.

[2] M. Ern´e, General stone duality, Topology and Its Applications, 137 (2004), 125-158. https://doi.org/10.1016/
S0166-8641(03)00204-9

[3] M. H. Escard´o, R. C. Flagg, Semantic domains, injective spaces and monad: Extended abstract, Electronic Notes in
Theoretical Computer Science, 20 (1999), 229-244. https://doi.org/10.1016/S1571-0661(04)80077-X

[4] S. P. Franklin, Some results on order convexity, The American Mathematical Monthly, 69 (1962), 357-359. https:
//doi.org/10.1080/00029890.1962.11989897

[5] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott, Continuous lattices and domains,
Cambridge University Press, Cambridge, 2003.

[6] J. Goubault-Larrecq, Non-Hausdorff topology and domain theory, Cambridge University Press, Cambridge, 2013.

[7] U. H¨ohle, Many-valued topology and its applications, Kluwer Academic Publishers, Boston, Dordrecht, London,
2001.

[8] H. Komiya, Convexity on a topological space, Fundamenta Mathematicae, 111 (1981), 107-113. https://doi.org/
10.4064/fm-111-2-107-113

[9] M. Y. Liu, Y. L. Yue, The reflectivity of the category of stratified L-algebraic closure spaces, Iranian Journal of
Fuzzy Systems, 21 (2024), 117-127. https://doi.org/10.22111/IJFS.2024.47190.8314

[10] M. Y. Liu, Y. L. Yue, X. W. Wei, Frame-valued Scott open set monad and its algebras, Fuzzy Sets and Systems,
460 (2023), 52-71. https://doi.org/10.1016/j.fss.2022.11.002

[11] E. Marczewski, Independence in abstract algebras results and problems, Colloquium Mathematicum, 14 (1966),
169-188. https://doi.org/10.4064/cm-14-1-169-188

[12] Y. Maruyama, Lattice-valued fuzzy convex geometry, Computational Geometry and Discrete Mathematics, 164
(2009), 22-37.

[13] Y. Maruyama, Fundamental results for pointfree convex geometry, Annals of Pure and Applied Logic, 161 (2010),
1486-1501. https://doi.org/10.1016/j.apal.2010.05.002

[14] J. Nieminen, The ideal structure of simple ternary algebras, Colloquium Mathematicum, 40 (1978), 23-29. https:
//doi.org/10.4064/cm-40-1-23-29

[15] R. Noor, A. K. Srivastava, The categories L-Top0 and L-Sob as epireflective hulls, Soft Computing, 18 (2014),
1865-1871. https://doi.org/10.1007/s00500-014-1315-8

[16] B. Pang, Quantale-valued convex structures as lax algebras, Fuzzy Sets and Systems, 473 (2023), 108737. https:
//doi.org/10.1016/j.fss.2023.108737

[17] B. Pang, F. G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets and Systems, 313 (2017), 61-74.
https://doi.org/10.1016/j.fss.2016.02.014

[18] B. Pang, F. G. Shi, Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces,
Fuzzy Sets and Systems, 369 (2019), 20-39. https://doi.org/10.1016/j.fss.2018.05.012

[19] M. V. Rosa, A study of fuzzy convexity with special reference to separation properties, Cochin University of Science
and Technology, Cochin, India, 1994.

[20] K. I. Rosenthal, Quantales and their applications, Longman House, Burnt Mill, Harlow, 1990.

[21] C. Shen, F. G. Shi, Characterizations of L-convex spaces via domain theory, Fuzzy Sets and Systems, 380 (2020),
44-63. https://doi.org/10.1016/j.fss.2019.02.009

[22] C. Shen, S. J. Yang, D. S. Zhao, F.G. Shi, Lattice-equivalence of convex spaces, Algebra University, 80 (2019), 26.
https://doi.org/10.1007/s00012-019-0600-x

[23] S. H. Su, Q. G. Li, The category of algebraic L-closure systems, Journal of Intelligent and Fuzzy Systems, 33
(2017), 2199-2210. https://doi.org/10.3233/JIFS-16452

[24] L. C. Sun, B. Pang, Completely Scott closed set and its applications, Topology and Its Applications, Preprint, 2024.

[25] M. Van De Vel, Binary convexities and distributive lattices, Proceedings of the London Mathematical Society, 48
(1984), 1-33. https://doi.org/10.1112/plms/s3-48.1.1

[26] M. Van De Vel, On the rank of a topological convexity, Fundamenta Mathematicae, 119 (1984), 17-48. https:
//doi.org/10.4064/fm-119-2-101-132

[27] M. Van De Vel, Theory of convex spaces, North-Holland, Amsterdam, 1993.

[28] M. Y. Wu, L. K. Guo, Q. G. Li, New representatons of algebraic domains and algebraic L-domains via closure
systems, Semigroup Forum, 103 (2021), 700-712. https://doi.org/10.1007/s00233-021-10209-7

[29] G. J. Wu, W. Yao, Sober L-convex spaces and L-join-semilattices, Iranian Journal of Fuzzy Systems, 21 (2024),
163-177. https://doi.org/10.22111/ijfs.2024.48576.8572

[30] C. C. Xia, Some further results on pointfree convex geometry, Algebra University, 85 (2024), 20. https://doi.
org/10.1007/s00012-024-00847-7

[31] L. S. Xu, X. X. Mao, Formal topological characterizations of various continuous domains, Computational and
Applied Mathematics, 56 (2008), 444-452. https://doi.org/10.1016/j.camwa.2007.12.010

[32] W. Yao, Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed-complete poset, Fuzzy Sets and
Systems, 161 (2010), 983-987. https://doi.org/10.1016/j.fss.2009.06.018

[33] W. Yao, A survey of fuzzifications of frames, the Papert-Papert-Isbell adjunction and sobriety, Fuzzy Sets and
Systems, 190 (2012), 63-81. https://doi.org/10.1016/j.fss.2011.07.013

[34] W. Yao, S. E. Han, A stone-type duality for sT0 stratified Alexandrov L-topological spaces, Fuzzy Sets and Systems,
282 (2016), 1-20. https://doi.org/10.1016/j.fss.2014.12.012

[35] W. Yao, L. X. Lu, Fuzzy Galois connections on fuzzy posets, Mathematical Logic Quarterly, 55 (2009), 84-91.
https://doi.org/10.1002/malq.200710079

[36] W. Yao, C. J. Zhou, Representation of sober convex spaces by join-semilattices, Journal of Nonlinear and Convex
Analysis, 21 (2020), 2715-2724.

[37] Y. L. Yue, W. Yao, W. K. Ho, Applications of Scott-closed sets in convex structures, Topology and Its Applications,
314 (2022), 108093. https://doi.org/10.1016/j.topol.2022.108093

[38] Z. X. Zhang, A general categorical reflection for various completions of closure spaces and Q-ordered sets, Fuzzy
Sets and Systems, 473 (2023), 1-18. https://doi.org/10.1016/j.fss.2023.108736

[39] Q. Y. Zhang, L. Fan, Continuity in quantitative domains, Fuzzy Sets and Systems, 154 (2005), 118-131. https:
//doi.org/10.1016/j.fss.2005.01.007

[40] B. Zhao, Y. J. Zhang, Epireflections in the category of T0 stratified Q-cotopological spaces, Soft Computing, 27
(2023), 2443-2451. https://doi.org/10.1007/s00500-022-07809-y