Sober L-convex spaces and L-join-semilattices

Document Type : Research Paper

Authors

Nanjing University of Information Science and Technology

Abstract

With a complete residuated lattice $L$ as the truth value table, we extend the definition of sobriety of classical convex spaces to the framework of $L$-convex spaces. We provide a specific construction for the sobrification of an $L$-convex space and demonstrate that the full subcategory of sober $L$-convex spaces is reflective in the category of $L$-convex spaces with convexity-preserving mappings. Additionally, we introduce the concept of Scott $L$-convex structures on $L$-ordered sets. {\color{blue} As an application of this type of sobriety}, we obtain a characterization for the $L$-join-semilattice completion of an $L$-ordered set: an $L$-ordered set $Q$ is an $L$-join-semilattice completion of an $L$-ordered set $P$ if and only if the Scott $L$-convex space $(Q, \sigma^{\ast}(Q))$ is a sobrification of the Scott $L$-convex space $(P, \sigma^{\ast}(P))$.

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References

[1] J. Ad´amek, H. Herrlich, G. E. Strecker, Abstract and concrete categories, Wiley, New York, 1990.
[2] R. Bˇelohl´avek, Some properties of residuated lattices, Czechoslovak Mathematical Journal, 53 (2003), 161-171.
https://doi.org/10.1023/a:1022935811257
[3] M. Farber, R. E. Jamison, Convexity in graphs and hypergraphs, SIAM Journal on Discrete Mathematics, 7 (1986),
433-444. https://doi.org/10.1137/0607049
[4] M. Farber, R. E. Jamison, On local convexity in graphs, Discrete Mathematics, 66 (1987), 231-247. https://doi.
org/10.1016/0012-365x(87)90099-9
[5] S. P. Franklin, Some results on order convexity, The American Mathematical Monthly, 69 (1962), 357-359. https:
//doi.org/10.1080/00029890.1962.11989897
[6] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott, Continuous lattices and domains,
Cambridge University Press, New York, 2003.
[7] P. H´ajek, Metamathematics of fuzzy logic, Kluwer Academic Publishers, Dordrecht, 1998.
[8] F. Harary, J. Nieminen, Convexity in graphs, Journal of Differential Geometry, 16 (1981), 185-190. https://doi.
org/10.4310/jdg/1214436096
[9] U. H¨ohle, Many-valued topology and its applications, Kluwer Academic Publishers, New York, 2001.
[10] A. W. Jankowski, Some modifications of Scott’s theorem on injective spaces, Studia Logica, 45 (1986), 155-166.
https://doi.org/10.1007/BF00373271
[11] P. T. Johnstone, Stone spaces, Cambridge University Press, Cambridge, 1982.
[12] H. Komiya, Convexity on a topological space, Fundamenta Mathematicae, 111 (1981), 107-113. https://doi.org/
10.4064/fm-111-2-107-113
[13] 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
[14] 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
[15] X. X. Mao, L. S. Xu, Representation theorems for directed completions of consistent algebraic L-domains, Algebra
University, 54 (2005), 435-447. https://doi.org/10.1007/s00012-005-1953-x
[16] 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
[17] Y. Maruyama, Lattice-valued fuzzy convex geometry, Computational Geometry Discrete Mathematics, 164 (2009),
22-37.
[18] K. Menger, Untersuchungen ¨uber allgemeine metrik, Mathematische Annalen, 100 (1928), 75-163. https://doi.
org/10.1007/978-3-7091-6110-4_20
[19] 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
[20] 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
[21] 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
[22] B. Pang, F. G. Shi, Fuzzy counterparts of hull operators and interval operators in the framework of L-convex spaces,
Fuzzy Sets and System, 369 (2019), 20-39. https://doi.org/10.1016/j.fss.2018.05.012
[23] B. Pang, Z. Y. Xiu, An axiomatic approach to bases and subbases in L-convex spaces and their applications, Fuzzy
Sets and Systems, 369 (2019), 40-56. https://doi.org/10.1016/j.fss.2018.08.002
[24] M. V. Rosa, A study of fuzzy convexity with special reference to separation properties, Cochin University of Science
and Technology, Cochin, India, 1994.
[25] K. I. Rosenthal, Quantales and their applications, Longman House, Burnt Mill, Harlow, 1990.
[26] C. Shen, F. G. Shi, L-convex systems and the categorical isomorphism to Scott-hull operators, Iranian Journal of
Fuzzy Systems, 15 (2018), 23-40. https://doi.org/10.22111/ijfs.2017.3296
[27] 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
[28] F. G. Shi, Z. Y. Xiu, (L,M)-fuzzy convex structures, Journal of Nonlinear Sciences and Applications, 10 (2017),
3655-3669. https://doi.org/10.22436/jnsa.010.07.25
[29] M. Van De Vel, Binary convexities and distributive lattices, Proc. London Mathematical Society, 48 (1984), 1-33.
https://doi.org/10.1112/plms/s3-48.1.1
[30] 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
[31] M. Van De Vel, Theory of convex spaces, North-Holland, Amsterdam, 1993.
[32] K. Wang, F. G. Shi, Many-valued convex structures induced by fuzzy inclusion orders, Journal of Intelligent and
Fuzzy Systems, 36 (2019), 2705-2713. https://doi.org/10.3233/jifs-181103
[33] C. C. Xia, A categorical isomorphism between injective balanced L-S0-convex spaces and fuzzy frames, Fuzzy Sets
and Systems, 437 (2022), 114-126. https://doi.org/10.1016/j.fss.2021.09.018
[34] C. C. Xia, Some further results on pointfree convex geometry, Algebra University, 85 (2024), 20. https://doi.
org/10.1007/s00012-024-00847-7
[35] L. S. Xu, Continuity of posets via Scott topology and sobrifiction, Topology and its Applications, 153 (2006),
1886-1894. https://doi.org/10.1016/j.topol.2004.02.024
[36] 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
[37] W. Yao, A categorical isomorphism between injective stratified fuzzy T0-spaces and fuzzy continuous lattices, IEEE
Transactions on Fuzzy Systems, 24 (2016), 131-139. https://doi.org/10.1109/tfuzz.2015.2428720
[38] W. Yao, F. G. Shi, Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on fuzzy directed complete
posets, Fuzzy Sets and Systems, 173 (2011), 60-80. https://doi.org/10.1016/j.fss.2011.02.003
[39] W. Yao, Y. L. Yue, Algebraic representation of frame-valued continuous lattices via open filter monad, Fuzzy Sets
and Systems, 420 (2021), 143-156. https://doi.org/10.1016/j.fss.2021.02.004
[40] W. Yao, C. J. Zhou, Representation of sober convex spaces by join-semilattices, Journal Nonlinear Convex Analysis,
21 (2020), 2715-2724.
[41] W. Yao, C. J. Zhou, A lattice-type duality of lattice-valued fuzzy convex spaces, Journal Nonlinear Convex Analysis,
21 (2021), 2843-2853.
[42] 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
[43] D. X. Zhang, Sobriety of quantale-valued cotopological spaces, Fuzzy Sets and Systems, 350 (2018), 1-19. https:
//doi.org/10.1016/j.fss.2017.09.005
[44] 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
[45] Y. Zhang, K. Wang, Bounded sobriety and k-bounded sobriety of Q-cotopological spaces, Filomat, 33 (2019), 2095-
2106. https://doi.org/10.2298/fil1907095z
[46] Z. X. Zhang, F. G. Shi, Q. G. Li, K. Wang, On fuzzy monotone convergence Q-cotopological spaces, Fuzzy Sets
and Systems, 425 (2021), 18-33. https://doi.org/10.1016/j.fss.2020.11.021
[47] D. S. Zhao, T. H. Fan, Dcpo-completion of posets, Theoretical Computer Science, 411 (2010), 2167-2173. https:
//doi.org/10.1016/j.tcs.2010.02.020
Iranian Journal of Fuzzy Systems
Volume 21, Issue 4
July and August 2024
Pages 163-177

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