The reflexivity of the category of stratified L-algebraic closure spaces

Document Type : Research Paper

Authors

1 Department of Mathematis, Ocean Universiy of China

2 Department of Mathematics, Ocean University of China, 238 Songling Road, 266100, Qingdao, P.R.China

Abstract

In this paper, for each commutative and integral quantale, we give the stratified Sierpinski $L$-algebraic closure space and a sobrification of stratified $L$-algebraic closure spaces. Furthermore, we show that $\bf{S}$$L$-$\bf{AC}_0$---the category of stratified $S_0$-$L$-algebraic closure spaces is epireflective in $\bf{S}$$L$-$\bf{AC}$---the category of stratified $L$-algebraic closure spaces, and $\bf{Sob}$$L$-$\bf{AC}$---the category of sober $L$-algebraic closure spaces is epireflective and $\mathcal{E}$-firm epireflective in the category $\bf{S}$$L$-$\bf{AC}_0$.

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References

[1] J. Ad´amek, H. Herrlich, G. E. Strecker, Abstract and concrete categories, Dover Publications, Inc. New York, 1990.
[2] M. Barr, C. Wells, Toposes, triples and theories, Springer-Verlag, Berlin, 1985.
[3] M. M. Bonsangue, B. Jacobs, J. N. Kok, Duality beyond sober spaces: Topological spaces and observation frames,
Theoretical Computer Science, 151 (1995), 79-124. https://doi.org/10.1016/0304-3975(95)00048-2
[4] G. C. L. Br¨ummer, E. Giuli, H. Herrlich, Epireflections which are completions, Cahiers de Topologie et G´eom´etrie
Diff´erentielle Cat´egoriques, 33 (1992), 71-93.
[5] M. Ern´e, Closure, Chapter 5 in Beyond topology (Frederic Mynard, Elliott Pearl, eds.), American Mathematical
Society, 2009. https://doi.org/10.1090/conm/486/09510
[6] U. H¨ohle, Locales and L-topologies, Mathematik-Arbeitspapiere, 48 (1997), 223-250.
[7] P. T. Johnstone, Stone spaces, Cambridge University Press, Cambridge, 1982.
[8] L. D. Nel, R. G. Wilson, Epireflections in the caregory of T0-spaces, Fundamenta Mathematicae, 75 (1972), 69-74.
https://doi.org/10.4064/fm-75-1-69-74
[9] 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
[10] 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
[11] E. Riehl, Category theory in context, Dover Publications, Inc. New York, 2016.
[12] D. S. Scott, Outline of a mathematical theory of computation, in: The Fourth Annual Princeton Conf. on Information
Sciences and Systems, Princeton University Press, Princeton, 1970.
[13] D. S. Scott, Continuous lattices, topos, algebraic geometry and logic, in: Lecture Notes in Mathematics, Springer,
Berlin, 274 (1972), 97-136.
[14] 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
[15] C. Shen, S. J. Yang, D. S. Zhao, F. G. Shi, Lattice-equivalence of convex spaces, Algebra Universalis, 80(3) (2019),
1-19. https://doi.org/10.1007/s00012-019-0600-x
[16] S. K. Singh, A. K. Srivastava, On Q-sobriety, Quaestiones Mathematicae, 39 (2016), 179-188. https://doi.org/
10.29\89/16073606.2015.1068241
[17] A. K. Srivastava, A. S. Khastgir, On fuzzy sobriety, Information Sciences, 110 (1998), 195-205. https://doi.org/
10.1016/S0020-0255(97)10038-X
[18] M. V. D. Vel, Theory of convex structures, North-Holland, Amsterdam, 1993. https://doi.org/10.1016/
s0924-6509(09)x7015-7
[19] S. Vickers, Topology via logic, Cambridge University Press, Cambridge, 1989.
[20] W. Yao, C. J. Zhou, A lattice-type duality of lattice-valued fuzzy convex spaces, Journal of Nonlinear and Convex
Analysis, 21(12) (2020), 2843-2853.
[21] 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
[22] 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
[23] Y. J. Zhang, K. Y. Wang, Bounnded sobriety and k-bounded sobriety of Q-cotopological spaces, Filomat, 33(7)
(2019), 2095-2106. https://doi.org/10.2298/fil1907095z
[24] D. X. Zhang, G. Zhang, Sober topological spaces valued in a quantale, Fuzzy Sets and Systems, 444 (2022), 30-48.
https://doi.org/10.1016/j.fss.2021.11.007
[25] 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
 
Iranian Journal of Fuzzy Systems
Volume 21, Issue 2
March and April 2024
Pages 117-127

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