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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 55 Issue: 4

Coreflective subcategories and classical embedding in \(L\)-measurable spaces category

Xiu Xin(Tianjin University of Technology)0000-0001-8667-1494
Huan Lian(Tianjin Normal University)0009-0001-7130-0188
Pages
—
Publication Date
2025-12-30
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

This research addresses thedefects of traditional frameworks, improves the categorical systemof lattice-valued measurable spaces, and provides support for thecross-application of the two types of measure theories.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:To meet the measurement needs in scenarios with fuzziness anduncertainty, fuzzy measure theory needs to extend the framework ofclassical measure theory to the lattice-valued context.
📊
Key Findings & Data:Based on cut sets, it establishesembedding/reduction functors, confirming that the category ofclassical measurable spaces {\bf MS} can be embedded into \(L\)-{\bfSMS} and \(L\)-{\bf WIMS}, and is isomorphic to the category ofgenerated \(L\)-\(\sigma\)-algebras and the category of induced\(L\)-measurable spaces \(L\)-{\bf IMS}.
💡
Conclusion & Impact:This research addresses thedefects of traditional frameworks, improves the categorical systemof lattice-valued measurable spaces, and provides support for thecross-application of the two types of measure theories.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:\(L\)-$\sigma$-algebra · Galois correspondence · coreflective subcategory

Abstract

To meet the measurement needs in scenarios with fuzziness anduncertainty, fuzzy measure theory needs to extend the framework ofclassical measure theory to the lattice-valued context. However,existing studies have limitations such as difficulties in verifyingthe closedness of complement operations and insufficient connectionsat the categorical level. Focusing on Shi$'$s\(L\)-\(\sigma\)-algebras and combining cut set theory with categorytheory, this paper defines subclasses of \(L\)-measurable spaces andconstructs the relevant categorical framework. By using Galoiscorrespondence and closure functors, it proves that the category ofstratified \(L\)-measurable spaces \(L\)-{\bf SMS} and the categoryof weakly induced \(L\)-measurable spaces \(L\)-{\bf WIMS} are bothcoreflective full subcategories of the category of \(L\)-measurablespaces \(L\)-{\bf MS}. Based on cut sets, it establishesembedding/reduction functors, confirming that the category ofclassical measurable spaces {\bf MS} can be embedded into \(L\)-{\bfSMS} and \(L\)-{\bf WIMS}, and is isomorphic to the category ofgenerated \(L\)-\(\sigma\)-algebras and the category of induced\(L\)-measurable spaces \(L\)-{\bf IMS}. This research addresses thedefects of traditional frameworks, improves the categorical systemof lattice-valued measurable spaces, and provides support for thecross-application of the two types of measure theories.

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Cite This Article

Xin, X., Lian, H. (2025). Coreflective subcategories and classical embedding in \(L\)-measurable spaces category. Hacettepe Journal of Mathematics and Statistics, Volume: 55 Issue: 4. https://doi.org/10.15672/hujms.1784919

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References (19)

  1. J. Ad´amek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990.
  2. R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, MO, 1974.
  3. L. Biacino, A. Lettieri, $L$-$\sigma$-algebras and $L$-measures, Fuzzy Sets Syst. 44, 219–225, 1991.
  4. D.L. Cohn, Measure theory (2nd ed.), Boston, MA: Birkhauser Boston, 2013.
  5. Y. Gao, B. Pang, Subcategories of the category of $\top$-convergence spaces, Hacet. J. Math. Stat. 53, 88–106, 2024.
  6. Y. Gao, B. Pang, The largest $T_2$ $\top$-compactification of $\top$-convergence spaces, Topol. Appl. 2026, https://doi.org/10.1016/j.topol.2025.109673.
  7. H.L. Huang and F.G. Shi, L-fuzzy numbers and their properties, Inform. Sci. 178, 1141–1151, 2008.
  8. E.P. Klement, Fuzzy $\sigma$-algebras and fuzzy measurable functions, Fuzzy Sets Syst. 4, 83–93, 1980.
  9. B. Pang, F.G. Shi, Subcategories of the category of L-convex spaces, Fuzzy Sets Syst. 313, 61–74, 2016.
  10. B. Pang. L-fuzzifying convex structures as L-convex structures, J. Nonlinear Convex Anal. 21, 2831–2841, 2020.
  11. F.G. Shi, The theory and applications of $L_\beta$-nested sets and $L_\alpha$-nested sets and its applications, Fuzzy Syst. Math. 9, 65–72, 1995. (Chinese)
  12. F.G. Shi, L-fuzzy relations and L-fuzzy subgroups, J. Fuzzy Math. 8, 491–499, 2000.
  13. F.G. Shi, A new approach to the fuzzification of matroids, Fuzzy Sets Syst. 160, 696–705, 2009.
  14. F.G. Shi, $(L, M)$-fuzzy $\sigma$-algebras, Int. J. Math. Math. Sci. 2010, 1–11, 2010.
  15. F.G. Shi, Z. Y. Xiu, (L,M)-fuzzy convex structures, J. Nonlinear Sci. Appl. 10, 3655– 3669, 2017.
  16. T. TAO, An introduction to measure theory, Providence, RI: American Mathematical Society, 2011.
  17. X.Y. Wu, E.Q. Li, Category and subcategories of (L,M)-fuzzy convex spaces, Iran. J. Fuzzy Syst. 16, 173–190, 2019.
  18. X. Xin, F.G. Shi, Categories of bi-fuzzy pre-matroids, Comput. Math. Appl. 59, 1548– 1558, 2010.
  19. X.S. Zhang, Representations of Image and Inverse of L-Fuzzy Mappings, Fuzzy Syst. Math. 16, 24–28, 2002. (Chinese)