Research Article
BibTex RIS Cite

Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces

Year 2025, Volume: 54 Issue: 3, 894 - 911, 24.06.2025

Yu Shen , C. H. Yan

https://doi.org/10.15672/hujms.1482251

Abstract

This paper aims to introduce the concepts of Mackey convergence degree for sequences and separation degree for spaces in $(L, M)$-fuzzy bornological vector spaces. Additionally, the paper presents the concept of bornological closure degree for fuzzy sets. Moreover, the paper discusses various characteristics of these concepts. Furthermore, the paper examines the degree relationships among a Mackey convergence sequence, a separated space, and a bornologically closed fuzzy set. Finally, the paper analyzes the properties of functors $\omega$ and $\iota$ between $M$-fuzzifying bornological vector spaces and $(L, M)$-fuzzy bornological vector spaces in terms of Mackey convergence degree and separation degree.

Keywords

(L M)-fuzzy bornological vector spaces Mackey convergence separation M-fuzzifying bornological vector spaces quotient (L M)-fuzzy bornological vector spaces

Ethical Statement

The authors declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supporting Institution

The National Natural Science Foundation of China

Project Number

NO. 12071225

References

  • [1] M. Abel and A. Šostak, Towards the theory of L-bornological spaces, Iranian Journal of Fuzzy Systems, 8(1), 19–28, 2011.
  • [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, The Journal of Fuzzy Mathematics, 11(3), 687–705, 2003.
  • [3] G. Beer and S. Levi, Gap, excess and bornological convergence, Set-Valued Analysis, 16, 489–506, 2008.
  • [4] G. Beer and S. Levi, Total boundedness and bornology, Topology and its Applications, 156, 1271–1288, 2009.
  • [5] G. Beer and S. Levi, Strong uniform continuity, Journal of Mathematical Analysis and Applications, 350, 568–589, 2009.
  • [6] G. Beer, S. Naimpally and J. Rodrigues-Lopes, S-topologies and bounded convergences, Journal of Mathematical Analysis and Applications, 339, 542–552, 2008.
  • [7] A. Caserta, G. Di Maio and L. Holá, Arzelá’s theorem and strong uniform convergence on bornologies, Journal of Mathematical Analysis and Applications, 371, 384–392, 2010.
  • [8] A. Caserta, G. Di Maio and Lj.D.R. Kočinac, Bornologies, selection principles and function spaces, Topology and its Applications, 159, 1847–1852, 2012.
  • [9] J.X. Fang and C.H. Yan, L-fuzzy topological vector spaces, The Journal of Fuzzy Mathematics, 5(1), 133–144, 1997.
  • [10] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.
  • [11] S.T. Hu, Boundedness in a topological space, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 28, 287–320, 1949.
  • [12] S.T. Hu, Introduction to general topology, Holden-Day, San-Francisko, 1966.
  • [13] Z.Y. Jin and C.H. Yan, Induced L-bornological vector spaces and L-Mackey convergence, Journal of Intelligent & Fuzzy Systems, 40, 1277–1285, 2021.
  • [14] Z.Y. Jin and C.H. Yan, Fuzzifying bornological linear spaces, Journal of Intelligent & Fuzzy Systems, 42, 2347–2358, 2022.
  • [15] A. Lechicki, S. Levi and A. Spakowski, Bornological convergence, The Australian Journal of Mathematical Analysis and Applications, 297, 751–770, 2004.
  • [16] C.Y. Liang, F.G. Shi and J.Y. Wang, (L,M)-fuzzy bornological spaces, Fuzzy Sets and Systems, 467, 108496, 2023.
  • [17] Y. Liu and M. Luo, Fuzzy Topology, World Scientific Publishing, Singapore, 1998.
  • [18] S. Özçağ, Bornologies and bitopological function spaces, Filomat, 27(7), 1345–1349, 2013.
  • [19] J. Paseka, S. Solovyov and M. Stehlík, On the category of lattice-valued bornological vector spaces, Journal of Mathematical Analysis and Applications, 419, 138–155, 2014.
  • [20] G. N. Raney, A subdirect-union representation for completely distributive complete lattices, Proceeding of American Math Society, 4, 518–522, 1953.
  • [21] M. Saheli, Fuzzy topology generated by fuzzy norm, Iranian Journal of Fuzzy Systems, 13(4), 113–123, 2016.
  • [22] Y. Shen and C.H. Yan, Fuzzifying bornologies induced by fuzzy pseudo-norms, Fuzzy Sets and Systems, 467, 108436, 2023.
  • [23] A. Šostak and I. Uljane, L-valued bornologies on powersets, Fuzzy Sets and Systems, 294, 93–104, 2016.
  • [24] A. Šostak and I. Uljane, Bornological structures on many-valued sets, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., 21, 143–168, 2017.
  • [25] G.J. Wang, Order-homomorphisms on Fuzzes, Fuzzy Sets and Systems, 12, 281–288, 1984.
  • [26] L. A. Zadeh, Fuzzy sets, Information and Control, 8, 238–353, 1965.
  • [27] H. Zhang and H. Zhang, The construction of I-bornological vector spa

Year 2025, Volume: 54 Issue: 3, 894 - 911, 24.06.2025

Yu Shen , C. H. Yan

https://doi.org/10.15672/hujms.1482251

Abstract

Project Number

NO. 12071225

References

  • [1] M. Abel and A. Šostak, Towards the theory of L-bornological spaces, Iranian Journal of Fuzzy Systems, 8(1), 19–28, 2011.
  • [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, The Journal of Fuzzy Mathematics, 11(3), 687–705, 2003.
  • [3] G. Beer and S. Levi, Gap, excess and bornological convergence, Set-Valued Analysis, 16, 489–506, 2008.
  • [4] G. Beer and S. Levi, Total boundedness and bornology, Topology and its Applications, 156, 1271–1288, 2009.
  • [5] G. Beer and S. Levi, Strong uniform continuity, Journal of Mathematical Analysis and Applications, 350, 568–589, 2009.
  • [6] G. Beer, S. Naimpally and J. Rodrigues-Lopes, S-topologies and bounded convergences, Journal of Mathematical Analysis and Applications, 339, 542–552, 2008.
  • [7] A. Caserta, G. Di Maio and L. Holá, Arzelá’s theorem and strong uniform convergence on bornologies, Journal of Mathematical Analysis and Applications, 371, 384–392, 2010.
  • [8] A. Caserta, G. Di Maio and Lj.D.R. Kočinac, Bornologies, selection principles and function spaces, Topology and its Applications, 159, 1847–1852, 2012.
  • [9] J.X. Fang and C.H. Yan, L-fuzzy topological vector spaces, The Journal of Fuzzy Mathematics, 5(1), 133–144, 1997.
  • [10] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.
  • [11] S.T. Hu, Boundedness in a topological space, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 28, 287–320, 1949.
  • [12] S.T. Hu, Introduction to general topology, Holden-Day, San-Francisko, 1966.
  • [13] Z.Y. Jin and C.H. Yan, Induced L-bornological vector spaces and L-Mackey convergence, Journal of Intelligent & Fuzzy Systems, 40, 1277–1285, 2021.
  • [14] Z.Y. Jin and C.H. Yan, Fuzzifying bornological linear spaces, Journal of Intelligent & Fuzzy Systems, 42, 2347–2358, 2022.
  • [15] A. Lechicki, S. Levi and A. Spakowski, Bornological convergence, The Australian Journal of Mathematical Analysis and Applications, 297, 751–770, 2004.
  • [16] C.Y. Liang, F.G. Shi and J.Y. Wang, (L,M)-fuzzy bornological spaces, Fuzzy Sets and Systems, 467, 108496, 2023.
  • [17] Y. Liu and M. Luo, Fuzzy Topology, World Scientific Publishing, Singapore, 1998.
  • [18] S. Özçağ, Bornologies and bitopological function spaces, Filomat, 27(7), 1345–1349, 2013.
  • [19] J. Paseka, S. Solovyov and M. Stehlík, On the category of lattice-valued bornological vector spaces, Journal of Mathematical Analysis and Applications, 419, 138–155, 2014.
  • [20] G. N. Raney, A subdirect-union representation for completely distributive complete lattices, Proceeding of American Math Society, 4, 518–522, 1953.
  • [21] M. Saheli, Fuzzy topology generated by fuzzy norm, Iranian Journal of Fuzzy Systems, 13(4), 113–123, 2016.
  • [22] Y. Shen and C.H. Yan, Fuzzifying bornologies induced by fuzzy pseudo-norms, Fuzzy Sets and Systems, 467, 108436, 2023.
  • [23] A. Šostak and I. Uljane, L-valued bornologies on powersets, Fuzzy Sets and Systems, 294, 93–104, 2016.
  • [24] A. Šostak and I. Uljane, Bornological structures on many-valued sets, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., 21, 143–168, 2017.
  • [25] G.J. Wang, Order-homomorphisms on Fuzzes, Fuzzy Sets and Systems, 12, 281–288, 1984.
  • [26] L. A. Zadeh, Fuzzy sets, Information and Control, 8, 238–353, 1965.
  • [27] H. Zhang and H. Zhang, The construction of I-bornological vector spa
There are 27 citations in total.

Details

Primary Language English
Subjects Topology
Journal Section Mathematics
Authors

Yu Shen 0009-0002-9808-9404

C. H. Yan 0000-0002-6500-7807

Project Number NO. 12071225
Early Pub Date August 27, 2024
Publication Date June 24, 2025
Submission Date May 11, 2024
Acceptance Date July 19, 2024
Published in Issue Year 2025 Volume: 54 Issue: 3

Cite

APA Shen, Y., & Yan, C. H. (2025). Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces. Hacettepe Journal of Mathematics and Statistics, 54(3), 894-911. https://doi.org/10.15672/hujms.1482251
AMA Shen Y, Yan CH. Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces. Hacettepe Journal of Mathematics and Statistics. June 2025;54(3):894-911. doi:10.15672/hujms.1482251
Chicago Shen, Yu, and C. H. Yan. “Mackey Convergence and Separation in $(L, M)$-Fuzzy Bornological Vector Spaces”. Hacettepe Journal of Mathematics and Statistics 54, no. 3 (June 2025): 894-911. https://doi.org/10.15672/hujms.1482251.
EndNote Shen Y, Yan CH (June 1, 2025) Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces. Hacettepe Journal of Mathematics and Statistics 54 3 894–911.
IEEE Y. Shen and C. H. Yan, “Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 3, pp. 894–911, 2025, doi: 10.15672/hujms.1482251.
ISNAD Shen, Yu - Yan, C. H. “Mackey Convergence and Separation in $(L, M)$-Fuzzy Bornological Vector Spaces”. Hacettepe Journal of Mathematics and Statistics 54/3 (June 2025), 894-911. https://doi.org/10.15672/hujms.1482251.
JAMA Shen Y, Yan CH. Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces. Hacettepe Journal of Mathematics and Statistics. 2025;54:894–911.
MLA Shen, Yu and C. H. Yan. “Mackey Convergence and Separation in $(L, M)$-Fuzzy Bornological Vector Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 3, 2025, pp. 894-11, doi:10.15672/hujms.1482251.
Vancouver Shen Y, Yan CH. Mackey convergence and separation in $(L, M)$-fuzzy bornological vector spaces. Hacettepe Journal of Mathematics and Statistics. 2025;54(3):894-911.

For more information about the journal, please visit: https://dergipark.org.tr/en/pub/hujms