ISSN: A/F
JANOLI International Journal of
Mathematical Science ( JIJMS )
Login/Register
For Reference
About Journal Aim and Scope Editorial Board Publication Ethics Indexing And Abstracting Peer Review Process About Journal Aim and Scope Editorial Board Publication Ethics Indexing And Abstracting Peer Review Process

Topological Insights into Prime and Minimal Prime Filters on Paradistributive Latticoids

Authors: Vishwash Singh

Published: 2025-03-05 23:07:06.283

Download PDF

Abstract

In this paper, we investigate prime filters and minimal prime filters on a paradistributive latticoid (PDL) and establish significant theoretical results. We demonstrate that the annihilator filter S∙S^\bullet is equal to the intersection of all prime filters not containing SS, providing a novel perspective on filter structures in PDLs. Furthermore, we explore minimal prime filters, analyzing their unique properties and relationships within the PDL framework. Our study also presents equivalent conditions for a PDL to be relatively complemented, offering insights into its algebraic structure. Additionally, we derive and discuss the topological properties of the spaces of prime filters and minimal prime filters, emphasizing their interactions and contributions to the broader understanding of PDLs. These results expand the foundational knowledge of PDLs and provide new tools for analyzing their algebraic and topological characteristics.

Keywords

Minimal Prime Filters, Paradistributive Latticoids (PDLs), Lattice Theory, Annihilator Filters, Topological Properties, Filter Theory

References

  1. Birkhoff, G. (1940). Lattice Theory. American Mathematical Society.
  2. Gratzer, G. (2011). Lattice Theory: Foundation. Springer Science & Business Media.
  3. Balbes, R., & Dwinger, P. (1974). Distributive Lattices. University of Missouri Press.
  4. Johnstone, P. T. (1982). Stone Spaces. Cambridge University Press.
  5. Simmons, H. (1982). "The lattice of filters of a distributive lattice." Algebra Universalis, 15, 158–164.
  6. Ward, M., & Dilworth, R. P. (1939). "Residuated lattices." Transactions of the American Mathematical Society, 45(3), 335-354.
  7. Priestly, H. A. (1970). "Representation of distributive lattices by means of ordered topological spaces." Bulletin of the London Mathematical Society, 2(2), 186-190.
  8. Lakser, H. (1971). "Injective hulls of semilattices." Canadian Journal of Mathematics, 23(5), 836-844.
  9. Rudeanu, S. (1974). Lattice Functions and Equations. Springer-Verlag.
  10. G. Birkhof, Lattice Teory (Providence: American Mathe-matical Society Colloquium Publications XXV, 1967).
  11. U. M. Swamy and G. C. Rao, “Almost Distributive Lattices,”Journal of the Australian Mathematical Society. Series A. PureMathematics and Statistics 31, no. 1 (1981): 77–91.
  12. ] G. N. Rao and T. A. Habtamu, “Filters in Almost Lattices,”Bulletin of the Intentional Mathenmatical Virtual Institute 10,no. 1 (2020): 69–82.[4] M. Benado, “Partially Ordered Sets and the Schreir Re-fnement Teorem, II.Teory of Multistructures,” Czecho-slovak Mathematical Journal 5, no. 3 (1955): 308–344.
  13. I. P. Cabrera, P. Cordero, G. Gutierrez, J. Martinez, andM. Ojeda-Aciego, “On Residuation in Multilattices: Filters,Congruences and Homomorphisms,” Fuzzy Sets and Systems234, no. 1 (2014): 1–21, https://doi.org/10.1016/j.fss.2013.04.002.
  14. D. C. Awouafack, E. Fouotsa, and C. L´el´e, “Te Properties ofMaximal Filters in Multilattices,” Journal of Mathematics2022, no. 1 (2022): https://doi.org/10.1155/2022/9714656.[7] M. F. Atiyah and I. Macdonald, Introduction to CommutativeAlgebra (New York: Longman Higher Education, 1969).
  15. S. Rasouli, “Te Pure Spectrum of a Residuated Lattice”(2022).
  16. R. Bandaru and S. Ajjarapu, “Paradistributive Latticoids,” Eu-ropean Journal of Pure and Applied Mathematics 17, no. 2 (2024):819–834, https://doi.org/10.29020/nybg.ejpam.v17i2.5125.
  17. S. Ajjarapu, R. Bandaru, R. Shukla, and Y. B. Jun,“Parapseudo-Complementation on Paradistributive Latti-coids,” European Journal of Pure and Applied Mathematics 17,no. 2 (2024): 1129–1145,
Download PDF

How to Cite

Vishwash Singh, (2025-03-05 23:07:06.283). Topological Insights into Prime and Minimal Prime Filters on Paradistributive Latticoids. JANOLI International Journal of Mathematical Science, Volume OM6fuUNmywfCu7IAA7wn, Issue 1.

Follow Us
Get In Touch

Vitthal Namdev Nagar, Opp Jain Temple Sanganer Sanganer Bazar Jaipur 302029 Rajasthan (India)

mail@janolijournals.org

© JANOLI International Journal. All Rights Reserved.