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Journals / Hacettepe Journal of Mathematics and Statistics / 2019 / Cilt: 48 - Sayı: 1

A note on distributivity of the lattice of L-ideals of a ring

Dilek BAYRAK
Sultan YAMAK
Pages
180–185
Publication Date
2022-10-18
DOI
—
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⚡ TL;DR(Too Long; Didn't Read)

In this study, we prove that the lattice of L-ideals of a ring is distributive if and only if the lattice of its ideals is distributive, for an infinitely ∨distributive lattice L.

📌 3 Key Takeaways

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Objective & Hypothesis:Many studies have investigated the lattice of fuzzy substructures of algebraic structures such as groups and rings.
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Key Findings & Data:Many studies have investigated the lattice of fuzzy substructures of algebraic structures such as groups and rings.
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Conclusion & Impact:In this study, we prove that the lattice of L-ideals of a ring is distributive if and only if the lattice of its ideals is distributive, for an infinitely ∨distributive lattice L.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

Many studies have investigated the lattice of fuzzy substructures of algebraic structures such as groups and rings. In this study, we prove that the lattice of L-ideals of a ring is distributive if and only if the lattice of its ideals is distributive, for an infinitely ∨distributive lattice L.

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Cite This Article

BAYRAK, D., YAMAK, S. (2022). A note on distributivity of the lattice of L-ideals of a ring. Hacettepe Journal of Mathematics and Statistics, Cilt: 48 - Sayı: 1, 180–185. https://ajindex.com/articles/529777

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References (15)

  1. N. Ajmal and K.V. Thomas, The lattices of fuzzy subgroups and fuzzy normal sub-groups, Inform. Sci. 76, 1–11, 1994.
  2. N. Ajmal and K.V. Thomas, The lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 74, 371–379, 1995.
  3. N. Ajmal and K.V. Thomas, The lattices of fuzzy normal subgroups is modular, In-form. Sci. 83, 199–218, 1995.
  4. D. Bayrak and S. Yamak, The lattice of generalized normal L-subgroups, J. Intell. Fuzzy Syst. 27, 1143–1152, 2014. D. Bayrak and S. Yamak Distributivity and pseudocomplementation of lattices of generalized L-subgroups, Int. J. Algebra Stat. 5, 107–114, 2016. G. Birhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., Rhode Island, 1967.
  5. N. Gao, Q. Li and Z. Li, When do L-fuzzy ideals of a ring generate a distributive lattice?, Open Math. 4 (1), 531–542, 2016.
  6. J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18, 145–174, 1967.
  7. T. Head, A meta theorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems 73, 349–358, 1995.
  8. I. Jahan, Modularity of Ajmal for the lattices of fuzzy ideals of a ring, Iran. J. Fuzzy Syst. 5, 71–78, 2008. I. Jahan,The lattice of L-ideals of a ring is modular, Fuzzy Sets and Systems 199, 121–129, 2012.
  9. R. Kumar, Non-distributivity of the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 97, 393–394, 1998.
  10. S. Majumdar and Q.S. Sultana, The lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 81, 271–273, 1996.
  11. J.N. Mordeson and D.S. Malik, Fuzzy Commutative Algebra, World Scientific, 1998.
  12. M. Tarnauceanu, Distributivity in lattices of fuzzy subgroups, Inform. Sci. 179, 1163– 1168, 2009.
  13. L.A. Zadeh, Fuzzy sets, Inform. Control 8, 338–353, 1965.
  14. Q. Zhang, The lattice of fuzzy (left, right) ideals of a ring is modular, Fuzzy Sets and Systems 125, 209–214, 2002.
  15. Q. Zhang and G. Meng, On the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 112, 349–353, 2000.