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

Dilek Bayrak; Sultan Yamak

Research Article

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

Year 2019, Volume: 48 Issue: 1, 180 - 185, 01.02.2019

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 $\vee$-distributive lattice $L$.

Keywords

lattice, distributive lattice, fuzzy ideal

References

N. Ajmal and K.V. Thomas, The lattices of fuzzy subgroups and fuzzy normal subgroups, Inform. Sci. 76, 1–11, 1994.
N. Ajmal and K.V. Thomas, The lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 74, 371–379, 1995.
N. Ajmal and K.V. Thomas, The lattices of fuzzy normal subgroups is modular, Inform. Sci. 83, 199–218, 1995.
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.
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.
J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18, 145–174, 1967.
T. Head, A meta theorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems 73, 349–358, 1995.
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.
R. Kumar, Non-distributivity of the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 97, 393–394, 1998.
S. Majumdar and Q.S. Sultana, The lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 81, 271–273, 1996.
J.N. Mordeson and D.S. Malik, Fuzzy Commutative Algebra, World Scientific, 1998.
M. Tarnauceanu, Distributivity in lattices of fuzzy subgroups, Inform. Sci. 179, 1163– 1168, 2009.
L.A. Zadeh, Fuzzy sets, Inform. Control 8, 338–353, 1965.
Q. Zhang, The lattice of fuzzy (left, right) ideals of a ring is modular, Fuzzy Sets and Systems 125, 209–214, 2002.
Q. Zhang and G. Meng, On the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 112, 349–353, 2000.

Year 2019, Volume: 48 Issue: 1, 180 - 185, 01.02.2019

Dilek Bayrak Sultan Yamak

Abstract

References

N. Ajmal and K.V. Thomas, The lattices of fuzzy subgroups and fuzzy normal subgroups, Inform. Sci. 76, 1–11, 1994.
N. Ajmal and K.V. Thomas, The lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 74, 371–379, 1995.
N. Ajmal and K.V. Thomas, The lattices of fuzzy normal subgroups is modular, Inform. Sci. 83, 199–218, 1995.
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.
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.
J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18, 145–174, 1967.
T. Head, A meta theorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems 73, 349–358, 1995.
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.
R. Kumar, Non-distributivity of the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 97, 393–394, 1998.
S. Majumdar and Q.S. Sultana, The lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 81, 271–273, 1996.
J.N. Mordeson and D.S. Malik, Fuzzy Commutative Algebra, World Scientific, 1998.
M. Tarnauceanu, Distributivity in lattices of fuzzy subgroups, Inform. Sci. 179, 1163– 1168, 2009.
L.A. Zadeh, Fuzzy sets, Inform. Control 8, 338–353, 1965.
Q. Zhang, The lattice of fuzzy (left, right) ideals of a ring is modular, Fuzzy Sets and Systems 125, 209–214, 2002.
Q. Zhang and G. Meng, On the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 112, 349–353, 2000.

There are 18 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	Dilek Bayrak Sultan Yamak
Publication Date	February 1, 2019
Published in Issue	Year 2019 Volume: 48 Issue: 1

Cite

APA	Bayrak, D., & Yamak, S. (2019). A note on distributivity of the lattice of $L$-ideals of a ring. Hacettepe Journal of Mathematics and Statistics, 48(1), 180-185.
AMA	Bayrak D, Yamak S. A note on distributivity of the lattice of $L$-ideals of a ring. Hacettepe Journal of Mathematics and Statistics. February 2019;48(1):180-185.
Chicago	Bayrak, Dilek, and Sultan Yamak. “A Note on Distributivity of the Lattice of $L$-Ideals of a Ring”. Hacettepe Journal of Mathematics and Statistics 48, no. 1 (February 2019): 180-85.
EndNote	Bayrak D, Yamak S (February 1, 2019) A note on distributivity of the lattice of $L$-ideals of a ring. Hacettepe Journal of Mathematics and Statistics 48 1 180–185.
IEEE	D. Bayrak and S. Yamak, “A note on distributivity of the lattice of $L$-ideals of a ring”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 1, pp. 180–185, 2019.
ISNAD	Bayrak, Dilek - Yamak, Sultan. “A Note on Distributivity of the Lattice of $L$-Ideals of a Ring”. Hacettepe Journal of Mathematics and Statistics 48/1 (February 2019), 180-185.
JAMA	Bayrak D, Yamak S. A note on distributivity of the lattice of $L$-ideals of a ring. Hacettepe Journal of Mathematics and Statistics. 2019;48:180–185.
MLA	Bayrak, Dilek and Sultan Yamak. “A Note on Distributivity of the Lattice of $L$-Ideals of a Ring”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 1, 2019, pp. 180-5.
Vancouver	Bayrak D, Yamak S. A note on distributivity of the lattice of $L$-ideals of a ring. Hacettepe Journal of Mathematics and Statistics. 2019;48(1):180-5.