International Journal of Nonlinear Analysis and Applications
Fuzzy $\alpha$-ideals and $\alpha$-congruences
Document Type : Research Paper
Author
Department of Mathematics, College of Natural and Computational Sciences, University of Gondar, Ethiopia
Abstract
In this paper, we introduce $L$-fuzzy $\alpha-$ideals and $L$-fuzzy $\alpha-$congruences on ADL $A$ and we discuss the properties of these. Also, we prove that the homomorphic image and pre-image of an $L$-fuzzy $\alpha-$congruence is also an $L$-fuzzy $\alpha-$congruence on $A$ under certain conditions. Finally, we establish a one-to-one correspondence between $L$-fuzzy $\alpha-$ideals and $L$-fuzzy $\alpha-$congruences on $A$.
Keywords
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[7] S. Lehmke, Some properties of fuzzy ideals of lattices, Fuzzy-IEEE 97 (1997), 813–818.
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[9] D.S. Malik and J.N. Mordeson, Extensions of fuzzy subrings and fuzzy ideals, Fuzzy Set Syst. 45 (1992), 245–251.
[10] T.K. Mukharjee and M.K. Sen, On fuzzy ideals of a ring, Proc. Seminar Fuzzy Syst. Non-Standard Logic, Calcutta,
India, 1984.
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457–466.
[12] A. Rosenfeld, Fuzzy groups, J. Maths. Anal. Appl. 35 (1971), 512–517.
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(2017), no. 4, 371–379.
[14] U.M. Swamy and D.V. Raju, Fuzzy ideals and congruences of lattices, Fuzzy Sets Syst. 95 (1998), 249–253.
[15] U.M. Swamy and D.V. Raju, Algebraic Fuzzy systems, Fuzzy Sets Syst. 41 (1991), 187–194.
[16] U.M. Swamy and D.V. Raju, Irreducibility in algebraic fuzzy systems, Fuzzy Sets Syst. 41 (1991), 233–241.
[17] U.M. Swamy and G.C. Rao, Almost distributive lattices, J. Aust. Math. Soc. 31 (1981), 77–91.
[18] U.M. Swamy, T. Rama Rao and Ch. Prabhakara Rao, On the Truth values of fuzzy statements, Int. J. Appl.
Comput. Sci. Math. 3 (2013), 1–6.
[19] L.A. Zadeh, Fuzzy set, Inf. Control 8 (1965), 338–353.
[2] T.K. Dutta, On fuzzy pseudo-ideals in semigroups, Proc. Seminar Fuzzy Syst. Non-Standard Logic, Calcutta,
India, 1984.
[3] L. Filep, Study on fuzzy algebra and relations from a general view point, Acta Math. Acad. Paedagog. Nyh´azi.(NS)
14 (1998), 49–55.
[4] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174.
[5] A.K. Katsaras and D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58
(1977), 135–146.
[6] N. Kuroki, Fuzzy ideals and fuzzy bi-ideals in semi-groups, Fuzzy Sets Syst. 5 (1981), 203–215.
[7] S. Lehmke, Some properties of fuzzy ideals of lattices, Fuzzy-IEEE 97 (1997), 813–818.
[8] W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Set Syst. 8 (1982), 133–139.
[9] D.S. Malik and J.N. Mordeson, Extensions of fuzzy subrings and fuzzy ideals, Fuzzy Set Syst. 45 (1992), 245–251.
[10] T.K. Mukharjee and M.K. Sen, On fuzzy ideals of a ring, Proc. Seminar Fuzzy Syst. Non-Standard Logic, Calcutta,
India, 1984.
[11] G.C. Ro and M.S. Rao, α−ideals in almost distributive lattices, Int. J. Contemp. Math, Sci. 4 (2009), no. 10,
457–466.
[12] A. Rosenfeld, Fuzzy groups, J. Maths. Anal. Appl. 35 (1971), 512–517.
[13] U.M. Swamy, C.S.S. Raj and N. Teshale, Fuzzy ideals of almost distributive lattices, Ann. Fuzzy Math. Inf. 14
(2017), no. 4, 371–379.
[14] U.M. Swamy and D.V. Raju, Fuzzy ideals and congruences of lattices, Fuzzy Sets Syst. 95 (1998), 249–253.
[15] U.M. Swamy and D.V. Raju, Algebraic Fuzzy systems, Fuzzy Sets Syst. 41 (1991), 187–194.
[16] U.M. Swamy and D.V. Raju, Irreducibility in algebraic fuzzy systems, Fuzzy Sets Syst. 41 (1991), 233–241.
[17] U.M. Swamy and G.C. Rao, Almost distributive lattices, J. Aust. Math. Soc. 31 (1981), 77–91.
[18] U.M. Swamy, T. Rama Rao and Ch. Prabhakara Rao, On the Truth values of fuzzy statements, Int. J. Appl.
Comput. Sci. Math. 3 (2013), 1–6.
[19] L.A. Zadeh, Fuzzy set, Inf. Control 8 (1965), 338–353.
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Volume 13, Issue 2
July 2022Pages 1455-1464
- Receive Date: 01 April 2021
- Revise Date: 29 May 2021
- Accept Date: 12 June 2021