Fuzzy Initial and Final Segments in ADL’s

Main Article Content

G. Srikanya, G. Prakasam Babu, K. Ramanuja Rao, Ch. Santhi Sundar Raj


In this paper, we define the concepts of fuzzy initial and final segments in an Almost Distributive Lattice (ADL) and certain properties of these are discussed. It is proved that the set of fuzzy initial segments forms a complete lattice and that the set of fuzzy final segments of an ADL A forms a complete lattice if and only if A is a bounded distributive lattice.

Article Details


  1. J.A. Goguen, L-Fuzzy Sets, J. Math. Anal. Appl. 18 (1967), 145-174. https://doi.org/10.1016/0022-247x(67)90189-8.
  2. N. Kuroki, On Fuzzy Ideals and Fuzzy Bi-Ideals in Semigroups, Fuzzy Sets Syst. 5 (1981), 203-215. https://doi.org/10.1016/0165-0114(81)90018-x.
  3. W. Liu, Fuzzy Invariant Subgroups and Fuzzy Ideals, Fuzzy Sets Syst. 8 (1982), 133-139. https://doi.org/10.1016/0165-0114(82)90003-3.
  4. D.S. Malik, J.N. Mordeson, Extensions of Fuzzy Subrings and Fuzzy Ideals, Fuzzy Sets Syst. 45 (1992), 245-251. https://doi.org/10.1016/0165-0114(92)90125-n.
  5. T.K. Mukherjee, M.K. Sen, On Fuzzy Ideals of a Ring I, Fuzzy Sets Syst. 21 (1987), 99-104. https://doi.org/10.1016/0165-0114(87)90155-2.
  6. N. Teshale Amare, S. Gonnabhaktula, Ch. Santhi Sundar Raj, L-Fuzzy Prime Spectrums of ADLs, Adv. Fuzzy Syst. 2021 (2021), 5520736. https://doi.org/10.1155/2021/5520736.
  7. A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl. 35 (1971), 512-517. https://doi.org/10.1016/0022-247x(71)90199-5.
  8. Ch.S.S. Raj, N.T. Amare, U.M. Swamy, Fuzzy Prime Ideals of ADL’s, Int. J. Comput. Sci. Appl. Math. 4 (2018), 32-36. https://doi.org/10.12962/j24775401.v4i2.3187.
  9. Ch.S.S. Raj, N.T. Amare, U.M. Swamy, Prime and Maximal Fuzzy Filters of ADLs, Palestine J. Math. 9 (2020), 730-739.
  10. M.H. Stone, The Theory of Representation for Boolean Algebras, Trans. Amer. Math. Soc. 40 (1936), 37-111. https://doi.org/10.2307/1989664.
  11. U.M. Swamy, Ch.S.S. Raj, A.N. Teshale, Fuzzy Ideals of Almost Distributive Lattices, Ann. Fuzzy Math. Inf. 14 (2017), 371-379.
  12. U.M. Swamy, D.V. Raju, Algebraic Fuzzy Systems, Fuzzy Sets Syst. 41 (1991), 187-194. https://doi.org/10.1016/0165-0114(91)90222-c.
  13. U.M. Swamy, D.V. Raju, Fuzzy Ideals and Congruences of Lattices, Fuzzy Sets Syst. 95 (1998), 249-253. https://doi.org/10.1016/s0165-0114(96)00310-7.
  14. U.M. Swamy, G.C. Rao, Almost Distributive Lattices, J. Aust. Math. Soc. A. 31 (1981), 77-91. https://doi.org/10.1017/s1446788700018498.
  15. U.M. Swamy, K.L.N. Swamy, Fuzzy Prime Ideals of Rings, J. Math. Anal. Appl. 134 (1988), 94-103. https://doi.org/10.1016/0022-247x(88)90009-1.
  16. R.V. Babu, B. Venkateswarlu, Initial and Final Segments in ADL’s, Southeast Asian Bull. Math. 41 (2017), 127–131.
  17. L.A. Zadeh, Fuzzy Sets, Inf. Control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65)90241-x.