Korean J. Math.  Vol 29, No 2 (2021)  pp.305-320
DOI: https://doi.org/10.11568/kjm.2021.29.2.305

On $L$-fuzzy semi-prime ideals of a poset and separation theorems

Derso Abeje Engidaw, Tilahun Bimerew Alemu

Abstract


In this paper, the relations between  $ L $-fuzzy semi-prime (respectively, $ L $-fuzzy prime) ideals of a poset and  $ L $-fuzzy semi-prime (respectively, $ L $-fuzzy prime) ideals of the lattice of all ideals of a  poset are established. A result analogous to Separation Theorem is obtained using $ L $-fuzzy semi-prime ideals.


Keywords


Poset, Prime Ideal, Semiprime Ideal, L-Fuzzy prime Ideal, L-fuzzy Semi-prime Ideal

Subject classification

06A11, 06D72, 06A99, 08A72

Sponsor(s)



Full Text:

PDF

References


U. Acar, On L-fuzzy prime submodules, Hacet. J. Math. Stat., 34 (2005) 17–25. (Google Scholar)

N. Ajmal and K. V. Thomas, Fuzzy lattices, Inform. Sci, 79 (1994) 271-291. (Google Scholar)

(Google Scholar)

B. A. Alaba and G. M. Addis, L-Fuzzy ideals in universal algebras, Ann. Fuzzy Math. Inform., 17(1), (2019) 31-39. (Google Scholar)

(Google Scholar)

B. A. Alaba and G. M. Addis, L-Fuzzy prime ideals in universal algebras, Advances in Fuzzy Systems, Vol. 2019, Article ID 5925036, 7 pages, 2019. (Google Scholar)

(Google Scholar)

B. A. Alaba and G. M. Addis, L-Fuzzy semi-prime ideals in universal algebras, Korean J. Math., 27(2), (2019), 327-340. (Google Scholar)

(Google Scholar)

B.A Alaba and T.G. Alemayehu, Closure Fuzzy Ideals of MS-algebras, Ann. Fuzzy Math. Inform., 16 (2018) 247-260. (Google Scholar)

(Google Scholar)

B.A Alaba and T.G. Alemayehu, Fuzzy ideals in demipseudocomplemented MS-algebras, Ann. Fuzzy Math. Inform., 18(2019) 123-143. (Google Scholar)

(Google Scholar)

B. A. Alaba, M. A. Taye and D. A. Engidaw, L-Fuzzy ideals of a poset, Ann. Fuzzy Math. Inform. 16 (2018), 285–299. (Google Scholar)

(Google Scholar)

B. A. Alaba, M. Alamneh and D. Abeje, L-Fuzzy filters of a poset , Int. J. Comp. Sci. Appl. Math. 5(1) (2019), 23–29 (Google Scholar)

(Google Scholar)

B. A. Alaba, M. A. Taye and D. A. Engidaw, L-fuzzy prime ideals and maximal L-fuzzy ideals of a poset , Ann. Fuzzy Math. Inform. 18(1) (2019) 1–13. (Google Scholar)

(Google Scholar)

B. A. Alaba and D. A. Engidaw, L-Fuzzy semiprime ideals of a poset, Advances in Fuzzy Systems, Vol. 2020, Article ID 1834970, 10 pages, 2020. (Google Scholar)

(Google Scholar)

B. A. Alaba and W. Z. Norahun, α− fuzzy ideals and space of prime α-fuzzy ideals in distributive lattices, Ann. Fuzzy Math. Inform. 17 (2019), 147-163. (Google Scholar)

(Google Scholar)

B. A. Alaba and W. Z. Norahun, Fuzzy ideals and fuzzy filters of pseudo-complemented semi- lattices, Advances in Fuzzy Systems, vol. 2019, Article ID 4263923, 13 pages, 2019. (Google Scholar)

(Google Scholar)

R. Ameri and R. Mahjoob, The spectrum of prime L− submodules, Fuzzy Sets and Systems 159, (2008) 1107–1115. (Google Scholar)

(Google Scholar)

Y. Bo and W. Wu, Fuzzy ideals on a distributive lattice, Fuzzy Sets and Systems 35 (1990), 231–240. (Google Scholar)

(Google Scholar)

P. Das, Fuzzy vector spaces under triangular norms, Fuzzy Sets and Systems 25 (1988) 73-85. (Google Scholar)

(Google Scholar)

Davey, B. A., Priestley, H. A., Introduction to Lattices and Order , Cambridge University Press, 1990. (Google Scholar)

(Google Scholar)

V. N. Dixit, R. Kumar and N. Ajamal, On fuzzy rings, Fuzzy Sets and Systems, 49 (1992) 205-213. (Google Scholar)

(Google Scholar)

V. N. Dixit, R. Kumar and N. Ajmal, Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy Sets and Systems, 44(1991) 127–138. (Google Scholar)

(Google Scholar)

J. A. Goguen, L-fuzzy sets , J. Math. Anal. Appl. 18 (1967) 145–174. (Google Scholar)

(Google Scholar)

G.Gr ̈atzer, General Lattice Theory , Academic Press, New York, 1978. (Google Scholar)

(Google Scholar)

R. Hala ́s, Decomposition of directed sets with zero, Math. Slovaca 45 (1) (1995) 9-17. (Google Scholar)

(Google Scholar)

R. Hala ́s, Annihilators and ideals in ordered sets , Czechoslovak Math. J. 45(120) (1995), 127– 134. (Google Scholar)

(Google Scholar)

R. Hala ́s and J.Rach ̊unek, Polars and prime ideals in ordered sets, Discuss. Math., Algebra and Stochastic Methods 15 (1995), 43—59. (Google Scholar)

(Google Scholar)

A. K. Katsaras and D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58 (1977) 135–146. (Google Scholar)

(Google Scholar)

V. S. Kharat and K. A. Mokbel, Semi-prime ideals and separation theorem in posets, Order, 25(3) (2008), 195 - 210. (Google Scholar)

(Google Scholar)

V. S. Kharat and K. A. Mokbel, Primeness and semiprimeness in posets , Math. Bohem. 134(1) (2009), 19–30. (Google Scholar)

(Google Scholar)

B. B. N. Koguep and C. Lele, On fuzzy prime ideals of lattices, SJPAM 3(2008) 1–11. (Google Scholar)

(Google Scholar)

Kumar, R., Fuzzy ideals and fuzzy semiprime ideals, Inform. Sci. 66 (1992) 43–52. (Google Scholar)

(Google Scholar)

H.V. Kumbhojkar and M.S. Bapat, On semiprime fuzzy ideals, Fuzzy Sets and Systems 60 (1993) 219–223. (Google Scholar)

(Google Scholar)

J. Larmerov ́a and J. Rach ̊unek, Translations of distributive and modular ordered sets , Acta. Univ. Palack. Olomuc. Fac. Rerum.Natur. Math., 91 (1988), 13–23. (Google Scholar)

(Google Scholar)

B. B. Makamba, and V. Murali, On prime fuzzy submodules and radicals, J. Fuzzy Math. 8(4) (2000) 831–843. (Google Scholar)

(Google Scholar)

Y. Rav, Semiprime ideals in general lattices , J. Pure Appl. Algebra, 56 (1989) 105—118. (Google Scholar)

(Google Scholar)

A. Rosenfeld, Fuzzy groups , J. Math. Anal. Appl. 35 (1971) 512–517. (Google Scholar)

(Google Scholar)

M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936) 37-111. (Google Scholar)

(Google Scholar)

U. M. Swamy and D. V. Raju, Fuzzy Ideals and congruences of lattices, Fuzzy Sets and Systems 95 (1998) 249-253. (Google Scholar)

(Google Scholar)

K.L.N. Swamy and U.M. Swamy, Fuzzy prime ideals of rings , J. Math. Anal. and Appl. 134 (1988) 94–103. (Google Scholar)

(Google Scholar)

L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338–353. (Google Scholar)

(Google Scholar)


Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr