Fuzzy Identity, Fuzzy Zero of Po Semigroup
and Completely Prime Fuzzy, Prime Fuzzy
Ideal of Po Semigroup

RamyaLatha P; A. Gangadhara Rao; J.M.Pradeep; K.Aruna

doi:https://doi.org/10.14445/22315373/IJMTT-V61P501

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 61 | Number 1 | Year 2018 | Article Id. IJMTT-V61P501 | DOI : https://doi.org/10.14445/22315373/IJMTT-V61P501

Fuzzy Identity, Fuzzy Zero of Po Semigroup and Completely Prime Fuzzy, Prime Fuzzy Ideal of Po Semigroup

RamyaLatha P, A. Gangadhara Rao, J.M.Pradeep, K.Aruna

Citation :

RamyaLatha P, A. Gangadhara Rao, J.M.Pradeep, K.Aruna, "Fuzzy Identity, Fuzzy Zero of Po Semigroup and Completely Prime Fuzzy, Prime Fuzzy Ideal of Po Semigroup," International Journal of Mathematics Trends and Technology (IJMTT), vol. 61, no. 1, pp. 1-9, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V61P501

Abstract

In this Paper the terms [𝑓), fuzzy left(right) identity, fuzzy left(right) zero of a posemigroup are introduced. It is proved that fuzzy left identity and fuzzy right identity of a posemigroup if it exists then both are same. Also proved that fuzzy left zero and fuzzy right zero of a posemigroup exists then both are equal. Next proved that intersection of arbitrary family of fuzzy posemigroups of a posemigroup S is a fuzzy posemigroup of S. Furthermore, proved that intersection of all fuzzy posemigroups of S containing f is ideal generated by fuzzy subset f. Next proved that f is a fuzzy ideal of a posemigroup S iff𝑓𝑜𝑆 ⊆ 𝑓, 𝑆𝑜𝑓 ⊆ 𝑓 and 𝑓 ⊆ 𝑓. Next demonstrate the union, intersection of arbitrary family of fuzzy ideals of posemigroup S is fuzzy ideal of S. Next proved that every completely prime fuzzy ideal is weakly completely prime fuzzy ideal. Also proved that if f is completely prime fuzzy ideal then 𝑓𝑡 is sub semigroup and completely prime ideal of S. Next proved that 𝑓𝑡 is completely prime ideal of S then f is weakly completely prime fuzzy and completely prime fuzzy ideal of S. Next proved that equivalent conditions of prime fuzzy ideal, f is prime fuzzy ideal iff 1-f is a fuzzy m-system of S if 1 − 𝑓 ≠ ∅. Finally proved that every maximal fuzzy ideal of S is a prime fuzzy ideal of S.

Keywords

Fuzzy posemigroup,[𝑓), fuzzy identity of a po semigroup, fuzzy zero of a semigroup, fuzzy posemigroup generated by 𝑓, fuzzy ideal, completely prime fuzzy ideal, weakly completely prime fuzzy ideal, prime fuzzy ideal, msystem, Maximal ideal.

References

[1] Anjaneyulu A., Structure and ideal theory of semigroups – Thesis, ANU (1980).
[2] Clifford A.H and Preston G.B., The algebraic theory of semigroupsvol – I (American Math. Society, Province (1961)).
[3] Clifford A.H and Preston G.B., The algebraic theory of semigroupsvol – II (American Math. Society, Province (1967)).
[4] G.Mohanraj, D.KrishnaSwamy,R.Hema, On fuzzy m-systems and n-systems of ordered semigroup, Annals of Fuzzy Mathematics and Informatics, Volume X, Number X,2013.
[5] J.N.Mordeson, D.S.Malik, N.Kuroki, Fuzzy Semigroups, Springer-Verlag Berlin Heidelberg Gmbh, 2003(E.Book)
[6] L.A.Zadeh, Fuzzy Sets, Inform.Control.,8(1965) 338-353.
[7] N. Kehayopulu, M.Tsingelis, Fuzzy Sets in Ordered Groupoids, Semigroup forum 65(2002) 128-132.
[8] N.Kehayopulu, M.Tsingelis,On weakly Prime ideals of ordered Semigroups, Math. Japan. 35(1990) 1051-1056.
[9] N.Kehayopulu, On Prime, weakly prime ideals in ordered semigroups, Semigroup Forum 44(1992) 341-346.
[10] Padmalatha and A.Gangadhara Rao, Anjaneyulu A., Po Ideals in partially orderedsemigroups, International Research Journal of Pure Algebra-4(6),2014.
[11] P.M.Padmalatha and A.Gangadhara Rao, Simplepartially ordered semigroups, Global Journal of Pure and Applied Mathematics, Volume 10,, Number 3(2014).
[12] P.M.Padmalatha, A.Gangadhara Rao,P.RamyaLatha, Completely prime po ideals in ordered semigroups, Global Journal of Pure and Applied Mathematics, Volume 10,, Number 4(2014).
[13] Xiang-Yun Xie,,Jian Tang, Prime fuzzy radicals and fuzzy ideals of ordered semigroups, Information Sciences 178 (2008), 4357–4374.
[14] X.Y.Xie, Fuzzy ideals in Semigroups, J.Fuzzy math.,7(1999)357-365.
[15] X.Y.Xie, On prime fuzzy ideals of a Semigroups,J.Fuzzy math.,8(2000)231-241.