We define intuitionistic fuzzy idempotent, prime, strongly irreducible and t-pure ideals on a semiring, and investigate some properties of such ideals. We show that all intuitionistic fuzzy ideals are idempotent in a fully idempotent semirings, and right weakly regular semirings. We establish that an intuitionistic fuzzy set of a semiring is an intuitionistic fuzzy ideal if and only if (α, β)-cut is an ideal. We prove that if all intuitionistic fuzzy ideals of a semiring are t-pure, then an intuitionistic fuzzy ideal is prime if and only if it is strongly irreducible. We also establish that for a fully idempotent semirings, an intuitionistic fuzzy ideal is prime if and only if it is strongly irreducible.
AhsanJ., LatifR.M. and ShabirM., Representation of weakly regular semirings, Communication in Algebra21 (1993), 2819–2835.
2.
AhsanJ., MordesonJ.N. and ShabirM., Fuzzy semirings with application to Automata theory, Springer278 (2012), 18–43.
3.
AhsanJ., SaifullahK. and KhanM.F., Fuzzy semirings, Fuzzy Set and Syts60 (1993), 302–309.
4.
AhsanJ., SaifullahK. and ShabirM., Fuzzy prime ideals of a semiring and Fuzzy prime subimodules of semi modules over a semiring, New Math and Natural Computation2 (2006), 219–236.
5.
AhsanJ., Fully idempotent semirings, Proc Japan Acad Ser Math Sci69 (1993), 185–188.
6.
AkramM. and DudekW.A., Intuitionistic fuzzy left k-ideals of semirings, Soft Computing12 (2008), 881–890.
7.
AtanassovK.T., On intuitionistic fuzzy sets theory, Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 283, 2012, pp. 1–16.
8.
AtanassovK.T., Intuitionistic fuzzy sets, Fuzzy Sets and Syst20 (1986), 87–96.
9.
BiswasR., Intuitionistic fuzzy subgroups, Math Forum10 (1989), 37–46.
10.
BrownB. and McN.H., Coy, Some theory on groups with application to ring theory, Amer Math Soc69 (1950), 302–311.
11.
CamilloV. and XiaoY., Weakly regular rings, Communication in Algebra22 (1994), 4095–4112.
12.
CristeaI., DavvazB. and SadrabadiE.H., Special intuitionistic fuzzy subhypergroups of complete hypergroups, Journal Intelligent and Fuzzy Syst28 (2015), 237–245.
13.
DheenaP. and MahanrajG., On intuitiontic fuzzy k-ideals of semiring, International Journal of Computational Cognition9 (2011), 45–50.
14.
DuttaT.K. and BiswasB.K., Fuzzy prime ideals of a semiring, Bull Malays Math Soc17 (1994), 9–16.
15.
DuttaT.K. and KarS., On prime ideals and prime radical of ternary semirings, Bull Cal Math Soc97 (2005), 445–454.
16.
GhoshS., Fuzzy k-ideals of semirings, Fuzzy Sets and Syst95 (1998), 103–108.
17.
KimK.H. and LeeJ.G., On intuitionistic fuzzy bi-ideals of semi-group, Turk Journal of Math29 (2005), 201–210.
18.
KurokiN., Fuzzy bi-ideals in semi group, Comment Math Univ St Paul24 (1975), 21–26.
19.
MalikA.S. and MordesonJ.N., Fuzzy prime ideals of a ring, Fuzzy Set and Syst37 (1990), 93–98.
20.
MukherjeeT.K. and SenM.K., Prime fuzzy ideals in rings, Fuzzy Sets and Syst32 (1989), 337–341.
21.
RahmanS., SaikiaH.K. and DavvazB., On the definition of Atanassov’s intuitionistic fuzzy subrings and ideals, Bull Malays Math Sci Soc36 (2013), 401–418.
22.
RahmanS. and SaikiaH.K., Atanassov’s intuitionictic fuzzy sub modules with respect to a t-norm, Soft Computing17 (2013), 1253–1262.
23.
RahmanS. and SaikiaH.K., Some Aspects of Atanassov’s intuitionistic fuzzy sub module, International Journal of Pure and Appl Math77 (2012), 369–383.
24.
RamamurthyV.S., Weakly regular rings, Canad Math Bull16 (1973), 317–517.
25.
SaikiaH.K. and KalitaM.C., On fuzzy essential submodules, Journal of Fuzzy Math17 (2009), 109–117.
26.
SwamyU.M. and SwamyK.L.N., Fuzzy prime ideals of rings, Journal of Math Anal Appl134 (1988), 94–103.
27.
TanJ., DavvazB. and LuoY., Hyperfilters and fuzzy hyperfilters of ordered semihypergroups, Journal of Intelligent and Fuzzy Syst29 (2015), 75–84.
28.
TepavčevićA. and
TrajkovskiG., L-fuzzy lattices: An introduction, Fuzzy Sets and Syst123 (2001), 209–216.
29.
VandiverH.S., Note on a simple type of algebra in which the cancellation law of addition not hold, Bull Amer Math Soc40 (1934), 916–920.
