In this paper, as a generalization of the concept of quasi-Γ-hyperideals of Γ-semihypergroups to ordered Γ-semihypergroup theory, the concept of quasi-Γ-hyperideals of ordered Γ-semihypergroups is introduced, and related properties are discussed. Furthermore, we define and study fuzzy quasi-Γ-hyperideals of ordered Γ-semihypergroups. In particular, we investigate the structure of fuzzy quasi-Γ-hyperideal generated by a fuzzy subset in an ordered Γ-semihypergroup. In addition, we also introduce the concepts of completely prime, weakly completely prime and completely semiprime fuzzy quasi-Γ-hyperideals of ordered Γ-semihypergroups, and characterize bi-regular ordered Γ-semihypergroups in terms of completely semiprime fuzzy quasi-Γ-hyperideals. Finally, several characterizations of regular and intra-regular ordered Γ-semihypergroups by the properties of fuzzy quasi-Γ-hyperideals are given.
AbdullahS., HilaK. and AslamM., On bi-Γ-hyperideals of Γ-semihypergroups, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics74 (2012), 79–90.
2.
AnvariyehS.M., MirvakiliS. and DavvazB., On Γ-hyperideals in Γ-semihypergroups, Carpathian J Math26 (2010), 11–23.
3.
ChangphasT. and DavvazB., Properties of hyperideals in ordered semihypergroups, Italian J Pure Appl Math33 (2014), 425–432.
4.
ChattopadhyayS., Right inverse Γ-semigroups, Bull Cal Math Soc93 (2001), 435–442.
5.
ChattopadhyayS., Right orthodox Γ-semigroups, Southeast Asian Bull Math29 (2005), 23–30.
6.
ChinramR., Rough prime ideals and rough fuzzy prime ideals in Γ-semigroups, Commun Korean Math Soc24 (2009), 341–351.
7.
ChvalinaJ. and Hošková-MayerováŠ, On certain proximities and preorderings on the transposition hypergroups of linear first-order partial differential operators, An, Ştiinţ Univ “Ovidius” Constanţa Ser Mat22 (2014), 85–103.
8.
ChvalinaJ., Hošková-Mayerovአand NezhadA.D, General actions of hypergroups and some applications, An, Ştiinţ Univ “Ovidius” Constanţa Ser Mat21 (2013), 59–82.
9.
CorsiniP., Prolegomena of Hypergroup Theory, Aviani Editore, Italy, 1993.
10.
CorsiniP. and LeoreanuV., Applications of Hyperstructure Theory, Advances in Mathematics, Kluwer Academic Publishers, Dordrecht, Hardbound, 2003.
11.
CorsiniP., ShabirM. and MahmoodT., semisimple ihypergroups in terms of hyperideals and fuzzy hyperideals, Iran J Fuzzy Syst8 (2011), 95–111.
DavvazB., Some results on congruences on semihypergroups, Bull Malays Math Sci Soc23 (2000), 53–58.
14.
DavvazB., Fuzzy hyperideals in semihypergroups, Ital J Pure Appl Math8 (2000), 67–74.
15.
DavvazB. and Leoreanu-FoteaV., Structures of fuzzyΓ-hyperideals of Γ-semihypergroups, J Mult-Valued Logic Soft Comput19 (2012), 519–535.
16.
DavvazB., CorsiniP. and ChangphasT., Relationship between ordered ihypergroups and ordered semigroups by using pseuoorders, European J Combinatorics44 (2015), 208–217.
17.
DavvazB. and PoursalavatiN.S., semihypergroups andS-hypersystems, Pure Math Appl11 (2000), 43–49.
18.
DavvazB. and Leoreanu-FoteaV., Structures of fuzzyΓ-hyperideals in Γ-semihypergroups, Multiple-Valued Logic and Soft Computing19 (2012), 519–535.
19.
DudekW.A., DavvazB. and JunY.B., On intuitionistic fuzzy sub-hyperquasigroups of hyperquasigroups, Inform Sci170 (2005), 251–262.
20.
ErsoyB.A., SaricaogluY., YenigunM. and DavvazB., On fuzzy interior Γ-hyperideals of Γ-semihypergroups, Utilitas Mathematica88 (2012), 157–170.
21.
JunY.B., Roughness of Γ-subigroups/semideals inΓ-semigroups, Bull Korean Math Soc40 (2003), 531–536.
22.
HeidariD. and DavvazB., On ordered hyperstructures, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics73 (2011), 85–96.
23.
HeidariD., DehkordiS.O. and DavvazB.,
Γ-semi-hypergroups and their properties, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics72 (2010), 197–210.
24.
HilaK., Filters in ordered Γ-semigroups, Rocky Mountain Journal of Mathematics41 (2011), 189–203.
25.
HilaK. and AbdullahS., A study on intuitionistic fuzzy sets in Γ-semihypergroups, J Intell Fuzzy Systems26 (2014), 1695–1710.
26.
HilaK., DavvazB. and DineJ., Study on the structure of Γ-ihypergroups, Comm Algebra40 (2012), 2932–2948.
27.
HilaK., DavvazB. and NakaK., On quasi-hyperideals in semihypergroups, Comm Algebra39 (2011), 4183–4194.
28.
KehayopuluN., Green’s relations and the relation in Γ-semigroups, Quasigroups and Related Systems22 (2014), 89–96.
29.
KehayopuluN. and TsingelisM., Fuzzy right, left, quasiideals, bi-semideals in ordered igroups, Lobachevskii J Math30 (2009), 17–22.
30.
KehayopuluN. and TsingelisM., Regular ordered semigroups in terms of fuzzy subsets, Inform Sci176 (2006), 3675–3693.
31.
KhanF.M., SarminN.H. and KhanA., A novel approach towards fuzzy Γ-ideals in ordered Γ-semigroups, Indian J Pure Appl Math45 (2014), 343–362.
32.
KwonY.I. and LeeS.K., On weakly prime ideals of ordered Γ-semigroups, Commun Korean Math Soc13 (1998), 251–256.
33.
LeoreanuV., About the simplifiable cyclic semihypergroups, Ital J Pure Appl Math7 (2000), 69–76.
34.
Leoreanu-FoteaV. and RosenbergI.G., Homomorphisms of hypergroupoids associated with L-fuzzy sets, J Mult-Valued Logic Soft Comput15 (2009), 537–545.
35.
MartyF., Sur une generalization de la notion de group, in: Proc 8th Congress Mathematics Scandenaves, Stockholm, 1934, pp. 45–49.
36.
NazS., ShabirM., On prime soft bi-hyperideals of semihypergroups, J Intell Fuzzy Systems26 (2014), 1539–1546.
37.
PibaljommeeB. and DavvazB., Some study of several kinds of fuzzy hyperideals in ordered semihypergroups, Xanthi, Greece, pp, International Congress on Algebraic Hyperstructures and its Applications (2014), 91–94.
38.
PibaljommeeaB. and DavvazB., Characterizations of (fuzzy) bi-hyperideals in ordered semihypergroups, J Intell Fuzzy Systems28 (2015), 2141–2148.
39.
SahaN.K., On Γ-semigroup II, Bull Calcutta Math Soc79 (1987), 331–335.
40.
SardarS.K., DavvazB. and MajumderS.K., A study on fuzzy interior ideals of Γ-semigroups, Comput Math Appl60 (2010), 90–94.
41.
SenM.K., On Γ-semigroups, Proceeding of International Symposium on Algebra and Its Applications, Decker Publication, New York, 1981, pp. 301–308.
42.
SenM.K. and SahaN.K., On Γ-semigroup I, Bull Calcutta Math Soc78 (1986), 180–186.
43.
SenM.K. and SethA., On po-Γ-semigroups, Bull Calcutta Math Soc85 (1993), 445–450.
44.
ShabirM. and KhanA., Fuzzy quasi-ideal in ordered semigroups, Bull Malays Math Sci Soc34 (2011), 211–225.
45.
StienfeldO., Quasi-ideals in rings and semigroups, Akademiakiado, Budapest, 1978.
46.
TangJ., DavvazB. and LuoY., Hyperfilters and fuzzy hyperfilters of ordered semihypergroups, J Intell Fuzzy Systems29 (2015), 75–84.
47.
TangJ., KhanA. and LuoY., Characterizations of isimple ordered semihypergroups in terms of fuzzy hyperideals, J Intell Fuzzy Systems. DOI: 10.3233/IFS-151884.
48.
VougiouklisT., Hyperstructures and Their Representations, Hadronic Press, Florida, 1994.
49.
XieX.Y., An introduction to ordered semigroup theory, Kexue Press, Beijing.2001.
50.
YaqoobN. and AslamM., Prime (m; n) bi-Γ-hyperideals in Γ-semihypergroups, Appl Math Inf Sci8 (2014), 2243–2249.
51.
YaqoobN., AslamaM., DavvazB. and GhareebbA., Structures of bipolar fuzzy Γ-hyperideals in Γ-semihypergroups, J Intell Fuzzy Systems27(6) (2014), 3015–3032.
