On systems, maximal Γ-hyperideals and complete prime Γ-radicals in Γ-semihypergroups

Document Type : Research Paper

Authors

1 Department of Mathematics, S.N.J.B.’s K.K.H.A. Arts, S.M.G.L. Commerce & S.P.H.J. Science College, Chandwad, District Nashik, 423 101, M.S., India

2 Department of Mathematics, School of Mathematical Sciences, Kavayitri Bahinabai Chaudhari North Maharashtra University, Jalgaon, 425 001 M.S., India

Abstract

In this paper c-system , n-system and complete prime Γ-radical in a Γ-semihypergroup are introduced and studied while
definition of m-system has been revised. Maximal Γ-hyperideals are studied and the conditions under which maximal Γ-hyperideals are prime and vice versa are investigated. It is proved that intersection of distinct maximal Γ-hyperideals in Γ-semihypergroup is non-empty. Relations between completely prime Γ-hyperideals, c-system and complete prime Γ-radicals are examined.

Keywords


[1] S. Abdullah, M. Aslam and T. Anwar, A note on M-hypersystems and N-hypersystems in Γ-semihypergroups, Quasigroups and Related Systems,19(2011), 169-172.
[2] R. Ameri, A. Kordi, S. Hoskova-Mayerova, Multiplicative hyperring of frac-tions and coprime hyperideals, An. Stiint. Univ. Ovidius Constanta, Ser. Mat.,25(1)(2017), 5-23.
[3] S. J. Ansari, K. F. Pawar,On Certain Γ-Hyperideals in Γ-Semihypergroups, Jour-nal of Hyperstructures (submitted), (2019).
[4] N. Antampoufi, S. Hoskova-Mayerova, A Brief survey on the two different ap-proaches of fundamental equivalent relations in hyperstructures, Ratio Mathe-matica, 33(2017), 47-60.
[5] S. M. Anvariyeh, S. Mirvakili and B. Davvaz, On Γ-hyperideals in Γ-semihypergroups, Carpathian J.Math.,26(1)(2010),11-23.
[6] P. Corsini, of Hypergroup Theory, 2nd ed., Aviani Editore Publisher, (1993).
[7] P. Corsini,Hypergraphs and hypergroups, Algebra Universalis, 35(4)(1996), 548-555.
[8] P. Corsini, L. Leoreanu, Applications of Hyperstructure Theory, Advances In Mathematics, Dordrecht: Kluwer Academic Publishers, (2003).
[9] Corsini, Piergiulio. Hypergroups associated with HX-groups, An. Stiint. Univ.Ovidius Constanta, Ser. Mat. 25(2017). 10.1515/auom-2017-0020.
[10] B. Davvaz, Semihypergroup Theory, Academic Press Elsevier, (2016).
[11] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, Florida,USA: International Academic Press, (2007).
[12] D. Heidari and B. Davvaz, Γ-hypergroups and Γ-semihypergroups associated to binary relations, Iranian Journal of Science & Technology, A2: (2011), 69-80.
[13] D. Heidari, S. Ostadhadi Dehkordi, B. Davvaz,Γ-Semihypergroups and Their Properties,U.P.B.Sci.Bull.,Series A,72(1)(2010).
[14] K. Hila, B. Davvaz, J. Dine,Study on the Structure of Γ-Semihypergroups,Com-munication in Algebra,40(2012),2932-2948.
[15] S. Hoskova, J. Chvalina, A survey of investigations of the Brno research group in the hyperstructure theory since the last AHA Congress, In:10th International Congress on Algebraic Hyperstructures and Applications: Brno 2008, Czech Re-public, Proceedings of AHA 2008, (2009), 7184. ISBN: 978-80-7231-688-5.
[16] S. Hoskova-Mayerova, A. Maturo, An analysis of social relations and social group behaviors with fuzzy sets and hyperstructures, International Journal of Algebraic Hyperstructures and its Applications, 2(2016), 91-99.
[17] S. Hoskova-Mayerova, A. Maturo, Hyperstructures in social sciences, AWER Pro-cedia Information Technology & Computer Science ,Barcelona, Spain, 3(2013), 547-552.
[18] S. Hoskova-Mayerova, A. Maturo, On Some Applications pf Algebraic Hyper-structures for the Management of Teaching and Relationships in Schools, Italian Journal of Pure and Applied Mathematics, 41(2019), 584-592.
[19] S. Hoskova-Mayerova, Quasi-order hypergroups determinated by T-hypergroups, Ratio Mathematica,32 (2017), 37-44.
[20] M. Jafarpour, I. Cristea, A. Tavakoli, A method to compute the number of regular reversible rosenberg hypergroup, Ars Combinatoria, 128(2016), 309-329.
[21] F. Marty, Sur une generalization de la notion de groupe, 8th congress Math. Scandinaves, Stockholm, (1934), 45-49.
[22] M. Nov´ak, EL-hyperstructures: an overview, Ratio Mathematica, 23(2012), 5-80.
[23] M. Nov´ak,n-ary hyperstructures constructed from binary quasi-orderer semi-groups, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 22(2014), 147-168.
[24] K. F. Pawar, J. J. Patil, B. Davvaz, On Regular Γ-semihypergrrings and idem-potent Γ-semihyperrings, KYUNGPOOK Math. J., 59(2019), 35-45.
[25] M. K. Sen and N. K. Saha, On Γ-semigroup I, Bulletin of the Calcutta Mathe-matical Society, 78(1986), 180-186.
[26] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Vh-Monographs in Mathematics, Palm Harbor Florida, (1994).