A Walk Through Completely Normal Bihyperideals in Hypersemi groups

**MLA Style:**Abul Basar, Poonam Kumar Sharma "A Walk Through Completely Normal Bihyperideals in Hypersemi groups" International Journal of Mathematics Trends and Technology 66.2 (2020):42-48.

**APA Style: **Abul Basar, Poonam Kumar Sharma(2020).A Walk Through Completely Normal Bihyperideals in Hypersemi groups International Journal of Mathematics Trends and Technology, 42-48.

**Abstract**

In this paper, we introduce completely normal hyperideal and bi-hyperideal in normal hypersemigroups. We prove that a hypersemigroup H is completely regular if and only if every bi-hyperideal of H is semiprime. Then, we prove that the center of a normal regular hypersemigroup is regular. We also prove several equivalent conditions connecting normal hypersemigroups, idempotent hypersemigroups, completely regular and viable hypersemigroups.

**Reference**

[1] H. Clifford and G. B. Preston: The algebraic Theory of Semigroups. Vol. I, Math. Surveys No. 7, Amer. Math. Soc, Providence, R. I., (1961).

[2] J. F. Griffith, An Introduction to Genetic Analysis, 7th edn. New York: W. H. Freeman, 1999.

[3] Abul Basar and M. Y. Abbasi, on generalized bi-Γ-ideals in Γ -semigroups, Quasigroups and related systems, 23(2015), 181–186.

[4] Abul Basar and M. Y. Abbasi, on some properties of normal Γ -ideals in normal Γ-semigroups, TWMS Journal of Applied Engineering and Mathematics, 9(3)(2019), 455-460.

[5] Abul Basar, M. Y. Abbasi and Sabahat Ali Khan, An introduction of theory of involutions in ordered semihypergroups and their weakly prime hyperideals, The Journal of the Indian Mathematical Society, 86(3-4)(2019), 230-240.

[6] Abul Basar, A note on (m, n)- Γ-ideals of ordered LA-Γ-semigroups, Konuralp Journal of Mathematics, 7(1)(2019), 107-111.

[7] Abul Basar, Application of (m, n)-Γ-Hyperideals in Characterization of LA-Γ-Semihypergroups, Discussion Mathematicae General Algebra and Applications, 39(1)(2019), 135-147.

[8] Abul Basar, M. Y. Abbasi and Bhavanari Satyanarayana, On generalized Γ -hyperideals in ordered Γ-semihypergroups, Fundamental Journal of Mathematics and Applications, 2(1)(2019), 18-23.

[9] Abul Basar, Shahnawaz Ali, Mohammad Yahya Abbasi, Bhavanari Satyanarayana and Poonam Kumar Sharma, On some hyperideals in ordered semihypergroups, Journal of New Theory, 29(2019), 42-48.

[10] Abul Basar, Shahnawaz Ali and Poonam Kumar Sharma, An excursion through some characterizations of hypersemigroups by normal hyperideals, International Journal of Mathematics Trends and Technology, 65(12)(2019), 142-147

.[11] Abul Basar, Shahnawaz Ali, Poonam Kumar Sharma, Bhavanari Satyanarayana and M. Y. Abbasi, A study of ordered bi-Γ-hyperideals in ordered Γ –semihypergroups, Ikonion Journal of Mathematics, 1(2)(2019), 34-45.

[12] Abul Basar, On some power joined Γ- semigroups, International Journal of Engineering, Science and Mathematics, 8(12) (2019), 53-61.

[13] Abul Basar, A mathematics letter lecture note on some variety of algebraic Γ -structures, International Journal of Science and Research, 9(1) (2020), 113-117.

[14] B. Davvaz, Semihypergroup Theory, 2016, Elsevier Ltd. All rights reserved.https://doi.org/10.1016/C2015-0-06681-3.

[15] B. Davvaz, Some results on congruences in semihypergroups, Bull. Malyas. Math. Sci. Soc., 23 (2000), 53–58.

[16] B. Davvaz, Characterization of subsemihypergroups by various triangular norms, Czech. Math. J., 55(4)(2005), 923–932.

[17] B. Davvaz and A. Dehghan-Nezhad, Chemical examples in hypergroups, Ratio Matematica, 14 (2003), 71-74.

[18] B. Davvaz, Semihypergroup Theory, 1st Edition, Elsevier, 2016.

[19] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, USA, 2007.

[20] B. Davvaz, A. D. Nezad and A. Benvidi, Chain reactions as experimental examples of ternary algebraic hyperstructures, MATCH Commun. Math. Comput. Chem., 65(2) (2011), 491-499.

[21] B. Davvaz, Characterizations of sub-semihypergroups by various triangular norms, Chechoslovak Math. J., 55(4) (2005), 923-932.

[22] B. Davvaz, Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ 2013., viii+200 pp.

[23] B. Davvaz, A. D. Nezad and M. Mazloum-Ardakani, Chemical hyperalgebra: redox reactions, MATCH Commun. Math. Comput. Chem., 71 (2014), 323-331.

[24] B. Davvaz, A. D. Nezhad, Dismutation reactions as experimental verifcations of ternary algebraic hyperstructures, MATCH Commun. Math. Comput. Chem., 68 (2012), 551-559.

[25] B. Davvaz, A. D. Nezhad and A. Benvidi, Chemical hyperalgebra: dismutation reactions, MATCH Commun. Math.Comput. Chem., 67 (2012), 55-63.

[26] B. Davvaz, A. D. Nezhad and M. Heidari, Inheritance examples of algebraic hyper-structures, Inform. Sci. 224 (2013), 180-187.

[27] B. Davvaz and N. S. Poursalavati, Semihypergroups and S-hypersystems, Pure Math. Appl., 11 (2000), 43-49.

[28] F. Marty, Sur uni generalization de la notion de groupe, 8th Congress Math. Scandinaves, Stockholm, (1934), 45–49.

[29] J. Chvalina, Commutative hypergroups in the sense of Marty and ordered sets, in: Proc. Summer School, Gen. Algebra ordered Sets, Olomouc (Czech Republic), (1994), 19-30.

[30] J. Chvalina and L. Chvalinova, Transposition hypergroups formed by transformation operators on rings of differentiable functions, Ital. J. Pure Appl. Math., 15 (2004), 93-106.

[31] K. Hilla, B. Davvaz and K. Naka, On quasi-hyperideals in semihypergroups, Commun. Algebra, 39 (2011), 41–83.

[32] M. Kondo and N. Lekkoksung, On intra-regular ordered Γ-semihypergroups, Int. J. Math. Anal. 7 (28) (2013) 1379–1386.

[33] M. S. Putcha and J. Weissglass, A semilattice decomposition into semigroups having at most one idempotent, Pacific J. Math., 38(1971), 225-228.

[34] M. D. Salvo, D. Freni and G. Lo Faro, Fully simple semihypergroups, J. Algebra, 399 (2014), 358–377.

[35] M. Novak, EL-hyperstructures, Ratio Math., 23 (2012), 65–80.

[36] M. Al Tahan and B. Davvaz, Algebraic hyperstructures associated to biological inheritance, Math. Biosci. 285 (2017), 112-118.

[37] M. S. Mitrovic, Semilattices of Archimedean Semigroups, University of Nis, Faculty of Mechanical Engineering, Nis, 2003. xiv+160pp.

[38] N. K. Funabashi, On Normal Semigroups, Czechoslovak Mathematical Journal, 27(1)(1977), 43-53.

[39] P. Bonansinga and P. Corsini, On semihypergroup and hypergroup homomorphisms, Boll. Un. Mat. Ital. B., 1(2)(1982), 717–727.

[40] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Advances in Mathematics (Dordrecht), Kluwer Academic Publishers, Dordrecht, 2003. xii+322 pp. ISBN: 1-4020-1222-5.

[41] P. Corsini, Sur les semi-hypergroupes, Atti Soc. Pelorit. Sci. Fis. Math. Nat. 26(4) (1980), 363-372.

[42] R. A. Good and D. R. Hughes, Associated groups for a semigroup, Bull. Amer. Math. Soc., 58 (1952), 624-625.

[43] S. Hoskova, Upper order hypergroups as a reflective subcategory of subquasiorder hypergroups, Ital. J. Pure Appl. Math., 20(2006), 2015–222.

[44] S. Lajos, Notes on generalized bi-ideals in semigroups, Soochow J. Math., 10(1984), 55–59.

[45] S. Lajos, Generalized ideals in semigroups, Acta Sci. Math., 2(1961), 217–222.

[46] S. Lajos, On the bi-ideals in semigroups, Proc. Japan Acad., 45 (1969), 710-712.

[47] S. Lajos, Note on (m, n)-ideals. II, Proc. Japan Acad., 40 (1964), 631-632.

[48] S. Lajos, F. A. Szasz, On the bi-ideals in associative ring, Proc. Japan Acad., 46 (1970), 505-507.

[49] S. Lajos, On semigroups that are semilattices of groups. II, Dept. Math. K. Marx Univ. of Economics, Budapest, (1971).

[50] S. Lajos, Characterizations of semilattices of groups, Math. Balkanica, 3 (1973), 310-311.

[51] S. Schwarz, A theorem on normal semigroups, Czechoslovak Math. J., 10 (85) (1960), 197-200.

[52] T. Changphas and B. Davvaz, Bi-hyperideals and quasi-hyperideals in ordered semihypergroups, Ital. J. Pure Appl. Math., 35 (2015), 493-508.

**Keywords**

hypersemigroup, normal hyperideal, bi-hyperideal.