Boletín de la Sociedad Matemática Mexicana

, Volume 24, Issue 2, pp 359–372 | Cite as

Transitivity of \(\rho _\mathfrak {R}\) relations in hyperrings using geometric spaces

  • M. Shirvani
  • S. MirvakiliEmail author
Original Article


In this paper, we determine a family \({\mathfrak {U}}_{\mathfrak {R}}\) of subsets of a hyperring R and sufficient conditions, such that the geometric space \((R,{\mathfrak {U}}_{\mathfrak {R}})\) is strongly transitive. Finally, we prove that in any hyperfield or any hyperring \((R,+,\cdot )\), such that \((R,+)\) has an identity element, \(\rho _\mathfrak {R}=\rho ^*_\mathfrak {R}\).


Hyperring Strongly transitive geometric space Equivalence relation 

Mathematics Subject Classification

20N20 16Y99 


  1. 1.
    Anvariyeh, S.M., Mirvakili, S., Davvaz, B.: Fundamental relation on \((m, n)\)-ary hypermodules over \((m, n)\)-ary hyperrings. ARS Comb. 94, 273–288 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anvariyeh, S.M., Davvaz, B.: Strongly transitive geometric spaces associated to hypermodules. J. Algebra 322, 1340–1359 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Babaei, H., Jafarpour, M., Mousavi, SSh: \(\Re \)-parts in hyperrings. Iran. J. Math. Sci. Inform. 7(1), 59–71 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Corsini, P.: Prolegomena of hypergroup theory. Supplement to Rivista di Matematica Pura ed Applicata. Aviani Editore, Tricesimo (1993). ISBN: 88-7772-025-5Google Scholar
  5. 5.
    Corsini, P., Leoreanu, V.: Applications of hyperstructure theory. Advanced in Mathematics. (Dordrecht), 5. Kluwer Academic Publisher, Dordrecht (2003). ISBN: 1-4020-1222-5CrossRefGoogle Scholar
  6. 6.
    Cristea, I., Stefanescu, M.: Hypergroups and \(n\)-ary relations. Eur. J. Comb. 31(3), 780–789 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Davvaz, B., Leoreanu-Fotea, V.: Hyperring Theory and Applications. International Academic Press, New York (2007)Google Scholar
  8. 8.
    Davvaz, B., Karimian, M.: On the \(\gamma ^*\)-complete hypergroups. Eur. J. Comb. 28(1), 86–93 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Davvaz, B., Vougiouklis, T.: Commutative rings obtained from hyperrings (\(H_v\)-rings) with \(\alpha ^*\)-relations. Commun. Algebra 35(11), 3307–3320 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Freni, D.: Une note sur le coeur d’un hypergroupe et sur la cloture \(\beta ^*\) de \(\beta \). Riv. Mat. Pura Appl. 8, 153–156 (1991)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Freni, D.: A new characterization of the derived hypergroup via strongly regular equivalences. Commun. Algebra 30(8), 3977–3989 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Freni, D.: Strongly transitive geometric spaces: applications to hypergroups and semigroups theory. Commun. Algebra 32(3), 969–988 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gutan, M.: Properties of hyperproducts and the relation \(\beta \) in quasihypergroups. Ratio Math. 12, 19–34 (1997)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Koskas, M.: Groupoides, Demi-hypergroupes et hypergroupes. J. Math. Pures Appl. 49, 155–192 (1970)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Leoreanu-Fotea, V., Davvaz, B.: \(n\)-Hypergroups and binary relations. Eur. J. Comb. 29(5), 1207–1218 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Leoreanu-Fotea, V., Rosenberg, I.G.: Hypergroupoids determined by lattices. Eur. J. Comb. 31(3), 925–931 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mirvakili, S., Davvaz, B.: Relations on Krasner \((m, n)\)-hyperrings. Eur. J. Comb. 31(3), 790–802 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mirvakili, S., Anvariyeh, S.M., Davvaz, B.: Rings derived from strongly U-regular relations. Bol. Soc. Mat. Mex. (2017). doi: 10.1007/s40590-016-0157-z MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mirvakili, S., Anvariyeh, S.M., Davvaz, B.: On \(\alpha \)-relation and transitivity conditions of \(\alpha \). Commun. Algebra 36(5), 1695–1703 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mirvakili, S., Anvariyeh, S.M., Davvaz, B.: Transitivity of \(\Gamma \)-relation on hyperfields. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 51(99)(3), 233–243 (2008)Google Scholar
  21. 21.
    Mirvakili, S., Davvaz, B.: Strongly transitive geometric spaces: application to hyperring. Rev. Univ. Mat. Argent. 53(1), 43–53 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mousavi, S.S.H., Leoreanu-Fotea, V., Jafarpour, M.: \(\Re \)-parts in (semi)hypergroups. Ann. Mat. Pura Appl. 190(4), 667–680 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pelea, C.: On the fundamental relation of a multialgebra. Ital. J. Pure Appl. Math. 10, 141–146 (2001)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Vougiouklis, T.: The fundamental relation in hyperrings. The general hyperfield. In: Proc. fourth int. congress on algebraic hyperstructures and applications (AHA 1990), World Scientific, pp 203–211 (1991)Google Scholar
  25. 25.
    Vougiouklis, T.: Hyperstructures and their representations, vol. 115. Hadronic Press Inc, Palm Harber (1994)zbMATHGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2017

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityTehranIran

Personalised recommendations