Abstract
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, \(\rho _\mathfrak {R}=\rho ^*_\mathfrak {R}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Anvariyeh, S.M., Davvaz, B.: Strongly transitive geometric spaces associated to hypermodules. J. Algebra 322, 1340–1359 (2009)
Babaei, H., Jafarpour, M., Mousavi, SSh: \(\Re \)-parts in hyperrings. Iran. J. Math. Sci. Inform. 7(1), 59–71 (2012)
Corsini, P.: Prolegomena of hypergroup theory. Supplement to Rivista di Matematica Pura ed Applicata. Aviani Editore, Tricesimo (1993). ISBN: 88-7772-025-5
Corsini, P., Leoreanu, V.: Applications of hyperstructure theory. Advanced in Mathematics. (Dordrecht), 5. Kluwer Academic Publisher, Dordrecht (2003). ISBN: 1-4020-1222-5
Cristea, I., Stefanescu, M.: Hypergroups and \(n\)-ary relations. Eur. J. Comb. 31(3), 780–789 (2010)
Davvaz, B., Leoreanu-Fotea, V.: Hyperring Theory and Applications. International Academic Press, New York (2007)
Davvaz, B., Karimian, M.: On the \(\gamma ^*\)-complete hypergroups. Eur. J. Comb. 28(1), 86–93 (2007)
Davvaz, B., Vougiouklis, T.: Commutative rings obtained from hyperrings (\(H_v\)-rings) with \(\alpha ^*\)-relations. Commun. Algebra 35(11), 3307–3320 (2007)
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)
Freni, D.: A new characterization of the derived hypergroup via strongly regular equivalences. Commun. Algebra 30(8), 3977–3989 (2002)
Freni, D.: Strongly transitive geometric spaces: applications to hypergroups and semigroups theory. Commun. Algebra 32(3), 969–988 (2004)
Gutan, M.: Properties of hyperproducts and the relation \(\beta \) in quasihypergroups. Ratio Math. 12, 19–34 (1997)
Koskas, M.: Groupoides, Demi-hypergroupes et hypergroupes. J. Math. Pures Appl. 49, 155–192 (1970)
Leoreanu-Fotea, V., Davvaz, B.: \(n\)-Hypergroups and binary relations. Eur. J. Comb. 29(5), 1207–1218 (2008)
Leoreanu-Fotea, V., Rosenberg, I.G.: Hypergroupoids determined by lattices. Eur. J. Comb. 31(3), 925–931 (2010)
Mirvakili, S., Davvaz, B.: Relations on Krasner \((m, n)\)-hyperrings. Eur. J. Comb. 31(3), 790–802 (2010)
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
Mirvakili, S., Anvariyeh, S.M., Davvaz, B.: On \(\alpha \)-relation and transitivity conditions of \(\alpha \). Commun. Algebra 36(5), 1695–1703 (2008)
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)
Mirvakili, S., Davvaz, B.: Strongly transitive geometric spaces: application to hyperring. Rev. Univ. Mat. Argent. 53(1), 43–53 (2012)
Mousavi, S.S.H., Leoreanu-Fotea, V., Jafarpour, M.: \(\Re \)-parts in (semi)hypergroups. Ann. Mat. Pura Appl. 190(4), 667–680 (2011)
Pelea, C.: On the fundamental relation of a multialgebra. Ital. J. Pure Appl. Math. 10, 141–146 (2001)
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)
Vougiouklis, T.: Hyperstructures and their representations, vol. 115. Hadronic Press Inc, Palm Harber (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shirvani, M., Mirvakili, S. Transitivity of \(\rho _\mathfrak {R}\) relations in hyperrings using geometric spaces. Bol. Soc. Mat. Mex. 24, 359–372 (2018). https://doi.org/10.1007/s40590-017-0162-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40590-017-0162-x