Document Type : Original Paper


1 1 School of Mathematics, Statistics and Computer Sciences, University of Tehran, Tehran, Islamic Repupblic of Iran

2 2 Department of Mathematics, Faculty of Mathematics, University of Payam-e- Noor, Tehran, Islamic Republic of Iran


The purpose of this paper is to construct Boolean rings from multirings. In this regards, a method to construct a multigroup(multiring) on a given non-empty set, are introduced and its properties has been investigated. Also, an equivalence relation on a multiring are introduced and it is extended to an smallest strongly regular  equivalence relation, such that its quotient  space be a commutative Boolean ring with identity. Finally, the transitivity of this relation based on complete parts are   proved.


  1. Marty F. Sur une generalization de la notion de groupe 8th Congres Math. Scandinaves, Stockholm. 1934; 45-49.
  2. Chvalina J,  Mayerova    SH, Nezhad  AD. General actions of hyperstructures and some applications. An St UnivOvidius Constanta. 2013;21(1):59-82.
  3. Corsini P, Leoreanu V. Applications of Hyperstructure Theory. Kluwer Academic Pub-lishers, 2003.
  4. Krasner M. Approximation des corps values complet de caractristique p>0 par ceux de caracteristique zero. colloqued′-Algebra superieure (Bruxells, December 1956).
  5. Corsini P. Prolegomena of Hypergroup theory. Second Edition, Aviani Editor, (1993).
  6. Hamidi M, Ashrafi AR. Fundamental relation and automorphism group of very thinHv-groups. Comm. Algebra. 2017; 45(1): 130-140.
  7. Hamidi M,  Borumand   S A, Leoreanu V. Divisible Groups Derived From Divisible Hy-pergroups. U.P.B. Sci. Bull., Series A. 2017; 79(2): 59-70.
  8. Freni D. A new characterization of the derived hypergroup via strongly regular equivalences. Communicationin algebra. 2002;30, No. 8: 3977-3989.
  9. Fotea, VL, Jafarpour M, Mousavi SS. The relation and multisemi-direct hyperproducts of hypergroups. Commun Algebra. 2012;40:3597-3608 .
  10. Vougiouklis T. The fundamental Relations in Hyperrings. The general th International congress in Algebraic Hyperstructures hyperfield Proc.4 and Its Applications (AHA 1990) World Scientific. 1990; 203-211.


  1. Davvaz B, Vougiouklis T. Commutative rings obtained from hyperrings (Hv-rings) with -relations. Commun Algebra. 2007; 35: 3307-3320.
  2. Ghiasvand P,  Mirvakili   S, Davvaz  B. Boolean Rings Obtained from Hyperrings with Relations. Iran. J. Sci. Technol Trans. Sci. 2017; 41(1): 69-79.
  3. Mirvakili S, Davvaz B. Applications of the -relation to Krasner hyperrings. J Algebra. 2012;362: 145-156.
  4. Ameri R, Nozari T. A New Characterization of Fundamental Relation on Hyperrings. Int.J. Contemp. Math. Sciences. 2010 ;5(15): 721-738.
  5. Glodki P, Marshall M. Orderings and Signatures of Higher Level on Multirings and Hyperfields. Journal of K-theory and its Applications to Algebra Geometry and Topology. 2012 10(3): 489-518.
  6. Marshall M. Real Reduced Multirings and Multifields. Journal of Pure and Applied Algebra. 2006; 295(2): 452-468.
  7. Andradas C,  Brocker  L,  Ruiz  J. Constructible sets in real geometry. Springer, 1996.
  8. Marshall M. Spaces of orderings and abstract real spectra. Springer Lecture Notes in Math. 1636;1996.
  9. Ribeiro H.R.D.O., Roberto K.M.D.A. Mariano   H.L. Functorial relationships between multirings and the various abstract theories of quadratic forms. São Paulo J. Math. Sci. 2020.
  10. Riberio DO,   Rafeal    H. L.   Von Neumann Regular Hyperring and applications to Real  Reduced Multirings, Arxiv, 2021.