Hyper vector spaces over Krasner hyperfields

Document Type : Original Article


1 Department of Mathematics, Lebanese International University, Bekaa, CZ-961, Lebanon

2 Department of Mathematics Yazd University, Yazd Iran


In this paper, we study hyper vector spaces over Krasner hyperfields. First, we introduce the notions of linearly independence (dependence) and basis for a hyper vector space. Second, we investigate their properties and prove some results for hyper vector spaces that are similar to that of vector spaces over fields. Then, we define linear transformations over hyper vector spaces and investigate their properties. Finally, we prove the dimension theorem for linear transformations.


[1] R. Ameri, O.R. Dehghan, On dimension of hypervector spaces, European Journal of Pure and
Applied Mathematics, 1(2) (2008), 32-50.
[2] M. Baker, N. Bowler, Matroids over hyper elds, (preprint). Available at: arxiv:math.CO/16
01.0124v5, 2017.
[3] P. Corsini, Prolegomena of hypergroup theory, Aviani Editore, (1993), 216 pp.
[4] P. Corsini, V. Leoreanu, Applications of hyperstructures theory, Advances in Mathematics,
Kluwer Academic Publisher, 2003.
[5] B. Davvaz, Polygroup theory and related systems, World Scienti c Publishing Co. Pte. Ltd.,
Hackensack, NJ, 2013. viii+200 pp.
[6] B. Davvaz, V. Leoreanu-Fotea, Hyperring theory and applications, International Academic
Press, USA, 2007.
[7] W.C. Hu man, V. Pless, Fundamentals of error correcting codes, Cambrige, 2003.
[8] M. Krasner, A class of hyperrings and hyper elds, International Journal of Mathematics and
Mathematical, 2 (1983), 307-312.
[9] F. Marty, Sur une generalization de la notion de group, In 8th Congress Mathematics Scan-
denaves, (1934), 45-49.
[10] H.K. Mirdar, S.M. Anvariyeh, A hypervaluation of a hyper eld onto a totally ordered canonical
hypergroup, Studia Scientiarum Mathematicarum Hungarica, 52(1) (2015), 87-101.
[11] S. Mirvakili, S.M. Anvariyeh, B. Davvaz, Transitivity of  -relation on hyper elds, Bulletin
Mathematique de la Societe des Sciences Roumanie (N.S.), 51(99) (2008), 233-243.
[12] S. Mirvakili, B. Davvaz, Relations on Krasner (m; n)-hyperrings, European Journal of Com-
binatorics, 31 (2010), 790-802.
[13] A. Nakassis, Expository and survey article of recent results in hyperring and hyper eld theory,
International Journal of Mathematics and Mathematical, 11 (1988), 209-220.
[14] M.S. Tallini, Hypervector spaces, Proceedings of the Fourth International Congress on Alge-
braic Hyperstructures and Applications, Xanthi, Greece, (1990), 167-174.
[15] M.S. Tallini, Characterization of remarkable hypervector spaces, Proc. 8-th int. Congress on
Algebraic Hyperstructures and Applications, Samotraki, Greece, (2002), Spanidis Press, Xan-
thi, (2003), 231-237.
[16] T. Vougiouklis, Hyperstructures and their representations, Aviani editor. Hadronic Press,
Palm Harbor, USA, 1994.
[17] T. Vougiouklis, The fundamental relation in hyperrings. The general hyper eld, Proc. Fourth
Int. Congress on Algebraic Hyperstructures and Applications (AHA 1990), World Scienti c,
(1991), 203-211.
[18] T. Vougiouklis, Hv-vector spaces, Algebraic hyperstructures and applications (Iasi, 1993),
Hadronic Press, Palm Harbor, FL, (1994), 181-190.