Convex, balanced and absorbing subsets of hypervector spaces

Document Type : Research Paper

Authors

Department of Mathematics, Vali-e-Asr University of Rafsanjan, P. O. Box 7713936417, Rafsanjan, Iran

10.29252/as.2020.1735

Abstract

In this paper, we define convex, balanced and absorbing subsets of a hypervector space $V$ over a field $K$, where $K$ is considered $\mathbb{R}$ or $\mathbb{C}$ and give some examples of them. We prove that every subspace of a hypervector space is a convex and balanced subset. Also, for every regular equivalence relation $\rho$ on a hypervector space $V$, we  show that if $A$ is a convex, balanced or an absorbing subset of $V$, then $A/\rho$ is respectively a convex, balanced or an absorbing subset of a hypervector space $V/\rho$.

Keywords

References

[1] R. Ameri, O. R. Dehghan, On dimension of hypervector spaces, European J. Pure and Appl. Math., 1 (2008) 32-50.
[2] S. M. Anvariyeh, S. Mirvakili and B. Davvaz, relation on hypermodules and fundamental modules over commutative fundamental rings. Comm. Alg., 36 (2008) 622-631.
[3] S. M. Anvariyeh, S. Mirvakili and B. Davvaz, R-parts and modules derived from strongly U-regular relations on hypermodules. Matematiche (Catania) 72 (2017), no. 1, 103-121.
[4] P. Corsini, Prolegomena of hypergroup theory, Second Edition, Aviani Editor, 1993.
[5] P. Corsini, V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publications, 2003.
[6] B. Davvaz, V. leoreanu-Fotea, Hyperring theory and applications. USA: International Academic Press, 2007.
[7] F. Marty. Sur nue generalization de la notion do group, 8th congress of the Scandinavic Mathematics , Stockholm, (1934), 45-49.
[8] P. Raja, S. M. Vaezpour, Normed hypervector spaces, Iran. J. Math. Sci. Inform., 2 (2007), no. 2, 35-44.
[9] A. Taghavi, R. Hosseinzadeh, Hahn-Banach theorem for functional hypervector space, The J. Math. comp. sci., 2 (2011), no. 4, 682-690.
[10] A. Taghavi, R. Hosseinzadeh, Operators on normed hypervector spaces, Southeast Asian Bull. Math., (2011) 367-372.
[11] A. Taghavi, R. Hosseinzadeh, Uniform boundenes principle for operators of hypervector spaces, Iran. J. Math. Sci. Inform., 7 (2012), no. 2, 9-16.
[12] A. Taghavi, R. Hosseinzadeh, Hyper algebras and quotient hyper algebras, Ital. J. Pure Appl. Math., 26 (2009), no. 26, 17-24.
[13] M. S. Tallini, A-ipermoduli e spazi ipervettoriali, Rivisita di Matematica Della Universita di Parma, 3 (1988) 39-48.
[14] T. Vougiuklis, The Fundamental relation in hyperrings. The general hyper eld. Alg. Hyperstruc. Appl.,(1991) 203-211.
[15] T. Vougiuklis, Hyperstructures and their Representations, Hadronic Press, Inc., 1994.
[16] E. Zangiabadi, Z. Nazari, Pseodo-topological hypervector spaces and their properties, Ital. J. Pure Appl. Math., 38 (2017) 643-652.