Topological vector space derived from a (Tallini) topological hypervector space
محورهای موضوعی : General algebraic systems
R. Ameri
1
,
M. Hamidi
2
,
A. Samadifam
3
1 - School of Mathematics, Statistics and Computer Sciences, University of Tehran, Tehran, Iran
2 - Department of Mathematics, Payame Noor University, Tehran, Iran
3 - Department of Mathematics, Payame Noor University, Tehran, Iran
کلید واژه: Topological hypervector space, fundamental relation, complete part, homeomorphic,
چکیده مقاله :
In this paper, we consider a hypervector space (in the sense of Tallini) $V$ over a field $K$. We use the fundamental relation $\varepsilon^*$ over $V$, as the smallest equivalence relation on $V$, to derived the fundamental vector space ${V}/\varepsilon^*$. In this regards, we prove that if $V$ is a (resp. quasi) topological hypervector space, then the fundamental vector space ${V}/\varepsilon^*$ with the property that each open subset of it is a complete part, then its fundamental vector space ${V}/\varepsilon^*$ is a topological vector space. Finally, we prove that for a topological vector space $(V,+,\cdot,K)$ and every subspace $W$ of $V$, the hypervector space $(\overline{V},+,\circ,K)$ is a topological hypervector space and we will prove $\overline{V}/\varepsilon^*$ and $V/W$ are homeomorphic, where $\overline{V}=V$.
In this paper, we consider a hypervector space (in the sense of Tallini) $V$ over a field $K$. We use the fundamental relation $\varepsilon^*$ over $V$, as the smallest equivalence relation on $V$, to derived the fundamental vector space ${V}/\varepsilon^*$. In this regards, we prove that if $V$ is a (resp. quasi) topological hypervector space, then the fundamental vector space ${V}/\varepsilon^*$ with the property that each open subset of it is a complete part, then its fundamental vector space ${V}/\varepsilon^*$ is a topological vector space. Finally, we prove that for a topological vector space $(V,+,\cdot,K)$ and every subspace $W$ of $V$, the hypervector space $(\overline{V},+,\circ,K)$ is a topological hypervector space and we will prove $\overline{V}/\varepsilon^*$ and $V/W$ are homeomorphic, where $\overline{V}=V$.
[1] H. Aghabozorgi, B. Davvaz, M. Jafarpour, Nilpotent groups derived from hypergroups, J. Algebra. 382 (2013), 177-184.
[2] R. Ameri, Topological (transposition) hypergroups, Italian J. Pure Appl. Math. 13 (2003), 171-176.
[3] R. Ameri, O. R. Dehghan, Dimension of fuzzy hypervector spaces, Iran. J. Fuzzy Syst. 8 (5) (2011), 149-166.
[4] R. Ameri, O. R. Dehghan, Fuzzy basis of fuzzy hypervector spaces, Iran. J. Fuzzy Syst. 7 (3) (2010), 97-113.
[5] R. Ameri, O. R. Dehghan, Fuzzy hypervector spaces, Adv. Fuzzy Syst. 1 (2008), 1-9.
[6] R. Ameri, O. R. Dehghan, Fuzzy hypervector spaces based on fuzzy singletons, Comp. Math. Appl. 61 (2011), 2933-2943.
[7] R. Ameri, O. R. Dehghan, On dimension of hypervector spaces, European J. Pure. Appl. Math. 1 (2) (2008), 32-50.
[8] R. Ameri, E. Mohammadzadeh, Engel groups derived from hypergroups, European J. Combin. 44 (2015), 191-197.
[9] S. Borhani-Nejad, B. Davvaz, On proximity spaces and topological hyper nearrings, Communic. Fac. Sci. Uni. Ankara. 69 (2) (2020), 1418-1427.
[10] P. Corsini, Prolegomena of Hypergroup Theory, Hecond Edition, Aviani Editor, 1993.
[11] P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, Springer, 5, 2013.
[12] I. Cristea, S. Hoskova, Fuzzy topological hypergroupoids, Iran. J. Fuzzy Syst. 6 (4) (2009), 1321.
[13] I. Cristea, S. Hoskova, Fuzzy pseudotopological hypergroupoids, Iran. J. Fuzzy Syst. 6 (4) (2009), 1119.
[14] L. Cristea, J. Zhan, Lower and upper fuzzy topological subhypergroups, Acta Math. Sin. 29 (2013), 315-330.
[15] O. R. Dehghan, R. Ameri, H. A. Ebrahim-Aliabadi, Some results on hypervectorspaces, Italian J. Pure Appl. Math. 41 (2019), 23-41.
[16] D. Freni, Une note sur le coeur dun hypergroupe et sur la cloture transitive β∗de β, Riv. diMat. Pura Appl. 8 (1991), 153-156.
[17] J. Jamalzadeh, Paratopological polygroups versus topological polygroups, Filomat. 32 (8) (2018), 2755-2761.
[18] M. Hamidi, A. Borumand Saeid, V. Leoreanu, Divisible groups derived from divisible hypergroups, U.P.B. Sci. Bull. Series A. 79 (2) (2017), 5970.
[19] D. Heidari, B. Davvaz, S. M. S. Modarres, Topological hypergroups in the sence of Marty, Comm. Algebra. 42 (2014), 4712-4721.
[20] D. Heidari, B. Davvaz, S. M. S. Modarres, Topological polygroups, Bull. Malays. Sci. Soc. 39 (2016), 707-721.
[21] S. Hoskova-Mayerova, Topological hypergroupoids, Comput. Math. Appl. 64 (9) (2012), 2845-2849.
[22] S. Hoskova-Mayerova, An overview of topological and fuzzy topological hypergroupoids, Ratio Math. 33 (2017), 21-38.
[23] M. Infusino, Topological Vector Spaces, Lecture Notes, University of Konstanz, 2016.
[24] M. Koskas, Groupoides, demi-hypergroupes et hypergroupes, J. Math. Pures Appl. 49 (9) (1970), 155-192.
[25] F. Marty, Sur une Generalization de la Notion de Groupe, 8th Congress Math. Scandinavia, Stockholm, 1934.
[26] M. Nodehi, M. Norouzi, O. R. Dehghan, An introduction to hyperrrings, CJMS. 9 (2) (2020), 210-223.
[27] M. Salehi Shadkami, M. R. Ahmadi Zand, B. Davvaz, Left big subsets of topological polygroups, Filomat. 30 (12) (2016), 3139-3147.
[28] M. Salehi Shadkami, M. R. Ahmadi Zand, B. Davvaz, The role of complete parts in topological polygroups, Int. J. Anal. Appl. 11 (1) (2016), 5460.
[29] M. Singh, K. Das, B. Davvaz, On topological complete hypergroups, Filomat. 31 (16) (2017), 5045-5056.
[30] M. Al Tahan, S. Hoskova-Mayerova, B. Davvaz, An overview of topological hypergroupoids, J. Intell. Fuzzy Syst. 34 (3) (2018), 1907-1916.
[31] M. S. Tallini, Hypervector Spaces, Proceedings of the Fourth International Congress on Algebraic Hyper-structures and Applications, Xanthi, Greece, 1990.
[32] M. S. Tallini, Weak Hypervector Spaces and Norms in such Spaces, Atti Convegno Algebraic Hyperstructures and Applications, Harbor, USA, 1994.
[33] T. Vougiouklis, Hyperstructures and their Representations, Hadronic Press Inc, 1994.
[34] E. Zangiabadi, Z. Nazari, Pseudo-topological hypervector spaces and their properties, Italian J. Pure. Appl. Math. 38 (2017), 643-652.
