Derived metabelian groups from hv-groups

Document Type : Research Paper

Authors

Department of Mathematics, Vali-e-Asr university of Rafsanjan, Iran

Abstract

In this paper first we introduce and analyze a new definition of left and right commutators in Hv-group. Secondly, using
commutators we introduce a new strongly equivalence relation πon an Hv-group H such that the quotient H/π, the set of all equivalence classes, is a metabelian group. Then we introduce metabelian Hv-groups and investigate some of their properties. Finally, we investigate some properties of commutators for the class of weak polygroups.

Keywords


[1] H. Aghabozorgi, B. Davvaz, M. Jafarpour, Solvable polygroups and derived sub-polygroups, Comm. Algebra, 41 (2013), 3098-3107.
[2] P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, Tricesimo, 1993.
[3] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Aca-demical Publications, Dordrecht, 2003.
[4] P. Corsini and V. Leoreanu, About the heart of a hypergroup, Acta Univ. Caroli-nae, 37 (1996) 17-28.
[5] B. Davvaz, Polygroup Theory and Related Systems, World Scientific, 2013.
[6] B., Davvaz, T., Vougiouklis, Commutative rings obtained from hyperrings (Hv-rings) with α∗ -relations, Comm. Algebra, 2007, 35:11, 3307-3320.
[7] D. Freni, A new characterization of the derived hypergroup via strongly regular equivalences, Comm. Algebra, 2002, 30:8, 3977-3989.
[8] M. Koskas, Groupoids, demi-hypergroupes et hypergroupes, J. Math. Pure Appl.,1970, 49:9, 155-192.
[9] F.Marty,Sur une Generalization de la Notion de Groupe,8th Congress Math.Scandenaves,Stockholm,Sweden,(1934)45-49.
[10] M. Norouzi, I. Cristea, Fundamental relation on m-idempotent hyperrings. Open Math. 2017; 15: 15581567.
[11] T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press,Palm Harbor, FL, 1994.
[12] T. Vougiouklis. The fundamental relation in hyperrings, The general hyperfield, Proc. 4th Internat. Congr. AHA, Xanthi, 1990 (World Scientific, Singapore,1991) 209-217.
[13] T. Vougiouklis, Groups in hypergroups, Combinatorics ’86 (Trento, 1986), 459–467, Ann. Discrete Math., 37,
North-Holland, Amsterdam, 1988.
[14] T. Vougiouklis, The fundamental relation in hyperrings. The general hyperfield, Algebraic hyperstructures and applications (Xanthi, 1990), World Sci. Publ.,Teaneck, NJ, 1991, 203-211.