U.P.B. Sci. Bull., Series A, Vol. 73, Iss. 1, 2011 ISSN 1223-7027 ON HYPERACTIONS OF HYPERGROUPS Jianming Zhan 1 , S.Sh. Mousavi 2 , M. Jafarpour 3 In this paper, we define the notion of hyperaction of a hyper- group on a nonempty set and also the notion of index of a subhypergroup in a hypergroup, as a generalization of the concept of action of a group on a nonempty set and the notion of index of a subgroup in a group, re- spectively. Some properties such as the generalized orbit-stabilizer theorem, are investigated. In particular, introduce a construction of a hypergroup from a hyperaction. Finally, we assign a generalized state hypergroup to a nondeterministic automata which can associated from a hyperaction. Keywords: (semi)hypergroup, index, hyperaction, nondeterministic au- tomata MSC2000: 20N20, 68Q70 1. Introduction Hyperstructure theory was born in 1934 at the 8th congress of Scandina- vian Mathematicians, where Marty [16] introduced the hypergroup notion as a generalization of groups and proved its utility in solving some problems of groups, algebraic functions and rational fractions. Surveys of the theory can be found in the books of Corsini [3], Vougiouklis [17], Corsini and Leoreanu [7]. Hypergroups are studied from the theoretical point of view and for their appli- cations to many subjects of pure and applied mathematics: geometry, topology, cryptography and code theory, graphs and hypergraphs, probability theory, binary and n-ary relations, theory of fuzzy and rough sets, automata theory, artificial intelligence, etc. See, for example [2, 5, 11, 13, 15, 19, 20]. Some related recent work which some of them overlap the topic of this paper can be found in [1, 4, 6, 10, 12, 18]. We recall here some basic notions of hypergroup theory. 1 Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei Province, 445000, P. R. China, e-mail: zhanjianming@hotmail.com 2 Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran, e-mail: smousavi@mail.uk.ac.ir 3 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran, e- mail: m.j@mail.vru.ac.ir 117