International Journal on Recent and Innovation Trends in Computing and Communication ISSN: 2321-8169 Volume: 2 Issue: 4 861 – 866 __________________________________________________________________________________________________ 861 IJRITCC | April 2014, Available @ http://www.ijritcc.org __________________________________________________________________________________________ A Note on Distributivity of a Poset of Subhypergroup of a Hyper Group A.D. Lokhande Department of mathematics, Yashwantrao chavan Warana Mahavidyalaya, Warana nagar,Kolhapur,India aroonlokhande@gmail.com Aryani Gangadhara Department of Mathematics,JSPM’s Rajarshi shahu college of Enginering, Pune,India. aryani.santosh@gmail.com MSC Code: Primary- 20N20, Secondary -06A11 Abstract: In this paper we defined distributive Property of Poset of Subhypergroup of hyper group and proved this property using principal filter. Also we proved distributivity using translation .we have defined annihilators and ideals of Poset of Subhypergroup of a hyper group. we proved some sufficient condition for Poset to be a distributive Poset. Keywords:Poset,Subhypergroup,DistributivePoset,Principlefilter,Annihilator,Ideals,Translations.homomorphism. __________________________________________****____________________________________________ Introduction The theory of hyper structures was introduced in 1934 by Marty at the 8th Congress of Scandinavian mathematicians [1]. This theory has been subsequently developed by Corsini[5],Mittas[3]and by various authors, Basic definitions and propositions are found in [4] .M Tarauceanu contributed to the study of Poset of Subhypergroup of a hyper group .He had drawn conclusions on Poset (Sub(H),⊆)and he had also given some open problems on above stated Poset and lattices . In this section, we study the distributivity of Poset of Subhypergroup of a Hyper group Principle Filter and we study the necessary condition for Poset to be a Distributive Poset Using the Concept of Translation,[7]. Finally we study the Annihilators [6] and ideals of Poset of Subhypergroup using Principle Filter. Notations and definitions are used from [4],[2],[6]. 1.Basic Notations and Terminology: Definition 1.1 [4]: A hyper Operation on H is a map : H x H P*(H). Definition 1.2[4]: If the hyper operation” “is associative and a H=H=H a, then (H, ) is a Hyper group. Definition 1.3 [4]: A hyper group (H, is called a join space if “ ” is commutative and a/b c/d a d b .where a/b ={x H/a b x} 2. Sub(L)=F(L) Example 2.1[4] : Let (L; л;v) be a complete lattice and, for every a 2 L, denote by F(a) the principal Filter of L generated by a (F(L)={x є L/a ≤ x})Then L is a join space under the hyper operation a ◦ b = F(a˄b); for all a; b є L. Proposition 2.2[4]: For the join space(L, ◦ ) given by the following equality holds Sub(L)=F(L)={F(a)/a єL} That is the subgroup of L coincides with the principal filter of L. In particular Sub(L) is a lattice anti isomorphic to L. Definition 2.3: Let Sub(L)=F(L) be a Poset. If F(A) Sub(L)=F(L) then we will denote L(F(A))={f(x) є Sub(L),f(x)≤f(a) for all f(a) є F(A)}