Bull. Math. Soc. Sci. Math. Roumanie Tome 51(99) No. 3, 2008, 233–243 Transitivity of Γ-relation on hyperfields by S. Mirvakili, S.M. Anvariyeh and B. Davvaz Abstract In this paper we introduce the complete parts on hyperrings and study the complete closure on hyperrings. Also, we consider the fundamental relation Γ on hyperrings and we prove that the relation Γ is transitive on hyperfields. Key Words: Hyperring, complete part, hyperfield, strongly regular relation, fundamental relation, transitive. 2000 Mathematics Subject Classification: Primary 16Y99, Se- condary 20N20. 1 Introduction A hypergroupoid (H, ◦) is a non-empty set H together with a hyperoperation ◦ defined on H, that is, a mapping of H × H into the family of non-empty subsets of H. If (x, y) ∈ H × H, its image under ◦ is denoted by x ◦ y. If A, B are non-empty subsets of H then A ◦ B is given by A ◦ B =  {x ◦ y | x ∈ A, y ∈ B}.x ◦ A is used for {x}◦ A and A ◦ x for A ◦{x}. A hypergroupoid (H, ◦) is called a hypergroup in the sense of Marty [10] if for all x, y, z ∈ H the following two conditions hold: (i) x ◦ (y ◦ z)=(x ◦ y) ◦ z, (ii) x ◦ H = H ◦ x = H, means that for any x, y ∈ H there exist u, v ∈ H such that y ∈ x ◦ u and y ∈ v ◦ x. If (H, ◦) satisfies only the first axiom, then it is called a semi-hypergroup. An exhaustive review updated to 1992 of hypergroup theory appears in [1]. A recent book [2] contains a wealth of applications. If H is a semi-hypergroup and ρ ⊆ H ×H is an equivalence relation then for all pairs (A, B) of non-empty subsets of H, we set A ρB if and only if aρb for all a ∈ A and b ∈ B. The relation ρ is said to be strongly regular to the right if xρy implies x ◦ a ρy ◦ a for all (x, y, a) ∈ H 3 . Analogously, we can define strongly regular to the left. Moreover ρ is called strongly regular if it is strongly regular to the right and to the left. Let H be a hypergroup and ρ an equivalence relation on H. Let ρ(a) be the equivalence class of a with respect to ρ and let H/ρ = {ρ(a) | a ∈ H}.