International Mathematical Forum, Vol. 9, 2014, no. 14, 651 - 659 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.312232 On Geometric Hyper-Structures 1 Mashhour I.M. Al Ali Bani-Ata, Fethi Bin Muhammad Belgacem and Abdulhamid Al-ibrahim Department of Mathematics, PAAET, Shamieh, Kuwait Copyright c  2014 Mashhour I.M. Al Ali Bani-Ata et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this paper is to introduce the notion of hyper-semifields, hyper-spread sets and hyper-spreads on one hand and to investigate the relation among these structures. 1. Introduction We recall some hyper-structures theory. A hyper-groupoid (H ; ⊗) is the set H endowed with a binary multi-valued operation (hyper-operation) i.e a function ⊗ from H × H to ρ ∗ (H ) the non empty set of subsets of H . A quasi- hyper-group is a hyper-groupoid such that x ⊗ H = H ⊗ x = H ∀x ∈ H , (the reproduction axiom), where H ⊗ x =  h∈H h ⊗ x. A semi-hyper-group is a hyper-groupoid (H, ⊗) such that (x ⊗ y ) ⊗ z = x ⊗ (y ⊗ z) ∀(x, y, z) ∈ H 3 . A semi-hyper-group (H, ⊗) is a hyper-group (or also multi-group) if H ⊗ x = x ⊗ H = H ∀x ∈ H , or equivalently if for all (a, b) ∈ H 2 , there exists (c, d) ∈ H 2 such that b ∈ c ⊗ a, b ∈ a ⊗ d. The condition (x ⊗ y ) ⊗ z = x ⊗ (y ⊗ z) can be rephrased as :  u∈x⊗y u ⊗ z = 1 This project is supported by Research adminstration-Paaet project no. BE-11-15