On Geometric Hyper-Structures 1

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1 International Mathematical Forum, Vol. 9, 2014, no. 14, HIKARI Ltd, 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 quasihyper-group is a hyper-groupoid such that x H = H x = H x H, (the reproduction axiom), where H x = h x. A semi-hyper-group is a h H 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 z = u x y 1 This project is supported by Research adminstration-paaet project no. BE-11-15

2 652 Mashhour I.M. Al Ali Bani-Ata et al. x v. For more information about hyper-structures one may refer to [4], v y z [2], [1], [3] and [5]. 2. Preliminaries and earlier studies Definition 1. If (H, ) and (H, ) are two hyper-groupoids, the function Φ : H H is called a good homomorphism if and only if Φ(x y) =Φ(x) Φ(y) for all (x, y) H 2. Definition 2. Let (G, ) be a hyper-groupoid. The action of (G, ) on a non empty set A is a map : G A ρ (A) such that for all (g 1,g 2 ) G 2,a A: (i) t a = g 1 s t g 1 g 2 s g 2 a (ii) There exists e G such that a e a. Remark 1. Let A ρ(a) be the set of all functions from A to ρ(a), endowed with the composition operation, then Φ:G G ρ(a) defined by Φ(g)(a) =g a is a homomorphism. Proof. (Φ(g 1 g 2 ))(a) = t a. From the above definition one obtains t g 1 g 2 t a g 1 s = g 1 (g 2 a) =Φ(g 1 )(Φ(g 2 ))(a) = (Φ(g 1 ) Φ(g 2 ))(a). t g 1 g 2 s g 2 a The homomorphism Φ : G A ρ(a) is called a hyper-representation associated with the hyper-groupoid action. This process is reversible in the sense that if Φ:G A ρ(a) is any good homomorphism, then the map from G A ρ(a) defined by g a = Φ(g)(a) satisfies the properties of hyper-groupoid action of G on A. 3. Hyper-geometric notations Definition 3. A hyper-vector space is a quadraplet (V,+,, K), where (V,+) is abelian group, K is a field and is a mapping (product times a scalar in K). : K V ρ (V ) where ρ (V ) is the non empty subsets of V, such that the following conditions hold: (i) For all a K and for all x, y V, then a (x + y) a x + a y. (ii) For all a, b K and for all x V, then (a + b) x a x + b x. (iii) For all a, b K and for all x V, then a (b x) =(a b) x. (iv) For all a, b K and for all x V, then a ( x) =( a) x = (a x).

3 (v) For all x V, then x 1 x. On geometric hyper-structures 653 In the right hand side of (i), the sum is meant in the sense of Frobenius, that is a x + a y = {w + u w a x, u a x}. Definition 4. A hyper-algebra V over a field K is a triple (V,,K) where (V,+,,K) is a hyper-vector space over K, and is a hyper-operation : V V ρ (V ) such that the following conditions hold: (i) There is no element e V, such that x x e e x for all x V, i.e e is a right unit element of V. (ii) x (y+z) = x y+x z for all x, y, z V and (x+y) z = x z+y z for all x, y, z V, where the right hand side of (ii) are meant the Frobenius. (iii) If t K, then (t a) b = t (a b) =a (t b) for all t K and for all a, b V, where (t a) b = w b, t (a b) = t u and w t a u a b a (t b) = a s. s t a Definition 5. A hyper-semifield over a field K, is a hyper-algebra (V,,K) such that V has no zero divisors, i.e if 0 a V and b V such that 0 a b, then 0 b e. Definition 6. Let (V,,K) and (W,,F) be two hyper-semifields over the fields K and F respectively, then a good hyper-homomorphism σ : V W is a mapping such that σ(x y) =σ(x) σ(y) for all x, y V. Definition 7. If (V,+,,K) is a hyper-vector space over K, then a hyperlinear transformation δ : V ρ (V ) is defined by: (i) δ(v + w) =δ(v)+δ(w) for all u, w V. (ii) δ(t a) =t δ(a) for all t K and a V. The set of all hyper-linear transformations of V is denoted by HL(V ). In particular if (V,,K) is a hyper-semifield over K, then a hyper-linear transformation δ of V is called a hyper-faithful-linear transformation if 0 δ(b), b V then 0 e b, i.e e is the unite element of (V,,K). Proposition 1. Let (V,,K) be a hyper-semifield over a field K. Fora V, let the hyper-mappings α a : V ρ (V ) and α a : V ρ (V ) defined by α a (v) =a v and α a(v) =v a respectively, where v V, are hyper-faithfullinear transformations. Proof. Let a V,a 0, then

4 654 Mashhour I.M. Al Ali Bani-Ata et al. 1. α(x + y) =a (x + y) =a x + a y = α a (x) +α a (y). (Use (ii) of Definition 4) 2. α a (t x) =a (t x) =t (a x) =t (α a (x)) for all t K and x V (use (iii) of Definition 4). If 0 α a (b), this implies that 0 a b. As a, then from the non-zero divisors property, it follows that 0 b e, which means that α a is hyper-faithful linear transformation. Definition 8. Let (V,+,,K) be a hyper-vector space over K, and let Σ = {α a 0 a V } HL (V ) the set of all hyper-faithful-linear transformations of V over K such that: (i) t α S Σ where (t α S )(x) =t (α S (x)) for all S, x V and for all t K. (ii) If α S,α T Σ, then α S + α T Σ where (α S + α T )(x) =α S (x)+α T (x), for all x V. (iii) There exists I Σ such that x I(x), for all x V. (iv) There is a fixed element e V and for all non-zero x V, there exists a unique T Σ such that x T (e). (v) For all t K and for a, b V, the following condition must hold: α z (b) = t w = α a (u) z t a w α a(b) u t b Then (Σ,, +) is called a hyper-spread set over K. Theorem 1. If (V,,K) is a hyper-semifield over K, then the hyper-regular representation ξ : a α a, 0 a V affords a hyper-spread set Σ over K. Proof. Let Σ = {α a 0 a V }, where α a : V ρ (V ) defined by α a (x) = a x is a hyper-faithful-linear transformation, by Proposition 1 (i) If α a, α b Σ, then (α a + α b )(x) =α a (x) +α b (x) =a x + b x = (a + b) x = α a+b (x). This implies that α a + α b Σ. (ii) (t α a )(x) =t (α a (x)) = t (a x) =(t a) x = α t a (x) =α w (x), where w t a, t K. This implies that t α a Σ, for all t K. (iii) I = α e Σ for: α e (x) =e x and as x e x for all x V, then it follows that α e (x) =I(x). (iv) As x x e for all x V, it follows that there exists a unique α x such that x α x (e). Setting α x = T, thus, there is a unique T Σ such that x T (e),x V. Hence the claim follows. Theorem 2. Let Σ be a hyper-spread set over a field K, then Σ affords a hyper-semifield over K.

5 On geometric hyper-structures 655 Proof. As Σ HL(V ) is a hyper-spread set, then pick some vector 0 e V. By ((iv) of Definition 8), and for any x V, there exists a unique T Σ (can be called T x ) such that x T x (e). Hence we can define a hyper-multiplication operation on V as follows: x y = T x (y), where x T x (e) for all x, y V. This turns V into a hyper-semifield over K for: x (y + z) =T x (y + z) = T x (y)+t x (z) =x y + x z. As x + y V, then there exists a unique T x+y Σ such that (x + y) (x + y) e = T (x+y) (e). Arguing similarly x + y x e + y e = T x (e)+t y (e). From the uniqueness property of T x+y,t x and T y, it follows that T x + T y = T x+y and hence (T x+y )(z) =(x + y) z =(T x + T y )(z) = T x (z)+t y (z) =x z + y z, x, y, z V Similarly one can show that x (y + z) =x y + x z for all x, y, z V. From ((iv) Definition 8), there is e V such that there exists a unique T Σ such that x T (e) =T x (e) =x e. This implies that this e is the unit element of V. From ((v) Definition 8), it follows that: (t a) x = T t a (x) = T z (x) = t w z t a w α a = t (α a (b)) = T a (u) =T a (t b) u t b for all t K and for all a, b V. The property of has no zero-divisors is an immediate consequence from the faithful condition of elements of Σ. This completes the proof of the theorem. Definition 9. Let (V,+,,K) be a hyper-vector space over K. Then H V is a hyper-subspace of V if (H, +,,K) is a hyper-vector space over K. Definition 10. Let (V,+,, K) be a hyper-vector space, a hyper-spread over K is a collection K of hyper-subspaces of V such that: 1. If X, Y K and X Y, then 0 X Y. 2. X = V. The elements of K are called components. X K Lemma 1. Let (V,+,,K) be a hyper-vector space, then A = V V can be turned into a hyper-vector space as follows: Define the hyper-operation : V V ρ (V ) (a, b) (c, d) =(a c, b + d), a, b, c, d V,

6 656 Mashhour I.M. Al Ali Bani-Ata et al. and define the hyper-operation ˆ : K (V V ) ρ (V ) tˆ (a, b) =(t a, t b), a, b V, t K. The proof of (V V,, ˆ,K) is a hyper-vector space is easy and can be omitted. Lemma 2. Let (V,,K) be a hyper-semifield, then the subsets L = {(x e, 0) x V } and R = {(0,y e) y V } of (V V,, ˆ,K) are hypersemifields. Proof. Define the hyper-operation ˆ : L L ρ (L )by (x e, 0) ˆ (y e, 0) = ((x y) e, 0), x, y V, where ρ (L ) is the set of non-empty subsets of L. Also, the hyper-operations ˆ, on K L and L L respectively by: tˆ (x e, 0) = ((t x) e, 0) and (x e, 0) (y,e,0) = ((x + y) e, 0) for all x, y V and for all t K. (i) The hyper-operation ˆ is distributive with respect to in L L for (x e, 0) ˆ ((y e), 0) (z e, 0) = (x e, 0) ˆ ((y z) e, 0) = ((x (y z)) e, 0) = ((x y) (x z)) e, 0) = (x y) e, 0) ((x z) e, 0) = (x e, 0) ˆ (y e, 0) (x e, 0) ˆ (z e, 0) Also, one can show that: (x e, 0) ((y e), 0) ˆ (z e, 0) = (x e, 0) ˆ (z e, 0) (y e, 0) ˆ (z e, 0) (ii) If t K, then tˆ ((x e, 0) ˆ ((y e), 0) = tˆ ((x y) e, 0) = ((t (x y) e, 0) = ((x ((t y) e, 0) = (x e) ˆ ((t y) e, 0) = (x e, 0) ˆ ((t y) e, 0) = (x e, 0) (tˆ (y e), 0) Also, as tˆ ((x y) e, 0) = t ((x y) e, 0) = (t (x y) e, 0) = (((t x) y) e, 0) = ((t x) e, 0) ˆ (y e, 0) = (tˆ (x e, 0) ˆ (y e), 0)

7 On geometric hyper-structures 657 (iii) (e e, 0) is the unit element of L L, for (x e, 0) ˆ (eė, 0) = ((x e) e, 0) for all (x e, 0) in L L, and as x x e e x for all x V, then x e w e =(x e) e. This implies that w x e (x e, 0) ((x e) e, 0) = (x e, 0) ˆ (e e, 0). Following a similar argument one can show that (x e, 0) (e e) ˆ (x e, 0) (iv) Assume that 0 (x e, 0) ˆ (y e, 0), where 0 (x e, 0), then 0 (x y) e, 0). This implies that 0 (x y) e. Since 0 e e, 0 (x y). But 0 x e, this implies that 0 y e and thus 0 (y e, 0). This completes the proof that L is a hyper-semifield. A similar argument can be followed to prove that R is a hyper-semifield. Lemma 3. Let (V,, ˆ,,K) be a hyper-semifield, then (L ˆ R, ˆ,K) is a hyper-vector space, where L ˆ R = {(x e, y e) x, y V }, (x e, y e) ˆ (a e, b e) = (((x e), 0) (a e, 0), (0, (y e) (0,b e))) and t (x e, y b) =((tˆ (x e), 0),tˆ (0, (y e))) for all x, y, a, b V, x e =(x e, 0). Proof. The proof is an immediate consequence of Lemma 1 and Lemma 2. Theorem 3. Let (V,,K) be a hyper-semifield, then (L ˆ R,,K) is a hyper-semifield, where (x e, y e) (a e, b e) = ( ((x e), 0) ˆ (a e, 0), (0,y e) ˆ (0,b e) ), where x e =(x e, 0) x V. Proof. 1. (x y, y e) ((a e, b e) ˆ (c e, d e)) = (x e, y e) ((a e, 0) (c e, 0), (0,b e) (0,d e)) = (x e, y e) (a + c) e, (b + d) e) = ( ((x e, 0) ˆ (a e, 0) (x e, 0) ˆ (c e, 0), (0,y e) ˆ (0,b e) (0,y e) ˆ (0,d e) ) =(x e, y e) (a e, c e) ˆ (x e, y e) (c e, d e) Arguing in a similar manner one can show that ((x e, y e) ˆ (a e, b e)) (c e, d e) = ((x e, y e) (c e, d e) ˆ (a e, b e) (c e, d e) From this it follows that is distributive with respect to ˆ.

8 658 Mashhour I.M. Al Ali Bani-Ata et al. 2. t ((x e, y e) (a e, c e)) = ( tˆ ((x e, 0) ˆ (a e, 0)),tˆ ((0,y e) ˆ (0,b e)) ). Then (2) follows from ((ii) of Lemma 2). 3. (e e, e e) is the unit element of L ˆ R. The proof directly follows from ((iii) of Lemma 2). 4. Assume that 0 ( (x e, y e) (a e, b e) ) such that 0 ((x e, 0) ˆ (a e, 0), (0,y e) ˆ (0,b e)). From ((iv) of Lemma 2), 0 (a e, 0) and 0 (0,b e)which means that 0 (a e, 0) ˆ (0,b e) and hence 0 (a e, b e). Remark 2. Let (V,,K) be a hyper-semifield, then the good hyper-homomorphism σ : V ρ (V ), keeps L ˆ R invariant if e σ = {e}. Proof. Let (x e, y e) L ˆ R then (x e, y e) σ =((x e) σ, (y e) σ ) = (((x e) σ, 0), (0, (y e) σ )) ((x σ y σ ), 0), (0, (y σ e σ ))=((z e, 0), (0, (w e)) = (z e, w e) L ˆ R, where z, w V. Definition 11. Let (V,,K) be a hyper-semifield. A left hyper-nucleus L V of V is defined by L V = {a V a (x y) =(a x) y, x, y V } Theorem 4. K e L V if and only if e L V. Proof. Let e L V and consider (t e) (x y) =t (e (x y)) = t ((e x) y) =(t (e x)) y =((t e) x) y (use (iii) of Definition 4), where t K, x, y V. From this it follows that K e L V. Conversely: Let K e L V, this implies that (t e) (x y) =((t e) x) y) for all t K and all x, y V. From this one has: t (e (x y)) = (t (e x)) y = t (e x) y which means that Hence e L V. t w = t z. w e (x y) z (e x) y

9 On geometric hyper-structures 659 Acknowledgment The authors thank are due to Paaet for the financial support, project no. BE and to ICTP for offering library facilities. References 1. M. Ibrahim Mohammad, On hypergroupoid actions, Riv. Mat. Univ. Parma (6) 4(2001), P. Corsini and J. Mittas, New topics in hypergroup theory, Multi-Valued log. 2 (1997), P. Corsini, Hypergraphs and hypergroups, Algebra Universalis, 35, (1996), P. Corsini, Prolegomena of hypergroup theory, Aviani Editori, Tricesimo, P. Dembowki, Finite Geometries Springer Verlag, Heidelberg Received: December 1, 2014