RATIO MATHEMATICA 26(2014), 113–128 ISSN:1592-7415 The Lie-Santilli admissible hyperalgebras of type An Pipina Nikolaidou, Thomas Vougiouklis Democritus University of Thrace, School of Education, 68 100 Alexandroupolis, Greece pnikolai@eled.duth.gr,tvougiou@eled.duth.gr Abstract The largest class of hyperstructures is the one which satisfy the weak properties. These are called Hv-structures introduced in 1990 and they proved to have a lot of applications on several applied sci- ences. In this paper we present a construction of the hyperstructures used in the Lie-Santilli admissible theory on square matrices. Key words: hyperstructures, Hv-structures, hopes, weak hopes, ∂-hopes, e-hyperstructures, admissible Lie-algebras. MSC 2010: 20N20, 17B67, 17B70, 17D25. 1 Introduction We deal with hyperstructures called Hv-structures introduced in 1990 [30], which satisfy the weak axioms where the non-empty intersection replaces the equality. Some basic definitions are the following: In a set H equipped with a hyperoperation (abbreviation hyperoperation = hope) · : H ×H → P(H) −{∅}, we abbreviate by WASS the weak associativity : (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ H and by COW the weak commutativity : xy ∩yx 6= ∅,∀x,y ∈ H. P. Nikolaidou, Th. Vougiouklis The hyperstructure (H, ·) is called an Hv-semigroup if it is WASS, it is called Hv-group if it is reproductive Hv-semigroup, i.e., xH = Hx = H,∀x ∈ H. The hyperstructure (R, +, ·) is called an Hv-ring if (+) and (·) are WASS, the reproduction axiom is valid for (+) and (·) is weak distributive with respect to (+): x(y + z) ∩ (xy + xz) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅, ∀x,y,z ∈ R. Motivations. The motivation for Hv-structures is the following: We know that the quotient of a group with respect to an invariant subgroup is a group. F. Marty from 1934, states that, the quotient of a group with respect to any subgroup is a hypergroup. Finally, the quotient of a group with respect to any partition (or equivalently to any equivalence relation) is an Hv-group. This is the motivation to introduce the Hv-structures [24]. In an Hv-semigroup the powers of an element h ∈ H are defined as follows: h1 = {h},h2 = h ·h,...,hn = h◦h◦ ...◦h, where (◦) denotes the n-ary circle hope, i.e. take the union of hyperproducts, n times, with all possible patterns of parentheses put on them. An Hv- semigroup (H, ·) is called cyclic of period s, if there exists an element h, called generator, and a natural number s, the minimum one, such that H = h1 ∪h2...∪hs. Analogously the cyclicity for the infinite period is defined [23]. If there is an element h and a natural number s, the minimum one, such that H = hs, then (H, ·) is called single-power cyclic of period s. For more definitions and applications on Hv-structures, see the books [2],[8],[24],[4],[1] and papers as [3],[28],[21],[22],[26],[9],[14],[13]. The main tool to study hyperstructures are the fundamental relations β*, γ* and �*, which are defined in Hv-groups, Hv-rings and Hv-vector spaces, resp., as the smallest equivalences so that the quotient would be group, ring and vector space, resp. These relations were introduced by T. Vougiouklis [30],[24],[29]. A way to find the fundamental classes is given by theorems as the following [24],[21],[25],[22],[7],[9],[20]: Theorem 1.1. Let (H, ·) be an Hv-group and denote by U the set of all finite products of elements of H. We define the relation β in H by setting xβy iff {x,y}⊂ u where u ∈ U. Then β* is the transitive closure of β. 114 The Lie-Santilli admissible hyperalgebras of type An Analogous theorems for the relations γ* in Hv-rings, �* in Hv-modules and Hv-vector spaces, are also proved. An element is called single if its fundamental class is singleton [24]. Fundamental relations are used for general definitions. Thus, an Hv-ring (R, +, ·) is called Hv-field if R/γ* is a field. Let (H, ·), (H,∗) be Hv-semigroups defined on the same set H. The hope (·) is called smaller than the hope (∗), and (∗) greater than (·), iff there exists an f ∈ Aut(H,∗) such that xy ⊂ f(x∗y), ∀x,y ∈ H. Then we write · ≤ ∗ and we say that (H,∗) contains (H, ·). If (H, ·) is a structure then it is called basic structure and (H,∗) is called Hb-structure and (∗) is called b-hope. Theorem 1.2. (The Little Theorem). Greater hopes than the ones which are WASS or COW, are also WASS or COW, respectively. Definition 1.1. [20],[25] Let (H, ·) be hypergroupoid. We remove h ∈ H, if we consider the restriction of (·) in the set H −{h}. h ∈ H absorbs h ∈ H if we replace h by h and h does not appear in the structure. h ∈ H merges with h ∈ H, if we take as product of any x ∈ H by h, the union of the results of x with both h, h, and consider h and h as one class with representative h, therefore, h does not appear in the hyperstructure. For several definitions and applications of hyperstructures in mathematics or in sciences and social sciences one can see [11],[15],[13],[3]. 2 The theta (∂) hopes In [19],[32],[11],[15] a hope, in a groupoid with a map f on it, denoted ∂f , is introduced. Since there is no confusion, we write simply theta ∂. The symbol ”∂” appears in Greek papyrus to represent the letter ”theta ”usually in middle rather than the beginning of the words. Definition 2.1. Let H be a set equipped with n operations (or hopes) ⊗1, ...,⊗n and a map (or multivalued map) f : H → H (or f : H → P(H) −∅, respec- tively), then n hopes ∂1,∂2,...,∂n on H can be defined, called theta-operations (we rename here theta-hopes and we write ∂-hope) by putting x∂iy = {f(x) ⊗i y,x⊗i f(y)},∀x,y ∈ H and i ∈{1, 2, ...,n} 115 P. Nikolaidou, Th. Vougiouklis or, in case where ⊗i is hope or f is multivalued map, we have x∂iy = (f(x) ⊗i y) ∪ (x⊗i f(y)),∀x,y ∈ H and i ∈{1, 2, ...,n} if ⊗i is associative then ∂i is WASS. Analogously one can use several maps f, instead than only one. Let (G, ·) be a groupoid and fi : G → G,i ∈ I, be a set of maps on G. Take the map f∪ : G → P(G) such that f∪(x) = {fi(x)|i ∈ I} and we call it the union of the fi(x). We call union ∂-hopes, on G if we consider the map f∪(x). A special case is to take the union of f with the identity, i.e. f = f ∪ (id), so f(x) = {x,f(x)},∀x ∈ G, which is called b-∂-hope. We denote the b-∂-hope by (∂), so x∂y = {xy,f(x) ·y,x ·f(y)},∀x,y ∈ G This hope contains the operation (·) so it is a b-hope. If f : G → P(G)− {∅}, then the b-∂-hope is defined by using the map f(x) = {x}∪f(x),∀x ∈ G. Motivation for the definition of the theta-hope is the map derivative where only the multiplication of functions can be used. Therefore, in these terms, for two functions s(x), t(x), we have s∂t = {s′t,st′} where (′) denotes the derivative. For several results one can see [19],[32]. Examples. (a) Taking the application on the derivative, consider all polynomials of up to first degree gi(x) = aix + bi. We have g1∂g2 = {a1a2x + a1b2,a1a2x + b1a2}, so this is a hope in the first degree polynomials. Remark that all polynomials x+c, where c be a constant, are units. (b) The constant map. Let (G, ·) be group and f(x) = a, thus x∂y = {ay,xa},∀x,y ∈ G. If f(x) = e, then we obtain x∂y = {x,y}, the smallest incidence hope. Properties. If (G, ·) is a semigroup then: (a) For every f, the ∂-hope is WASS. (b) For every f, the b-∂-hope (∂) is WASS. (c) If f is homomorphism and projection, then (∂) is associative. 116 The Lie-Santilli admissible hyperalgebras of type An Properties. Reproductivity. If (·) is reproductive then (∂) is also reproductive. Commutativity. If (·) is commutative then (∂) is commutative. If f is into the centre of G, then (∂) is commutative. If (·) is COW then, (∂) is COW. Unit elements. The elements of the kernel of f, are the units of (G,∂). Inverse elements. For given x, the elements x′ = (f(x))−1u and x′ = u(f(x))−1, are the right and left inverses, respectively. We have two-sided inverses iff f(x)u = uf(x). Proposition. Let (G, ·) be a group then, for all maps f : G → G, the hyperstructure (G,∂) is an Hv-group. Definition 2.2. Let (R, +, ·) be a ring and f : R → R, g : R → R be two maps. We define two hopes (∂+) and (∂−), called both theta-hopes, on R as follows x∂+y = {f(x) + y,x + f(y)} and x∂·y = {g(x) ·y,x ·g(y)},∀x,y ∈ G. A hyperstructure (R, +, ·), where (+), (·) are hopes which satisfy all Hv- ring axioms, except the weak distributivity, will be called Hv-near-ring. Propositions. (a) Let (R, +, ·) be a ring and f : R → R, g : R → R be maps. The (R,∂)+,∂·), called theta, is an Hv-near-ring. Moreover (∂+) is commu- tative. (b) Let (R, +, ·) be a ring and f : R → R, g : R → R maps, then (R,∂+,∂·), is an Hv-ring. Properties.(Special classes). The theta hyperstructure (R,∂+,∂·) takes a new form and has some properties in several cases as the following ones: (a) If f is a homomorphism and projection, then x∂·(y∂+z)∩(x∂·y)∂+(x∂·z) = {f(x)f(y)+f(x)z,f(x)y+f(x)f(z)} 6= ∅. Therefore, (R,∂)+,∂·) is an Hv-ring. (b) If f(x) = x,∀x ∈ R, then (R, +,∂·) becomes a multiplicative Hv-ring: x∂·(y + z) ∩ (x∂·y) + (x∂·z) = {g(x)y + g(x)z} 6= ∅. If, moreover, f is a homomorphism, then we have a ”more” strong distributivity: x∂·(y + z) ∩ ((x∂·y) + (x∂·z)) = {g(x)y + g(x)z,xg(y) + xg(z)} 6= ∅. 117 P. Nikolaidou, Th. Vougiouklis Now we can see theta hopes in Hv-vector spaces and Hv-Lie algebras: Theorem 2.1. Let (V, +, ·) be an algebra over the field (F, +, ·) and f : V → V be a map. Consider the ∂-hope defined only on the multiplication of the vectors (·), then (V, +,∂) is an Hv-algebra over F, where the related properties are weak. If, moreover f is linear then we have λ(x∂y) = (λx)∂y = x∂(λy). Another well known and large class of hopes is given as follows [23],[24]: Let (G, ·) be a groupoid then for every P ⊂ G, P 6= ∅, we define the following hopes called P-hopes : for all x,y ∈ G P : xPy = (xP)y ∪x(Py), P r : xP ry = (xy)P ∪x(yP), P l : xP ly = (Px)y ∪P(xy). The (G,P),(G,P r) and (G,P l) are called P-hyperstructures. The most usual case is if (G, ·) is semigroup, then xPy = (xP)y ∪x(Py) = xPy and (G,P) is a semihypergroup but we do not know about (G,P r) and (G,P l). In some cases, depending on the choice of P, the (G,P r) and (G,P l) can be associative or WASS. A generalization of P-hopes, introduced by Davvaz, Santilli, Vougiouklis in [7],[6] is the following: Construction 2.1. Let (G, ·) be an abelian group and P any subset of G with more than one elements. We define the hope ×P as follows: x×p y = { x×P y = x ·P ·y = {x ·h ·y|h ∈ P} if x 6= e and c 6= e x ·y if x = e and y = e we call this hope Pe-hope. The hyperstructure (G,×p) is an abelian Hv-group. Matrix Representations Hv-structures are used in Representation Theory of Hv-groups which can be achieved either by generalized permutations or by Hv-matrices [28],[24]. Representations by generalized permutations can be faced by translations. In this theory the single elements are playing a crucial role. Hv-matrix is called a matrix if has entries from an Hv-ring. The hyperproduct of Hv- matrices is defined in a usual manner. In representations of Hv-groups by Hv-matrices, there are two difficulties: To find an Hv-ring and an appropriate set of Hv-matrices. 118 The Lie-Santilli admissible hyperalgebras of type An Most of Hv-structures are used in Representation (abbreviate by rep) Theory. Reps of Hv-groups can be considered either by generalized permu- tations or by Hv-matrices [24]. Reps by generalized permutations can be achieved by using translations. In the rep theory the singles are playing a crucial role. The rep problem by Hv-matrices is the following: Hv-matrix is called a matrix if has entries from an Hv-ring. The hyper- product of Hv-matrices A= (aij ) and B= (bij ), of type m × n and n × r, respectively, is a set of m× r Hv-matrices, defined in a usual manner: A ·B = (aij ) · (bij ) = {C = (cij )|(cij ) ∈⊕ ∑ aik · bkj}, where (⊕) denotes the n-ary circle hope on the hyperaddition. Definition 2.3. Let (H, ·) be an Hv-group,(R, +, ·) be an Hv-ring R and consider a set MR = {(aij )|aij ∈ R} then any map T : H → MR : h 7→ T(h) with T(h1h2) ∩T(h1)T(h2) 6= ∅,∀h1,h2 ∈ H. is called Hv-matrix rep. If T(h1h2) ⊂ T(h1)T(h2), then T is an inclusion rep, if T(h1h2) = T(h1)T(h2), then T is a good rep. 3 The general Hv-Lie Algebra Definition 3.1. Let (F, +, ·) be an Hv-field, (V, +) be a COW Hv-group and there exists an external hope · : F ×V → P(V ) −{∅} : (a,x) → zx such that, for all a,b in F and x,y in V we have a(x + y) ∩ (ax + ay) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, then V is called an Hv-vector space over F. In the case of an Hv-ring instead of an Hv-field then the Hv-modulo is defined. In these cases the fundamental relation �* is the smallest equivalence relation such that the quotient V/�* is a vector space over the fundamental field F/γ*. The general definition of an Hv-Lie algebra was given in [31] as follows: Definition 3.2. Let (L, +) be an Hv-vector space over the Hv-field (F, +, ·), φ : F → F/γ* the canonical map and ωF = {x ∈ F : φ(x) = 0}, where 0 is the zero of the fundamental field F/γ. Similarly, let ωL be the core of the 119 P. Nikolaidou, Th. Vougiouklis canonical map φ′ : L → L/�* and denote by the same symbol 0 the zero of L/�*. Consider the bracket (commutator) hope: [, ] : L×L → P(L) : (x,y) → [x,y] then L is an Hv-Lie algebra over F if the following axioms are satisfied: (L1) The bracket hope is bilinear, i.e. ∀x,x1,x2,y,y1,y2 ∈ L,λ1,λ2 ∈ F [λ1x1 + λ2x2,y] ∩ (λ1[x1,y] + λ2[x2,y]) 6= ∅ [x,λ1y1 + λ2y2] ∩ (λ1[x,y1] + λ2[x,y2]) 6= ∅, (L2) [x,x] ∩ωL 6= ∅, ∀x ∈ L (L3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]]) ∩ωL 6= ∅, ∀x,y ∈ L Definition 3.3. Let (A, +, ·) be an algebra over the field F. Take any map f : A → A, then the ∂-hope on the Lie bracket [x,y] = xy−yx, is defined as follows x∂y = {f(x)y −f(y)x,f(x)y −yf(x),xf(y) −f(y)x,xf(y) −yf(x)}. Remark that if we take the identity map f(x) = x,∀x ∈ A, then x∂y = {xy −yx}, thus we have not a hope and remains the same operation. Proposition. Let (A, +, ·) be an algebra F and f : A → A be a linear map. Consider the ∂−hope defined only on the multiplication of the vectors (·), then (A, +, ·) is an Hv-algebra over F, with respect to the ∂-hopes on Lie bracket, where the weak anti-commutativity and the inclusion linearity is valid. Proposition. Let (A, +, ·) be an algebra and f : A → A : f(x) = a be a constant map. Consider the ∂-hope defined only on the multiplication of the vectors (·), then (A, +,∂) is an Hv-Lie algebra over F. In the above theorem if one take a=e, the unit element of the multiplica- tion, then the properties become more strong. 4 Santilli’s admissibility The Lie-Santilli isotopies born to solve Hadronic Mechanics problems. Santilli proposed [16] a ”lifting” of the trivial unit matrix of a normal theory into a nowhere singular, symmetric, real-valued, new matrix. The original theory is reconstructed such as to admit the new matrix as left and right unit. 120 The Lie-Santilli admissible hyperalgebras of type An The isofields needed correspond to Hv-structures called e-hyperfields which are used in physics or biology. Definition: Let (Ho, +, ·) be the attached Hv-field of the Hv-semigroup (H, ·). If (H, ·) has a left and right scalar unit e then (Ho, +, ·) is e-hyperfield, the attached Hv-field of (H, ·). The Lie-Santilli theory on isotopies was born in 1970’s to solve Hadronic Mechanics problems. Santilli proposed a ”lifting” of the n-dimensional triv- ial unit matrix of a normal theory into a nowhere singular, symmetric, real- valued, positive-defined, n-dimensional new matrix. The original theory is reconstructed such as to admit the new matrix as left and right unit. The isofields needed in this theory correspond into the hyperstructures were in- troduced by Santilli and Vougiouklis in 1996 [5],[17] and they are called e-hyperfields. The Hv-fields can give e-hyperfields which can be used in the isotopy theory in applications as in physics or biology. We present in the following the main definitions and results restricted in the Hv-structures. Definition 4.1. A hyperstructure (H, ·) which contain a unique scalar unit e, is called e-hyperstructure. In an e-hyperstructure, we assume that for every element x, there exists an inverse x−1, i.e. e ∈ x ·x−1 ∩x−1 ·x. Remark that the inverses are not necessarily unique. Definition 4.2. A hyperstructure (F, +, ·), where (+) is an operation and (·) is a hope, is called e-hyperfield if the following axioms are valid: 1. (F, +) is an abelian group with the additive unit 0, 2. (·) is WASS, 3. (·) is weak distributive with respect to (+), 4. 0 is absorbing element: 0 ·x = x · 0 = 0,∀x ∈ F , 5. exist a multiplicative scalar unit 1, i.e. 1 ·x = x · 1 = x,∀x ∈ F , 6. for every x ∈ F there exists a unique inverse x−1, such that 1 ∈ x · x−1 ∩x−1 ·x. The elements of an e-hyperfield are called e-hypernumbers. In the case that the relation: 1 = x ·x−1 = x−1 ·x, is valid, then we say that we have a strong e-hyperfield. Now we present a general construction which is based on the partial ordering of the Hv-structures and on the Little Theorem. 121 P. Nikolaidou, Th. Vougiouklis Definition 4.3. The Main e-Construction. Given a group (G, ·), where e is the unit, then we define in G, a large number of hopes (⊗) as follows: x⊗y = {xy,g1,g2, ...},∀x,y ∈ G−{e}, and g1,g2, ... ∈ G−{e} g1,g2,... are not necessarily the same for each pair (x,y). Then (G,⊗) be- comes an Hv-group, actually is an Hb-group which contains the (G, ·). The Hv-group (G,⊗) is an e-hypergroup. Moreover, if for each x,y such that xy = e, so we have x⊗y = xy, then (G,⊗) becomes a strong e-hypergroup The proof is immediate since we enlarge the results of the group by putting elements from G and applying the Little Theorem. Moreover one can see that the unit e is a unique scalar and for each x in G, there exists a unique inverse x−1, such that 1 ∈ x ·x−1 ∩x−1 ·x and if this condition is valid then we have 1 = x ·x−1 = x−1 ·x. So the hyperstructure (G,⊗) is a strong e-hypergroup. 5 Mathematical Realisation of type An The representation theory by matrices gives to researchers a flexible tool to see and handle algebraic structures. This is the reason to see Lie-Santilli’s admissibility using matrices or hypermatrices to study the multivalued (hy- per) case. Using the well known P-hyperoperations we extend the Lie- Santilli’s admissibility into the hyperstructure case. We present the problem and we give the basic definitions on the topic which cover the four following cases: Construction 5.1. [18] Suppose R, S be sets of square matrices (or hy- permatrices). We can define the hyper-Lie bracket in one of the following ways: 1. [x,y]RS = xRy −ySx (General Case) 2. [x,y]R = xRy −yx 3. [x,y]S = xy −ySx 4. [x,y]RR = xRy −yRx The question is when the conditions, for all square matrices (or hyperma- trices) x, y, z, [x,x]RS 3 0 [x, [y,z]RS ]RS + [y, [z,x]RS ]RS + [z, [x,y]RS ]RS 3 0 of a hyper-Lie algebra are satisfied [18]. We apply this generalization on the Lie algebras of the type An. 122 The Lie-Santilli admissible hyperalgebras of type An We deal with Lie-Algebra of type An, of traceless matrices M (Tr(M)=0), which is a graded algebra, using the principal realisation used in Infinite Dimensional Kac Moody Lie Algebras introduced in 1981[10] by Lepowsky and Wilson, Kac [12]. In this special algebra examples on the above described hyperstructure theory are being presented. Denote as Eij (i,j = 1, ...,n) the n×n matrix which is 1 in the ij-entry and 0 everywhere else and by ei = Eii −Ei+1,i+1, i = 1, ...,n− 1 The Simple base of the above type is the following: Base of Level 0 : ei, i = 1, 2, ...,n− 1 Base of Level 1 : Ei,i+1, i = 1, 2, ...,n Base of Level 2 : Ei,i+2, i = 1, 2, ...,n ... Base of Level n-1 : Ei,i+(n−1), i = 1, 2, ...,n Denote that all the subscripts are mod n. Therefore the levels are in bold as follows: Level 0 :   a11 0 0 . . . 0 0 a22 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 0 0 0 . . . ann   Level 1 :   0 a12 0 . . . 0 0 0 a23 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . an−1,n an1 0 0 . . . 0   Level 2 : 123 P. Nikolaidou, Th. Vougiouklis   0 0 a13 . . . 0 0 0 0 . . . a2n . . . . . . . . . . . . . . . . . . . . . . . . . . an−1,1 0 0 . . . 0 0 an2 0 . . . 0   .................................................. Level n-1 :   0 0 0 . . . a1n a21 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 0 0 . . . an,n−1 0   For our examples the Konstant’s Cyclic Element E is being used as the sum of First Level’s Simple Base [10]. E = E12 + E23 + E34 + ... + En−1,n + En1 This element is shifting every element of level L to the next level L + 1 [10],[27]. The base of the first level as well as for every level, except zero, has n elements. Level 0 has a n − 1 dimension because of the limitation of the zero trace. The cyclic element gets different element from the base and goes to different ellement of the next level, creating an 1-1 correspondance. The element E shifts level n − 1 to the Level 0 and because, as already remarked, Level 0 has n − 1 elements, contrary with every other level, the 1-1 correspondance is being corrupted. To summarize, according to the related theory, removing from every level (except Level-0), all the powers of E until n− 1 (E,E2, ...,En−1), an one to one complete correspondance between all levels, Level-0 included, is being created. We denote the first power : [E,En1] 1 = E ·En1 −En1 ·E = A1 the second power: [E,En1] 2 = [E,A1] = A2 ............................... and inductively by the n-power: [E,En1] n = [E,An−1] = An One can prove the following: 124 The Lie-Santilli admissible hyperalgebras of type An Theorem 5.1. [E,En1] n = = diag( ( n−1 0 ) , (−1)1 ( n−1 1 ) , (−1)2 ( n−1 2 ) , ..., (−1)n−2 ( n−1 n−2 ) , (−1)n−1 ( n−1 n−1 ) ) The above theorem helps as to find the basic element of first Level’s base and based on this theorem all the nth powers of the elements of the first level can also be found. Theorem 5.2. Based on this theory and P-hyperstructures a set P with two elements can be used, either from zero or first level, but only with two elements. In this case the shift is depending on the level, so if we take P from Level-0, the result will not change, although the result will be multivalued. In case of different level insted, the shift will be analogous to the level of P. In the general case in Construction 5.1(1), one can notice the possible cardinality of the result, checking the Jacoby identity is very big. Even in the small case when |R| = |S| = |P| = 2 in the anticommutativity xPx−xPx could have cardinality 4 and the left side of the Jacoby identity is (xP(yPz −zPy) − (yPz −zPy)Px) + (yP(zPx−xPz)− −(zPx−xPz)Py) + (zP (xPy −yPx) − (xPy −yPx)Pz) could have cardinality 218. The number is reduced in special cases. Theorem 5.3. In the case of the Lie-algebra of type An, of traceless matrices M, we can define a hyper-Lie-Santilli-admissible bracket hope as follows: [xy]p = xPy −yPx where P = {p,q}, with p,q elements of the zero level. Then we obtain a hyper-Lie-Santilli-algebra. Proof We need only to proof the anticommutativity and the Jacobi identity as in the hyperstructure case. Therefore we have (a) [xy]p = xPy − yPx = {0,xpx − xqx,xqx − xpx} 3 0, so the ”weak” anticommutativity is valid, and (b) [x, [y,z]p]p + [y, [z,x]p]p + [z, [x,y]p]p = (xP(yPz −zPy) − (yPz −zPy)Px) + (yP(zPx−xPz)− −(zPx−xPz)Py) + (zP (xPy −yPx) − (xPy −yPx)Pz). But this set contains the element xpypz −xpzpy −ypzpx + zpypx + ypzpx−ypxpz− −zpxpy + xpzpy + zpxpy −zpypx−xpypz + ypxpz = 0 125 P. Nikolaidou, Th. Vougiouklis So the ”weak” Jacobi identity is valid. Thus, zero belongs to the above results, as it has to be, but there are more elements because it is a multivalued operation. References [1] P. Corsini, Prolegomena of hypergroup theory, Aviani, 1994. [2] P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academic Publ.,2003. [3] B. Davvaz, A brief survey of the theory of Hv-structures, 8 th AHA Congress, Spanidis Press (2003), 39-70. [4] B. 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