Ratio Mathematica Vol. 33, 2017, pp. 151-165 ISSN: 1592-7415 eISSN: 2282-8214 Hyperstructures in Lie-Santilli Admissibility and Iso-Theories Ruggero Maria Santilli∗, Thomas Vougiouklis† ‡doi:10.23755/rm.v33i0.374 Abstract In the quiver of hyperstructures Professor R. M. Santilli, in early 90’es, tried to find algebraic structures in order to express his pioneer Lie-Santilli’s The- ory. Santilli’s theory on ’isotopies’ and ’genotopies’, born in 1960’s, desper- ately needs ’units e’ on left or right, which are nowhere singular, symmet- ric, real-valued, positive-defined for n-dimensional matrices based on the so called isofields.These elements can be found in hyperstructure theory, espe- cially in Hv-structure theory introduced in 1990. This connection appeared first in 1996 and actually several Hv-fields, the e-hyperfields, can be used as isofields or genofields so as, in such way they should cover additional prop- erties and satisfy more restrictions. Several large classes of hyperstructures as the P-hyperfields, can be used in Lie-Santilli’s theory when multivalued problems appeared, either in finite or in infinite case. We review some of these topics and we present the Lie-Santilli admissibility in Hyperstructures. Keywords: Lie-Santilli iso-theory, hyperstructures, hope, Hv-structures. 2010 AMS Mathematics Subject Classification: 20N20; 16Y99. ∗The Institute for Basic Research, 35246 US 19 North, P.O.Box 34684, CPalm Harbor, Florida 34684, USA; research@i-b-r.org †Democritus University of Thrace, School of Education, 68100 Alexandroupolis, Greece; tvou- gio@eled.duth.gr ‡ c©Ruggero Maria Santilli and Thomas Vougiouklis. Received: 31-10-2017. Accepted: 26- 12-2017. Published: 31-12-2017. 151 Ruggero Maria Santilli and Thomas Vougiouklis 1 Introduction In T. Vougiouklis, ”The Santilli’s theory ’invasion’ in hyperstructures” [24], there is a first description on how Santilli’s theories effect in hyperstructures and how new theories in Mathematics appeared by Santilli’s pioneer research. We continue with new topics in this direction. Last years hyperstructures have applications in mathematics and in other sci- ences as well. The applications range from biomathematics -conchology, inheritance- and hadronic physics or on leptons, in the Santilli’s iso-theory, to mention but a few. The hyperstructure theory is closely related to fuzzy theory; consequently, can be widely applicable in linguistic, in sociology, in industry and production, too. For all the above applications the largest class of the hyperstructures, the Hv-structures, is used, they satisfy the weak axioms where the non-empty inter- section replaces the equality. The main tools of this theory are the fundamental relations which connect, by quotients, the Hv-structures with the corresponding classical ones. These relations are used to define hyperstructures as Hv-fields, Hv- vector spaces and so on. Hypernumbers or Hv-numbers are called the elements of Hv-fields and they are important for the representation theory. The hyperstructures were introduced by F. Marty in 1934 who defined the hypergoup as a set equipped with an associative and reproductive hyperoperation. M. Koskas in 1970 was introduced the fundamental relation β∗, which it turns to be the main tool in the study of hyperstructures. T. Vougiouklis in 1990 was introduced the Hv-structures, by defining the weak axioms. The class of Hv- structures is the largest class of hyperstructures. Motivation for Hv-structures: The quotient of a group with respect to an invariant subgroup is a group. The quotient of a group with respect to any subgroup is a hypergroup. The quotient of a group with respect to any partition is an Hv-group. The Lie-Santilli theory on isotopies was born in 1970’s to solve Hadronic Mechanics problems. Santilli proposed a ’lifting’ of the n-dimensional trivial 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. According to Santilli’s iso-theory [14], [8] on a field F = (F,+, ·), a general isofield F̂ = F̂(â,+̂,×̂) is defined to be a field with elements â = a × 1̂, called isonumbers, where a ∈ F , and 1̂ is a positive-defined element generally outside F, equipped with two operations +̂ and ×̂ where +̂ is the sum with the conventional additive unit 0, and ×̂ is a new product â×̂b̂ := â× T̂ × b̂, with 1̂ = T̂−1,∀â, b̂ ∈ F̂. 152 Hyperstructures in Lie-Santilli Admissibility and Iso-Theories called iso-multiplication, for which 1̂ is the left and right unit of F̂, 1̂×̂â = â× 1̂ = â,∀â ∈ F̂ called iso-unit. The rest properties of a field are reformulated analogously. The isofields needed in this theory correspond into the hyperstructures were introduced by Santilli & Vougiouklis in 1996 [15], and called e-hyperfields. They point out that in physics the most interesting hyperstructures are the one called e-hyperstructures which contain a unique left ant right scalar unit. 2 Basic definitions on hyperstructures In what follows we present the related hyperstructure theory, enriched with some new results. However one can see the books and related papers for more definitions and results on hyperstructures and related topics: [2], [4], [17], [18], [19], [20], [23], [31], [33]. In a set H is called hyperoperation (abbreviated: hope) or multivalued oper- ation, any map from H ×H to the power set of H. Therefore, in a hope · : H ×H → ℘(H) : (x,y) → x ·y ⊂ H the result is subset of H, instead of element as we have in usually operations. In a set H equipped with a hope · : H ×H → ℘(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. The hyperstructure (H, ·) is called Hv-semigroup if it is WASS and it is called Hv-group if it is reproductive Hv-semigroup, i.e. xH = Hx = H,∀x ∈ H. The hyperstructure (R,+, ·) is called Hv-ring if (+) and (·) are WASS, the reproduc- tion axiom is valid for (+), and (·) is weak distributive to (+): x(y + z)∩ (xy + xz) 6= ∅,(x + y)z ∩ (xz + yz) 6= ∅,∀x,y,z ∈ R. An Hv-structure is very thin iff all hopes are operations except one, with all hy- perproducts singletons except one, which is set of cardinality more than one. The main tool to study all hyperstructures are the fundamental relations β*, γ* and �*, which are defined, in Hv-groups, Hv-rings and Hv-vector spaces, re- spectively, as the smallest equivalences so that the quotient would be group, ring and vector space, respectively [17], [18]. A way to find fundamental classes is given by analogous to the following: Theorem 2.1. Let (H, ·) be Hv-group and U all finite products of elements of H. Define the relation β by setting xβy iff {x,y} ⊂ u,u ∈ U. Then β* is the transitive closure of β. 153 Ruggero Maria Santilli and Thomas Vougiouklis Let (R,+, ·) be Hv-ring, U all finite polynomials of R. Define γ in R as fol- lows: xγy iff {x,y} ⊂ u where u ∈ U. Then γ* is the transitive closure of γ. An element is called single if its fundamental class is singleton. The fundamental relations are used for general definitions. Thus, to define the Hv-field the γ* is used [17], [18]: A Hv-ring (R,+, ·) is called Hv-field if R/γ* is a field. In the sequence the Hv-vector space is defined. Let (F,+, ·) be Hv-field, (V,+) a COW Hv-group and there exists an external hope · : F ×V → ℘(V) : (a,x) → ax such that, ∀a,b ∈ F and ∀x,y ∈ 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 Hv-field then the Hv-modulo is defined. In the above cases the fundamental relation �* is the smallest equivalence such that the quotient V/�* is a vector space over the fundamental field F/γ*. Let (H, ·), (H,∗) be Hv-semigroups defined on the same set H. (·) is called smaller than (∗), 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. The Little Theorem. Greater hopes than the ones which are WASS or COW, are also WASS or COW, respectively. The definition of Hv-field introduced a new class of hyperstructures: The Hv-semigroup (H, ·) is called h/v-group if the quotient H/β* is a group. In [20] the ’enlarged’ hyperstructures were examined if an element, outside the underlying set, appears in one result. In enlargement or reduction, most useful in representations are Hv-structures with the same fundamental structure. The Attach Construction. Let ((H, ·) be an Hv-semigroup and v /∈ H. We extend (·) into H = H∪{v} as follows: x·v = v ·x = v,∀x ∈ H, and v ·v = H. Then (H, ·) is an h/v-group where (H, ·)/β∗ ∼= Z2 and v is single element. We call the hyperstructure (H, ·) attach h/v-group of (H, ·) . Definition 2.1. Let (H, ·) be a hypergroupoid. We say that remove h ∈ H, if simply consider the restriction of (·) on H −{h}. We say that h ∈ H absorbs h ∈ H if we replace h, whenever it appears, by h. We say that h ∈ H merges with h ∈ H, if we take as product of x ∈ H by h, the union of the results of x with both h and h, and consider h and h as one class, with representative h. 154 Hyperstructures in Lie-Santilli Admissibility and Iso-Theories The uniting elements method was introduced by Corsini & Vougiouklis [3]. With this method one puts in the same class more elements. This leads, through hyperstructures, to structures satisfying additional properties. The uniting ele- ments method is the following: Let G be algebraic structure and d be a property, which is not valid and it is described by a set of equations; then, consider the partition in G for which it is put in the same partition class, all pairs that causes the non-validity of d. The quotient G/d is an Hv-structure. Then, quotient out the Hv-structure G/d by the fundamental relation β*, a stricter structure (G/d)/β* for which the property d is valid, is obtained. An application is when more than one properties are desired then: Theorem 2.2. [18] Let (G, ·) be a groupoid, and F = {f1, . . . ,fm,fm+1, . . . ,fm+n} be a system of equations on G consisting of two subsystems Fm = {f1, . . . ,fm} and Fn = {fm+1, . . . ,fm+n}. Let σ, σm be the equivalence relations defined by the uniting elements procedure using the systems F and Fm resp., and let σn be the equivalence relation defined using the induced equations of Fn on the groupoid Gm = (G/σm)/β∗. Then (G/σ)/β∗ ∼= (Gm/σn)/β∗. In a groupoid with a map on it, a hope is introduced [22]: Definition 2.2. Let (G, ·) be groupoid (resp., hypergroupoid) and f : G → G be map. We define a hope (∂), called theta and we write ∂-hope, on G as follows x∂y = {f(x) ·y,x ·f(y)},∀x,y ∈ G. (resp.x∂y = (f(x) ·y)∪ (x ·f(y),∀x,y ∈ G) If (·) is commutative then (∂) is commutative. If (·) is COW, then (∂) is COW. Motivation for a ∂-hope is the map derivative where only the product of func- tions is used. Thus for two functions s(x), t(x), we have s∂t = {s′t,st′} where (′) is the derivative. A large class of hyperstructures based on classical ones are defined by [18]: Definition 2.3. Let (G, ·) be groupoid, then for every P ⊂ G, P 6= ∅, we define the following hopes called P-hopes: ∀x,y ∈ G P : xPy = (xP)y ∪x(Py), Pr : xPry = (xy)P ∪x(yP), P l : xP ly = (Px)y ∪P(xy). The (G,P),(G,Pr) and (G,P l) are called P-hyperstructures. The usual case is for (G, ·) semigroup, then xPy = (xP)y ∪x(Py) = xPy and (G,P) is a semihypergroup. 155 Ruggero Maria Santilli and Thomas Vougiouklis 3 Representations. Hv-Lie algebras. Representations of Hv-groups, can be faced either by Hv-matrices or by gen- eralized permutations [18], [20], [31]. Hv-matrix (or h/v-matrix) is called a matrix with entries elements of an Hv- ring or Hv-field (or h/v-field). The hyperproduct 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 (⊕) is the n-ary circle hope on the hypersum: the sum of products of ele- ments is considered to be the union of the sets obtained with all possible parenthe- ses. In the case of 2 × 2 Hv-matrices the 2-ary circle hope which coincides with the hypersum in the Hv-ring. Notice that the hyperproduct of Hv-matrices does not nessesarily satisfy WASS. The representation problem by Hv-matrices is the following: Definition 3.1. Let (H, ·) be Hv-group, (R,+, ·) be Hv-ring and MR = {(aij)|aij ∈ R}, then any T : H → MR : h → T(h) with T(h1h2)∩T(h1)T(h2) 6= ∅,∀h1,h2 ∈ H, is called Hv-matrix representation If T(h1h2) ⊂ T(h1)T(h2), then T is an inclu- sion representation, if T(h1h2) = T(h1)T(h2), then T is a good representation. If T is one to one and good then it is a faithful representation. The main theorem of representations of Hv-structures is the following: Theorem 3.1. A necessary condition in order to have an inclusion representation T of an Hv-group (H, ·) by n × n Hv-matrices over the Hv-ring (R,+, ·) is the following: For all β∗(x),x ∈ H there must exist elements aij ∈ H,i,j ∈ {1, . . . ,n} such that T(β∗(a)) ⊂{A = (a′ij)|a ′ ij ∈ γ ∗(aij), i,j ∈{1, . . . ,n}} Therefore, every inclusion representation T : H → MR : a 7→ T(a) = (aij) in- duces an homomorphic representation T * of H/β* over R/γ* by setting T∗(β∗(a)) = [γ∗(aij)],∀β∗(a) ∈ H/β∗, where the element γ∗(aij) ∈ R/γ∗ is the ij entry of the matrix T∗(β∗(a)). Then T * is called fundamental induced representation of T . The helix hopes can be defined on any type of ordinary matrices [33], [34]: 156 Hyperstructures in Lie-Santilli Admissibility and Iso-Theories Definition 3.2. Let A = (aij) ∈ Mm×n be matrix and s,t ∈ N, with 1 ≤ s ≤ m, 1 ≤ t ≤ n. The helix-projection is a map st : Mm×n → Ms×t : A → Ast = (aij), where Ast has entries aij = {ai+κs,j+λt|1 ≤ i ≤ s,1 ≤ j ≤ t and κ,λ ∈ N,i + κs ≤ m,j + λt ≤ n} Let A = (aij) ∈ Mm×n,B = (bij) ∈ Mu×v be matrices and s = min(m,u), t = min(n,v). We define a hyper-addition, called helix-sum, by ⊕ : Mm×n ×Mu×v → ℘(Ms×t) : (A,B) → A⊕B = = Ast + Bst = (aij) + (bij) ⊂ Ms×t where (aij) + (bij) = {(cij) = (aij + bij)|aij ∈ aij and bij ∈ bij)}. Let A = (aij) ∈ Mm×n,B = (bij) ∈ Mu×v and s = min(n,u). Define the helix-product, by ⊗ : Mm×n ×Mu×v → ℘(Mm×v) : (A,B) → A⊗B = = Ams ·Bsv = (aij) · (bij) ⊂ Mm×v where (aij) · (bij) = {(cij) = ( ∑ aitbtj)|aij ∈ aij and bij ∈ bij)}. The helix-sum is commutative, WASS, not associative. The helix-product is WASS, not associative and not distributive to the helix-addition. Using several classes of Hv-structures one can face several representations. Some of those classes are as follows [18], [19], [7]: Definition 3.3. Let M = Mm×n, the set of m × n matrices on R and P = {Pi : i ∈ I}⊆ M. We define, a kind of, a P-hope P on M as follows P : M ×M → ℘(M) : (A,B)APB = {AP ti B : i ∈ I}⊆ M where P t is the transpose of P. P is bilinear Rees’ like operation where instead of one sandwich matrix a set is used. P is strong associative and inclusion distribu- tive to sum: AP(B + C) ⊆ APB + APC,∀A,B,C ∈ M. So (M,+,P) defines a multiplicative hyperring on non-square matrices. Definition 3.4. Let M = Mm×n be module of m×n matrices on R and take the sets S = {sk : k ∈ K}⊆ R,Q = {Qj : j ∈ J}⊆ M,P = {Pi : i ∈ I}⊆ M. 157 Ruggero Maria Santilli and Thomas Vougiouklis Define three hopes as follows S : R×M → ℘(M) : (r,A) → rSA = {(rsk)A : k ∈ K}⊆ M Q + : M×M → ℘(M) : (A,B) → AQ + B = {A + Qj + B : j ∈ J}⊆ M P : M×M → ℘(M) : (A,B) → APB = {AP ti B : i ∈ I}⊆ M Then (M,S,Q , P) is a hyperalgebra on R called general matrix P-hyperalgebra. The general definition of an Hv-Lie algebra is the following [26], [31], [16]: Definition 3.5. Let (L,+) be Hv-vector space on (F,+, ·), φ : F → F/γ*, 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 canonical map φ′ : L → L/�* and denote by the same symbol 0 the zero of L/�*. Consider the bracket hope (commutator): [, ] : L×L → ℘(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, and 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,z ∈ L 4 The Santilli’s: e-hyperstructures, iso-hyper the- ory. The e-hyperstructures where introduced in [15], [25] and where investigates in several aspects depending from applications [5], [6], [16], [31]. Definition 4.1. A hyperstructure (H, ·) which contains 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. 158 Hyperstructures in Lie-Santilli Admissibility and Iso-Theories Definition 4.2. A hyperstructure (F,+, ·), where (+) is an operation and (·) a hope, is called e-hyperfield if the following axioms are valid: (F,+) is an abelian group with the additive unit 0, (·) is WASS, (·) is weak distributive with respect to (+), 0 is absorbing element: 0·x = x·0 = 0,∀x ∈ F , there exists a multiplicative scalar unit 1, i.e. 1 · x = x · 1 = x,∀x ∈ F , and ∀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. Definition 4.3. Main e-Construction. Given a group (G, ·), where e is the unit, we define in G, an extremely 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). (G,⊗) is an Hv-group, it is an Hb-group which contains the (G, ·). (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 for both cases we enlarge the results of the group by putting elements from the set G and applying the Little Theorem. Moreover it is easy to see that the unit e is unique scalar element and for each x in G, there exists a unique inverse x−1, such that 1 ∈ x · x−1 ∩ x−1 · x. Finally if the last condition is valid then we have 1 = x · x−1 = x−1 · x, So the hyperstructure (G,⊗) is a strong e-hypergroup. Example 4.1. Consider the quaternion group Q = {1,−1, i,−i,j,−j,k,−k} with defining relations i2 = j2 = −1, ij = −ji = k. Denoting i = {i,−i},j = {j,−j},k = {k,−k} we may define a very large number (∗) hopes by enlarging only few products. For example, (−1)∗k = k,k∗i = j and i∗j = k. Then the hyperstructure (Q,∗) is a strong e-hypergroup. Construction 4.1. [31], [32]. On the ring (Z4,+, ·) we will define all the mul- tiplicative h/v-fields which have non-degenerate fundamental field and, moreover they are, (a) very thin minimal, (b) COW (non-commutative), (c) they have 0 and 1, scalars. 159 Ruggero Maria Santilli and Thomas Vougiouklis We have the isomorphic cases: 2⊗3 = {0,2} or 3⊗2 = {0,2}. The fundamental classes are [0] = {0,2}, [1] = {1,3} and we have (Z4,+,⊗)/γ∗ ∼= (Z2,+, ·). Thus it is isomorphic to (Z2 × Z2,+). In this Hv-group there is only one unit and every element has a unique double inverse. We can also define the analogous cases for the rings (Z6,+, ·),(Z9,+, ·), and (Z10,+, ·). In order to transfer Santilli’s iso-theory theory into the hyperstructure case we generalize only the new product ×̂ by replacing it by a hope including the old one [15], [27], [29], [32] and [1], [5], [6], [13], [14], [21], [24]. We introduce two general constructions on this direction as follows: Construction 4.2. General enlargement. On a field F = (F,+, ·) and on the isofield F̂ = F̂(â,+̂,×̂) we replace in the results of the iso-product â×̂b̂ = â× T̂ × b̂, with 1̂ = T̂−1 of the element T̂ by a set of elements Ĥab = {T̂, x̂1, x̂2, . . .} where x̂1, x̂2, . . . ∈ F̂, containing T̂ , for all â×̂b̂ for which â, b̂ /∈ {0̂, 1̂} and x̂1, x̂2, . . . ∈ F̂ −{0̂, 1̂}. If one of â, b̂, or both, is equal to 0̂ or 1̂, then Ĥab = {T̂}. Therefore the new iso-hope is â×̂b̂ = â× Ĥab × b̂ = â×{T̂, x̂1, x̂2, . . .}× b̂,∀â, b̂ ∈ F̂ F̂ = F̂(â,+̂,×̂) becomes isoHv-field. The elements of F are called isoHv- numbers or isonumbers. More important hopes, of the above construction, are the ones where only for few ordered pairs (â, b̂) the result is enlarged, even more, the extra elements x̂i, are only few, preferable one. Thus, this special case is if there exists only one pair (â, b̂) for which â×̂b̂ = â×{T̂, x̂}× b̂,∀â, b̂ ∈ F̂ and the rest are ordinary results, then we have a very thin isoHv-field. The assumption Ĥab = {T̂}, â or b̂, is equal to 0̂ or 1̂, with that x̂i, are not 0̂ or 1̂, give that the isoHv-field has one scalar absorbing 0̂, one scalar 1̂, and ∀â ∈ F̂ one inverse. A generalization of P-hopes, used in Santilli’s isotheory, is the following [5], [28], [31]: Let (G, ·) be abelian group and P a subset of G with #P > 1. We define the hope (×p) as follows: x×p y = { x ·P ·y = {x ·h ·y|h ∈ P} if x 6= e and y 6= e x ·y if x = e or y = e we call this hope Pe-hope. The hyperstructure (G,×p) is abelian Hv-group. 160 Hyperstructures in Lie-Santilli Admissibility and Iso-Theories Construction 4.3. The P-hope. Consider an isofield F̂ = F̂(â,+̂,×̂) with â = a× 1̂, the isonumbers, where a ∈ F , and 1̂ is positive-defined outside F, with two operations +̂ and ×̂, where +̂ is the sum with the conventional unit 0, and ×̂ is the iso-product â×̂b̂ = â× T̂ × b̂, with 1̂ = T̂−1,∀â, b̂ ∈ F̂ Take a set P̂ = {T̂, p̂1, ..., p̂s}, with p̂1, . . . , p̂s ∈ F̂ −{0̂, 1̂}, we define the isoP- Hv-field, F̂ = F̂(â,+̂,×̂p) where the hope ×̂P as follows: â×̂P b̂ := { â× ˆ̂P × b̂ = {â× ˆ̂h× b̂|ˆ̂h ∈ ˆ̂P} if â 6= 1̂ and b̂ 6= 1̂ â× ˆ̂T × b̂ if â = 1̂ or b̂ = 1̂ The elements of F̂ are called isoP-Hv-numbers. Remark. If P̂ = {T̂, p̂}, that is that P̂ contains only one p̂ except T̂ . The inverses in isoP-Hv-fields, are not necessarily unique. Example 4.2. Non degenerate example on the above constructions: In order to define a generalized P-hope on Ẑ7 = Ẑ7(â,+̂,×̂), where we take P̂ = {1̂, 6̂}, the weak associative multiplicative hope is described by the table: ×̂ 0̂ 1̂ 2̂ 3̂ 4̂ 5̂ 6̂ 0̂ 0̂ 0̂ 0̂ 0̂ 0̂ 0̂ 0̂ 1̂ 0̂ 1̂ 2̂ 3̂ 4̂ 5̂ 6̂ 2̂ 0̂ 2̂ 4̂,3̂ 6̂,1̂ 1̂,6̂ 3̂,4̂ 5̂, 2̂ 3̂ 0̂ 3̂ 6̂,1̂ 2̂,5̂ 5̂,2̂ 1̂,6̂ 4̂,3̂ 4̂ 0̂ 4̂ 1̂,6̂ 5̂,2̂ 2̂,5̂ 6̂,1̂ 3̂,4̂ 5̂ 0̂ 5̂ 3̂,4̂ 1̂,6̂ 6̂,1̂ 4̂,3̂ 2̂,5̂ 6̂ 0̂ 6̂ 5̂,2̂ 4̂,3̂ 3̂,4̂ 2̂,5̂ 1̂,6̂ The hyperstructure Ẑ7 = Ẑ7(â,+̂,×̂) is commutative and associative on the prod- uct hope. Moreover the β* classes on the iso-product ×̂ are {1̂, 6̂},{5̂, 2̂},{3̂, 4̂}. 5 The Lie-Santilli’s admissibility. Another very important new field in hypermathematics comes straightforward from Santilli’s Admissibility. We can transfer Santilli’s theory in admissibility for representations in two ways: using either, the ordinary matrices and a hope on them, or using hypermatrices and ordinary operations on them [10], [11], [12], [14], [16] and [7], [9], [30], [31], [34]. 161 Ruggero Maria Santilli and Thomas Vougiouklis Definition 5.1. Let L be 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∗. Let ωL be the core of the canonical map φ′ : L → L/�* and denote by the same symbol 0 the zero of L/�*. Take two subsets R,S ⊆ L then a Lie-Santilli admissible hyperalgebra is obtained by taking the Lie bracket, which is a hope: [, ]RS : L×L → ℘(L) : [x,y]RS = xRy−ySx = {xry−ysx|r ∈ R and s ∈ S} Special cases, but not degenerate, are the ’small’ and ’strict’ ones: (a) When only S is considered, then [x,y]S = xy −ySx = {xy −ysx|s ∈ S} (b) When only R is considered, then [x,y]R = xRy−yx = {xry−yx|r ∈ R} (c) When R = {r1,r2} and S = {s1,s2} then [x,y]RS = xRy−ySx = {xr1y−ys1x,xr1y−ys2x,xr2y−ys1x,xr2y−ys2x}. (d) We have one case which is ’like’ P-hope for any subset S ⊆ L: [x,y]S = {xsy −ysx|s ∈ S} On non square matrices we can define admissibility, as well: Construction 5.1. Let L = (Mm×n,+) be Hv-vector space of m × n hyper- matrices on the Hv-field (F,+, ·),φ : F → F/γ∗, canonical map and ωF = {x ∈ F : φ(x) = 0}, where 0 is the zero of the field F/γ*. Similarly, let ωL be the core of φ′ : L → L/�∗ and denote by the same symbol 0 the zero of L/�*. Take any two subsets R,S ⊆ L then a Santilli’s Lie-admissible hyperalgebra is obtained by taking the Lie bracket, which is a hope: [, ]RS : L×L → ℘(L) : [x,y]RS = xRty −yStx. Notice that [x,y]RS = xRty −yStx = {xrty −ystx|r ∈ R and s ∈ S} Special cases, but not degenerate, is the ’small’: R = {r1,r2} and S = {s1,s2} then [x,y]RS = xR ty −yStx = = {xrt1y −ys t 1x,xr t 1y −ys t 2x,xr t 2y −ys t 1x,xr t 2y −ys t 2x} 162 Hyperstructures in Lie-Santilli Admissibility and Iso-Theories References [1] R. Anderson, A. A. Bhalekar, B. Davvaz, P. S. Muktibodh, T. Vougiouk- lis, An introduction to Santilli’s isodual theory of antimatter and the open problem of detecting antimatter asteroids, NUMTA B., 6 (2012-13), 1–33. [2] P. Corsini, V. Leoreanu, Application of Hyperstructure Theory, Klower Ac. Publ.:(2003). [3] P. Corsini, T. Vougiouklis, From groupoids to groups through hypergroups, Rend. Mat. VII, 9, (1989), 173–181. [4] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, Int. Acad. Press, USA: (2007). [5] B. Davvaz, R.M. Santilli, T. Vougiouklis, Multi-valued Hypermathematics for characterization of matter and antimatter systems, J. Comp. Meth. Sci. Eng. (JCMSE) 13, (2013), 37–50. [6] B. Davvaz, R.M. Santilli, T. Vougiouklis, Algebra, Hyperalgebra and Lie- Santilli Theory, J. Generalized Lie Theory Appl., (2015), 9:2, 1–5. [7] A. Dramalidis, T. Vougiouklis, Lie-Santilli Admissibility on non-square ma- trices with the helix hope, CACAA, 4, N. 4, (2015), 353–360. [8] S. Georgiev, Foundations of Iso-Differential Carlculus, Nova Sc. Publ., V.1- 6: (2016). [9] P. Nikolaidou, T. Vougiouklis, The Lie-Santilli admissible hyperalgebras of type An, Ratio Math. 26, (2014), 113–128. [10] R. M. Santilli, Embedding of Lie-algebras into Lie-admissible algebras, Nuovo Cimento 51, 570: (1967). [11] R. M. Santilli, An introduction to Lie-admissible algebras, Suppl. Nuovo Cimento, 6, 1225: (1968). [12] R. M. Santilli, Dissipativity and Lie-admissible algebras, Mecc.1, 3: (l969). [13] R. M. Santilli, Representation of antiparticles via isodual numbers, spaces and geometries, Comm. Theor. Phys. 3, (1994), 153–181 [14] R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Volumes I, II, III, IV and V, International Academic Press, USA: (2007). 163 Ruggero Maria Santilli and Thomas Vougiouklis [15] R.M. Santilli, T. Vougiouklis, Isotopies, Genotopies, Hyperstructures and their Applications, New frontiers in Hyperstructures, Hadronic, (1996), 1– 48. [16] R. M. Santilli, T. Vougiouklis, Lie-admissible hyperalgebras, Italian J. Pure Appl. Math., N.31, (2013), 239–254. [17] T. Vougiouklis, The fundamental relation in hyperrings. The general hyper- field, 4th AHA, Xanthi 1990, World Scientific, (1991), 203–211 [18] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Inc.: (1994). [19] T. Vougiouklis, Some remarks on hyperstructures, Contemporary Math., Amer. Math. Society, 184, (1995), 427–431. [20] T. Vougiouklis, On Hv-rings and Hv-representations, Discrete Math., Else- vier, 208/209, (1999), 615–620. [21] T. Vougiouklis, Hyperstructures in isotopies and genotopies, Advances in Equations and Inequalities, Hadronic Press, (1999), 275–291. [22] T. Vougiouklis, A hyperoperation defined on a groupoid equipped with a map, Ratio Mat., N.1, (2005), 25–36. [23] T. Vougiouklis, ∂-operations and Hv-fields, Acta Math. Sinica, English S., V.23, 6, (2008), 965–972. [24] T. Vougiouklis, The Santilli’s theory ’invasion’ in hyperstructures, AGG, 28(1), (2011), 83–103. [25] T. Vougiouklis, The e-hyperstructures, J. Mahani Math. Research Center, V.1, N.1, (2012), 13–28. [26] T. Vougiouklis, The Lie-hyperalgebras and their fundamental relations, Southeast Asian Bull. Math., V.37(4), (2013), 601–614. [27] T. Vougiouklis, On the isoHv-numbers, Hadronic J., Dec.5, (2014), 1–18. [28] T. Vougiouklis, Lie-Santilli Admissibility using P-hyperoperations on matri- ces, Hadronic J., Dec.7, (2014), 1–14. [29] T. Vougiouklis, Iso-hypernumbers, Iso-Hv-numbers, ICNAAM 2014, AIP 1648, 510019, (2015); http://dx.doi.org/10.1063/1.4912724 164 Hyperstructures in Lie-Santilli Admissibility and Iso-Theories [30] T. Vougiouklis, Lie-Santilli Admissibility on non square matrices, Proc. IC- NAAM 2014, AIP 1648, (2015);http://dx.doi.org/10.1063/1.4912725 [31] T. Vougiouklis, Hypermathematics, Hv-structures, hypernumbers, hyper- matrices and Lie-Santilli admissibility, American J. Modern Physics, 4(5), (2015), 34–46. [32] T. Vougiouklis, Iso-Hv-numbers, Clifford Analysis, Clifford Alg. Appl. CA- CAA, V. 4, N. 4, (2015), 345–352. [33] T. Vougiouklis, S. Vougiouklis, The helix hyperoperations, Italian J. Pure Appl. Math., N.18, (2005), 197–206. [34] T. Vougiouklis, S. Vougiouklis, Hyper Lie-Santilli admisibility, AGG, 33, N.4, (2016), 427–442. 165