Ratio Mathematica Vol. 33, 2017, pp. 181-201 ISSN: 1592-7415 eISSN: 2282-8214 Hv-Fields, h/v-Fields Thomas Vougiouklis∗ †doi:10.23755/rm.v33i0.386 Abstract In the last decades, the hyperstructures have had a lot of applications in mathematics and in other sciences. These applications range from biomath- ematics and hadronic physics to linguistic and sociology. For applications the largest class of the hyperstructures, the Hv-structures, is used, they sat- isfy the weak axioms where the non-empty intersection replaces the equality. The main tools in the theory of hyperstructures are the fundamental relations which connect, by quotients, the Hv-structures with the corresponding clas- sical ones. These relations are used to define hyperstructures as Hv-fields, Hv-vector spaces and so on, as well. The extension of the reproduction axiom, from elements to fundamental classes, introduces the extension of Hv-structures to the class of h/v-structures. We focus our study mainly in the relation of these classes and we present some constructions on them. Keywords: hope; Hv-structure; h/v-structure; Hv-field; h/v-field. 2010 AMS subject classifications: 20N20, 16Y99. ∗Democritus University of Thrace, School of Education, 68100 Alexandroupolis, Greece; tvou- giou@eled.duth.gr † c©Thomas Vougiouklis. Received: 31-10-2017. Accepted: 26-12-2017. Published: 31-12- 2017. 181 Thomas Vougiouklis 1 Introduction The main object in this paper is the largest class of hyperstructures called Hv- structures introduced in 1990 [35], which satisfy the weak axioms where the non- empty intersection replaces the equality. Abbreviation: hyperoperation=hope. Definition 1.1. An algebraic hyperstructure is called a set H equipped with at least one 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 commu- tativity: xy ∩ yx 6= ∅,∀x,y ∈ H. 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. Motivation. The quotient of a group by an invariant subgroup, is a group. F. Marty (1934), ’Sur une generalization de la notion de groupe’. 8eme Congres Math. Scandinaves, Stockholm, pp.45-49, states: the quotient of a group by a subgroup is a hypergroup. The quotient of a group by a partition (or equivalently to any equivalence) is an Hv-group. In an Hv-semigroup the powers are defined by: h1 = {h},h2 = h ·h,...,hn = h◦h◦...◦h, where (◦) is 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 cyclic of period s, if there is an element h, called generator, and a natural number s, the minimum : H = h1 ∪ h2... ∪ hs. Analogously the cyclicity for the infinite period is defined [30], [33], [39]. If there is an h and s, the minimum: H = hs, then (H, ·), is called single-power cyclic of period s. Definition 1.2. An (R, +, ·) is called Hv−ring if (+) and (·) are WASS, the re- production 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. Let (R, +, ·) be an Hv-ring, (M, +) be a COW Hv-group and there exists an external hope · : R×M → P(M) : (a,x) → ax such that ∀a,b ∈ R and ∀x,y ∈ M we have a(x + y) ∩ (ax + ay) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, then M is called an Hv-module over F. In the case of an Hv-field F, which is defined later, instead of an Hv-ring R, then the Hv−vector space is defined. For more definitions and applications on hyperstructures one can see books [4], [5], [9], [10], [11], [39] and papers as [3], [7], [8], [15], [16], [20], [21], [27], [38], [40], [41], [43], [48], [55], [68]. 182 Hv-Fields, h/v-Fields Definition 1.3. Let (H, ·), (H,∗) be Hv-semigroups on the same set H, the hope (·) is called smaller than the (∗), 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 ones which are WASS or COW, are also WASS or COW, respectively. This Theorem leads to a partial order on Hv-structures and to posets [39], [42], [43], [21]. Let (H, ·) be hypergroupoid. We remove h ∈ H, if we take the restriction of (·) in the set H −{h}. h ∈ H absorbs h ∈ H if we replace h by h and h does not appear. 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. The main tool in hyperstructures is the fundamental relation. M. Koscas 1970, [20], defined in hypergroups the relation β and its transitive closure β*. This relation is defined in Hv-groups, as well, and connect hyperstructures with the classical structures. T. Vougiouklis [34], [35], [39], [40], [41], [53], [54], [60], introduced the γ* and �* relations, which are defined, in Hv-rings and Hv-vector spaces, respectively. He also named all these relations, fundamental. (see also [4], [5], [1], [8], [10], [11]). Definition 1.4. The fundamental relations β*, γ* and �*, are defined, in Hv- groups, Hv-rings and Hv-vector spaces, respectively, as the smallest equivalences so that the quotient would be group, ring and vector spaces, respectively. Specifying the above motivation we remark that: Let (G, ·) be a group and R be an equivalence relation (or a partition) in G, then (G/R, ·) is an Hv-group, therefore we have the quotient (G/R, ·)/β∗ which is a group, the fundamental one. The main Theorem to find the fundamental classes is the following: 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. Notation. We denote by [x] the fundamental class of the element x ∈ H. Therefore β∗(x) = [x]. Analogous theorems are for Hv-rings, Hv-vector spaces and so on. For proof, see [34], [39]. An element is called single [39] if its fundamental class is singleton so, [x] = {x}. 183 Thomas Vougiouklis More general structures can be defined by using the fundamental structures. An application in this direction is the general hyperfield. There was no general definition of a hyperfield, but from 1990 [35] there is the following [38], [39]: Definition 1.5. An Hv-ring (R, +, ·) is called Hv-field if R/γ* is a field. Since the algebras are defined on vector spaces, the analogous to Theorem 1.1, on Hv-vector spaces is the following: Let (V, +) be an Hv-vector space over the Hv-field F. Denote by U the set of all expressions consisting of finite hopes either on F and V or the external hope applied on finite sets of elements of F and V. We define the relation �, in V as follows: x�y iff {x,y} ∈ u where u ∈ U. Then the relation �* is the transitive closure of the relation �. Definition 1.6. [53], [54], [57]. 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*. 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 (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. [λ1x1 + λ2x2,y] ∩ (λ1[x1,y] + λ2[x2,y]) 6= ∅ [x,λ1y1 + λ2y2] ∩ (λ1[x,y1] + λ2[x,y2]) 6= ∅, ∀x,x1,x2,y,y1,y2 ∈ L,λ1,λ2 ∈ F (L2) [x,x] ∩ωL 6= ∅, ∀x ∈ L (L3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]]) ∩ωL 6= ∅, ∀x,y ∈ L In the Definition 1.5, was introduced a new class of which is the following [45] (for a preliminary report see: T. Vougiouklis. A generalized hypergroup, Abstracts AMS, Vol. 19.3, Issue 113, 1998, p.489): Definition 1.7. The Hv-semigroup (H, ·) is called h/v-group if H/β∗ is a group. An important and well known class of hyperstructures defined on classical structures are defined as follows [30], [33], [36], [57], [60]: Definition 1.8. 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), 184 Hv-Fields, h/v-Fields Pr : xPry = (xy)P ∪x(yP), P l : xP ly = (Px)y ∪P(xy). The (G,P),(G,Pr), (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. A generalization of P-hopes, used in Santilli’s isotheory, is the following [12], [13], [14]: 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 c 6= e x ·y if x = e or y = e we call this hope Pe-hope. The hyperstructure (G,×P ) is abelian Hv-group. Definition 1.9. [36]. An Hv-structure is called very thin if all hopes are opera- tions except one, which has all hyperproducts singletons except one, which is a subset of cardinality more than one. Therefore, in a very thin Hv-structure in H there exists a hope (·) and a pair (a,b) ∈ H2 for which ab = A, with cardA > 1, and all the other products, are singletons. From the properties of the very thin hopes the Attach Construction is obtained [43], [54]: Let (H, ·) be an Hv-semigroup and v /∈ H. We extend the (·) into H = H ∪{v} by: x ·v = v ·x = v,∀x ∈ H, and v ·v = H. The (H, ·) is an Hv-group, where (H, ·)/β∗ ∼= Z2 and v is a single. A class of Hv-structures is the following [47], [49], [57], [60]: Definition 1.10. Let (G, ·) be groupoid (resp. hypergroupoid) and f : G → G be a map. We define a hope (∂), called theta-hope, 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. If (G, ·) is a groupoid (or hypergroupoid) and f : G → P(G) −{∅} be any multivalued map. We define the ∂-hope on G as follows: x∂y = (f(x) ·y) ∪ (x ·f(y)), ∀x,y ∈ G. The ∂-hopes can be defined in Hv-vector spaces and Hv-Lie algebras: 185 Thomas Vougiouklis 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)}. 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. Motivation for the theta-hope is the map derivative where only the multiplica- tion of functions can be used. Basic property: if (G, ·) is semigroup then ∀f, the ∂-hope is WASS. Example. (a) In integers (Z, +, ·) fix n 6= 0, a natural number. Consider the map f such that f(0) = n and f(x) = x, ∀x ∈ Z − {0}. Then (Z,∂+,∂·), where ∂+ and ∂· are the ∂-hopes refereed to the addition and the multiplication respectively, is an Hv-near-ring, with (Z,∂+,∂·)/γ* ∼= Zn. (b) In (Z, +, ·) with n 6= 0, take f such that f(n) = 0 and f(x) = x, ∀x ∈ Z−{n}. Then (Z,∂+,∂·) is an Hv-ring, moreover, (Z,∂+,∂·)/γ* ∼= Zn. Special case of the above is for n = p, prime, then (Z,∂+,∂·) is an Hv-field. The uniting elements method was introduced by Corsini-Vougiouklis [6] in 1989. With this method one puts in the same class, two or more elements. This leads, through hyperstructures, to structures satisfying additional properties. The uniting elements method is the following: Let G be algebraic structure and d, a property which is not valid. Suppose that d is described by a set of equations; then, take the partition in G for which it is put together, in the same class, every pair of elements that causes the non-validity of the property d. The quotient by this partition 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. It is very important if more properties are desired, then we have the following [39]: Theorem 1.2. Let (R, +, ·) be a ring, and F = {f1, ...,fm,fm+1, ...,fm+n} be a system of equations on R 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 respectively, and let σn be the equivalence relation defined using the induced equations of Fn on the ring Rm = (R/σm)/γ*. Then, (R/σ)/γ∗ ∼= (Rm/σn)γ∗. 186 Hv-Fields, h/v-Fields Combining the uniting elements procedure with the enlarging theory or the ∂-theory, we can obtain analogous results [39], [51], [54], [60], [22]. Theorem 1.3. In the ring (Zn, +, ·), with n=ms we enlarge the multiplication only in the product of the special elements 0 · m by setting 0 ⊗ m = {0,m} and the rest results remain the same. Then (Zn, +,⊗)/γ∗ ∼= (Zm, +, ·). Remark that we can enlarge other products as well, for example 2·m by setting 2⊗m = {2,m + 2}, then the result remains the same. In this case 0 and 1 remain scalars. Corollary. In the ring (Zn, +, ·), with n=ps where p is prime, we enlarge only the product 0 ·p by 0⊗p = {0,p} and the rest remain the same. Then (Zn, +,⊗) is very thin Hv-field. 2 Constructions of Hv-fields and h/v-fields The class of h/v-groups is more general than the Hv-groups since in h/v-groups the reproductivity is not valid. The reproductivity of classes is valid, i.e. if H is partitioned into equivalence classes, then x[y] = [xy] = [x]y,∀x,y ∈ H, because the quotient is reproductive. In a similar way the h/v-rings, h/v-fields, h/v-modulus, h/v-vector spaces etc are defined. Remark 2.1. From definition of the Hv-field, we remark that the reproduction axiom in the product, is not assumed, the same is also valid for the definition of the h/v-field. Therefore, an Hv-field is an h/v-field where the reproduction axiom for the sum is also valid. We know that the reproductivity in the classical group theory is equivalent to the axioms of the existence of the unit element and the existence of an inverse element for any given element. From the definition of the h/v-group, since a generalization of the reproductivity is valid, we have to extend the above two axioms on the equivalent classes. Definition 2.1. Let (H, ·) be an Hv-semigroup, and denote [x] the fundamental, or equivalent class, of the element x ∈ H. We call unit class the class [e] if we have ([e] · [x]) ∩ [x] 6= ∅ and ([x] · [e]) ∩ [x] 6= ∅,∀x ∈ H, and for each element x ∈ H, we call inverse class of [x], the class [x′], if we have ([x] · [x′]) ∩ [e] 6= ∅ and ([x′] · [x]) ∩ [e] 6= ∅. 187 Thomas Vougiouklis The ’enlarged’ hyperstructures were examined in the sense that a new element appears in one result. In enlargement or reduction, most useful are those Hv- structures or h/v-structures with the same fundamental structure [43], [53]. Construction 2.1. (a) Let (H, ·) be an Hv-semigroup and v /∈ H. We extend the (·) into H = H ∪{v} as follows: x ·v = v ·x = v,∀x ∈ H, and v ·v = H. The (H, ·) is an h/v-group, called attach, where (H, ·)/β∗ ∼= Z2 and v is a single element. We have core (H, ·) = H. The scalars and units of (H, ·) are scalars and units (resp.) in (H, ·). If (H, ·) is COW (resp. commutative) then (H, ·) is also COW (resp. commutative). (b) Let (H, ·) be an Hv-semigroup and {v1, . . . ,vn}∩ H = ∅, is an ordered set, where if vi < vj, when i < j. Extend (·) in Hn = H ∪{v1, . . . ,vn} as follows: x ·vi = vi ·x = vi,vi ·vj = vj ·vi = vj,∀i < j and vi ·vi = H ∪{v1, . . . ,vi−1},∀x ∈ H,i ∈{1, . . . ,n}. Then (Hn, ·) is h/v-group, called attach elements, where (Hn, ·)/β∗ ∼= Z2 and vn is single. (c) Let (H, ·) be an Hv-semigroup, v /∈ H, and (H, ·) be its attached h/v-group. Take an element 0 /∈ H and define in Ho = H ∪{v, 0} two hopes: hypersum (+): 0 + 0 = x + v = v + x = 0, 0 + v = v + 0 = x + y = v, 0 + x = x + 0 = v + v = H, ∀x,y ∈ H hyperproduct (·): remains the same as in H moreover 0·0 = v ·x = x·0 = 0,∀x ∈ H Then (Ho, +, ·) is h/v-field with (Ho, +, ·)/γ∗ ∼= Z3. (+) is associative, (·) is WASS and weak distributive with respect to (+). 0 is zero absorbing and single but not scalar in (+). (Ho, +, ·) is called the attached h/v-field of the Hv-semigroup (H, ·). Let us denote by U the set of all finite products of elements of a hypergroupoid (H, ·). Consider the relation defined as follows: xLy iff there exists u ∈ U such that ux∩uy 6= ∅. Then the transitive closure L∗ of L is called left fundamental reproductivity rela- tion. Similarly, the right fundamental reproductivity relation R∗ is defined. 188 Hv-Fields, h/v-Fields Theorem 2.1. If (H, ·) is a commutative semihypergroup, i.e. the strong com- mutativity and the strong associativity is valid, then the strong expression of the above L relation: ux = uy, has the property: L∗ = L . Proof. Suppose that two elements x and y of H are L* equivalent. Therefore, there are u1, . . . ,un+1 elements of U, and z1, . . . ,zn elements of H, such that u1x = u1z1,u2z1 = u2z2, . . . ,unzn−1 = unzn,un+1zn = un+1y. From these relations, using the strong commutativity, we obtain un+1 . . .u2u1x = un+1 . . .u2u1z1 = un+1 . . .u1u2z1 = = un+1 . . .u2u1z2 = · · · = un+1 . . .u2u1y Therefore, setting u = un+1 . . .u2u1 ∈ U, we have ux = uy. 2 Corollary. Let (S, ·) be commutative semigroup which has an element w ∈ S such that the set wS is finite. Consider the transitive closure L* of the relation L defined by: xLy iff there exists z ∈ S such that zx = zy. Then < S/L∗,◦ /β∗ is a finite commutative group, where (◦) is the induced oper- ation on classes of S/L*. Open problem: Prove that L*, is the smallest equivalence: H/L*, is reproduc- tive. We present now the small non-degenerate Hv-fields on (Zn, +, ·) which sat- isfy the following conditions, appropriate in Santilli’s iso-theory: 1. multiplicative very thin minimal, 2. COW (non-commutative), 3. they have the elements 0 and 1, scalars, 4. when an element has inverse element, then this is unique. Remark that last condition means than we cannot enlarge the result if it is 1 and we cannot put 1 in enlargement. Moreover we study only the upper triangular cases, in the multiplicative table, since the corresponding under, are isomorphic since the commutativity is valid for the underline rings. From the fact that the reproduction axiom in addition is valid, we have always Hv-fields. Theorem 2.2. All multiplicative Hv-fields defined on (Z4, +, ·), which have non- degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: The only product which is set is 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, +, ·). 189 Thomas Vougiouklis Example. Let us denote by Eij the matrix with 1 in the ij-entry and zero in the rest entries. Then take the following 2×2 upper triangular Hv-matrices on the above Hv-field (Z4, +, ·) of the case that only 2 ⊗ 3 = {0, 2} is a hyperproduct: I = E11 +E22,a = E11 +E12 +E22,b = E11 + 2E12 +E22,c = E11 + 3E12 +E22, d = E11+3E22,e = E11+E12+3E22,f = E11+2E12+3E22,g = E11+3E12+3E22, then, we obtain for X = {I,a,b,c,d,e,f,g}, that (X,⊗) is non-COW Hv-group and the fundamental classes are a = {a,c},d = {d,f},e = {e,g} and the fun- damental group is isomorphic to (Z2 ×Z2, +). In this Hv-group there is only one unit and every element has a unique double inverse. Theorem 2.3. All multiplicative Hv-fields defined on (Z6, +, ·), which have non- degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: We have the only one hyperproduct, (I) 2 ⊗ 3 = {0, 3} or 2 ⊗ 4 = {2, 5} or 3 ⊗ 4 = {0, 3} or 3 ⊗ 5 = {0, 3} or 4 ⊗ 5 = {2, 5} Fundamental classes: [0] = {0, 3}, [1] = {1, 4}, [2] = {2, 5}, and (Z6, +, ·)/γ∗ ∼= (Z3, +, ·). (II) 2 ⊗ 3 = {0, 2} or 2 ⊗ 3 = {0, 4} or 2 ⊗ 4 = {0, 2} or 2 ⊗ 4 = {2, 4} or 2 ⊗ 5 = {0, 4} or 2 ⊗ 5 = {2, 4} or 3 ⊗ 4 = {0, 2} or 3 ⊗ 4 = {0, 4} or 3 ⊗ 5 = {3, 5} or 4 ⊗ 5 = {0, 2} or 4 ⊗ 5 = {2, 4} Fundamental classes: [0] = {0, 2, 4}, [1] = {1, 3, 5}, and (Z6, +,⊗)/γ∗ ∼= (Z2, +, ·). Theorem 2.4. All multiplicative Hv-fields defined on (Z9, +, ·), which have non- degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: We have the only one hyperproduct, 2 ⊗ 3 = {0, 6} or {3, 6}, 2 ⊗ 4 = {2, 8} or {5, 8}, 2 ⊗ 6 = {0, 3} or {3, 6}, 2 ⊗ 7 = {2, 5} or {5, 8}, 2 ⊗ 8 = {1, 7} or {4, 7}, 3 ⊗ 4 = {0, 3} or {3, 6}, 3 ⊗ 5 = {0, 6} or {3, 6}, 3 ⊗ 6 = {0, 3} or {0, 6}, 3 ⊗ 7 = {0, 3} or {3, 6}, 3 ⊗ 8 = {0, 6} or {3, 6}, 4 ⊗ 5 = {2, 5} or {2, 8}, 4 ⊗ 6 = {0, 6} or {3, 6}, 4 ⊗ 8 = {2, 5} or {5, 8}, 5 ⊗ 6 = {0, 3} or {3, 6}, 5 ⊗ 7 = {2, 8} or {5, 8}, 5 ⊗ 8 = {1, 4} or {4, 7}, 6 ⊗ 7 = {0, 6} or {3, 6}, 6 ⊗ 8 = {0, 3} or {3, 6}, 7 ⊗ 8 = {2, 5} or {2, 8}, Fundamental classes: [0] = {0, 3, 6}, [1] = {1, 4, 7}, [2] = {2, 5, 8}, and (Z9, +,⊗)/γ∗ ∼= (Z3, +, ·). 190 Hv-Fields, h/v-Fields Theorem 2.5. All Hv-fields defined on (Z10, +, ·), which have non-degenerate fundamental field, and satisfy the above 4 conditions, are the following isomorphic cases: (I) We have the only one hyperproduct, 2⊗4 = {3, 8}, 2⊗5 = {2, 5}, 2⊗6 = {2, 7}, 2⊗7 = {4, 9}, 2⊗9 = {3, 8}, 3⊗4 = {2, 7}, 3⊗5 = {0, 5}, 3⊗6 = {3, 8}, 3⊗8 = {4, 9}, 3⊗9 = {2, 7}, 4⊗5 = {0, 5}, 4⊗6 = {4, 9}, 4⊗7 = {3, 8}, 4⊗8 = {2, 7}, 5⊗6 = {0, 5}, 5⊗7 = {0, 5}, 5⊗8 = {0, 5}, 5⊗9 = {0, 5}, 6⊗7 = {2, 7}, 6⊗8 = {3, 8}, 6 ⊗ 9 = {4, 9}, 7 ⊗ 9 = {3, 8}, 8 ⊗ 9 = {2, 7}. Fundamental classes: [0] = {0, 5}, [1] = {1, 6}, [2] = {2, 7}, [3] = {3, 8}, [4] = {4, 9} and (Z10, +,⊗)/γ∗ ∼= (Z5, +, ·). (II) The cases where we have two classes [0] = {0, 2, 4, 6, 8} and [1] = {1, 3, 5, 7, 9}, thus we have fundamental field (Z10, +,⊗)/γ∗ ∼= (Z2, +, ·), can be described as follows: Taking in the multiplicative table only the results above the diagonal, we enlarge each of the products by putting one element of the same class of the results. We do not enlarge setting the element 1, and we cannot enlarge only the product 3 ⊗ 7 = 1. The number of those Hv-fields is 103. Example 2.1. In order to see how hard is to realize the reproductivity of classes and the unit class and inverse class, we consider the above Hv-field (Z10, +,⊗) where we have 2 ⊗ 4 = {3, 8}. Then the Multiplicative Table of the hyperproduct is the following: ⊗ 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 3,8 0 2 4 6 8 3 0 3 6 9 2 5 8 1 4 7 4 0 4 8 2 6 0 4 8 2 6 5 0 5 0 5 0 5 0 5 0 5 6 0 6 2 8 4 0 6 2 8 4 7 0 7 4 1 8 5 2 9 6 3 8 0 8 6 4 2 0 8 6 4 2 9 0 9 8 7 6 5 4 3 2 1 On this table it is easy to see that the reproductivity is not valid but it is very hard to see that the reproductivity of classes is valid. We can see the reproductivity of classes easier if we reformulate the Multiplicative Table according to the funda- mental classes, [0] = {0, 5}, [1] = {1, 6}, [2] = {2, 7}, [3] = {3, 8}, [4] = {4, 9}. Then we obtain: 191 Thomas Vougiouklis ⊗ 0 5 1 6 2 7 3 8 4 9 0 0 0 0 0 0 0 0 0 0 0 5 0 5 5 0 0 5 5 0 0 5 1 0 5 1 6 2 7 3 8 4 9 6 0 0 6 6 2 2 8 8 4 4 2 0 0 2 2 4 4 6 6 3,8 8 7 0 5 7 2 4 9 1 6 8 3 3 0 5 3 8 6 1 9 4 2 7 8 0 0 8 8 6 6 4 4 2 2 4 0 0 4 4 8 8 2 2 6 6 9 0 5 9 4 8 3 7 2 6 1 From this it is easy to see the unit class and the inverse class of each class. 3 The h/v-representations and applications Hv-structures are used in Representation Theory of Hv-groups which can be achieved either by generalized permutations or by Hv-matrices [31], [32], [38], [39], [44], [46], [57], [58]. The representations by generalized permutations can be faced by translations [37]. Moreover in hyperstructure theory we can define hyperproduct on non-square ordinary matrices by using the so called helix hopes where we use all entries of them [65], [28], [29] and [13], [14], [66], [67]. Thus, we face the representations of the hyperstructures by non-square matrices as well. Hv-matrix (or h/v-matrix) is a matrix with entries of an Hv-ring or Hv-field (or h/v-ring or h/v-field). The hyperproduct of two Hv-matrices (aij) and = (bij), of type m × n and n × r respectively, is defined in the usual manner and it is a set of m × r Hv-matrices. The sum of products of elements of the Hv-ring is considered to be the n-ary circle hope on the hyperaddition. The hyperproduct of Hv-matrices is not necessarily WASS. The problem of the Hv-matrix (or h/v-group) representations is the following: Definition 3.1. Let (H, ·) be an Hv-group (or h/v-group). Find an Hv-ring (or h/v- ring) (R, +, ·), a set MR={(aij)|aij∈R} and a map T : H → MR : h 7→ T(h) such that T(h1h2) ∩T(h1)T(h2) 6= ∅,∀h1,h2 ∈ H. T is an Hv-matrix (or h/v matrix) representation. If T(h1h2) ⊂ T(h1)T(h2),∀h1,h2 ∈ H, then T is an inclusion representation. If T(h1h2) = T(h1)T(h2),∀h1,h2 ∈ H, then T is a good representation and an induced representation T * of the hypergroup algebra is obtained. If T is one to one and the good condition is valid then it is called faithful representation. 192 Hv-Fields, h/v-Fields The main theorem of the theory of representations is the following [31], [32], [38]: Theorem 3.1. A necessary condition in order to have an inclusion representation T of an h/v-group (H, ·) by n×n, h/v-matrices over the h/v-ring (R, +, ·) is the following: For all classes 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}} Thus, inclusion representation T : H → MR : a 7→ T(a) = (aij) induces an homomorphic T * of H/β* over R/γ* by setting T *(β*(a)) = [γ*(aij)],∀β*(a) ∈ H/β*, where γ*(aij)R/γ* is the ij entry of T *(β*(a)). T * is called fundamental induced representation of T . Let T a representation of an h/v-group H by h/v-matrices and trφ(T(x)) = γ∗(Txii) be the fundamental trace, then is called fundamental character, the map- ping XT : H → R/γ* : x 7→ XT (x) = trφ(T(x)) = trT∗(x) In representations of Hv-groups there are two difficulties: First to find an Hv-ring or an Hv-field and second, an appropriate set of Hv-matrices. Notice that the more interesting cases are for the small Hv-fields, where the results have one or few elements. Example 3.1. In the case of the Hv-field (Z6, +,⊗) where the only one hyper- product is 2 ⊗ 4 = {2, 5} we consider the 2 × 2 h/v-matrices of type i = E11 + iE12 + 4E22, where i = 0, 1, 2, 3, 4, 5, then an h/v-group is obtained and the multiplicative table of the hyperproduct of those Hv-matrices is given by ⊗ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 4 5 0 1 2 3 2 2 0,3 1,4 2,5 0,3 1,4 3 0 1 2 3 4 5 4 4 5 0 1 2 3 5 2 3 4 5 0 1 where the fundamental classes are (0) = {0, 3}, (1) = {1, 4}, (2) = {2, 5} and the fundamental group is isomorphic to (Z3, +). Remark that (Z6,⊗) is an h/v- group which is cyclic where the elements 2 and 4 are generators of period 4. Notice that the hope (⊗) is a hyperproduct of h/v-matrices although (0) stands for the unit matrix, this is so because the symbolism follows the entry 12. 193 Thomas Vougiouklis Example 3.2. Let us denote by Eij the matrix with 1 in the ij-entry and zero in the rest entries. Then take the following 2 × 2 upper triangular h/v-matrices on the above h/v-field (Z4, +,⊗) of the case that only 2 ⊗ 3 = {0, 2} is a hyperproduct: I = E11 +E22,a = E11 +E12 +E22,b = E11 + 2E12 +E22,c = E11 + 3E12 +E22, d = E11+3E22,e = E11+E12+3E22,f = E11+2E12+3E22,g = E11+3E12+3E22, then, we obtain the following multiplicative table for the set X={I,a,b,c,d,e,f,g} ⊗ I a b c d e f g I I a b c d e f g a a b c I g d e f b b c I a d,f e,g d,f e,g c c I a b e f g d d d e f g I a b c e e f g d c I a b f f g d e I,b a,c I,b a,c g g d e f a b c I The (X,⊗) is non-COW, Hv-group and we can see that the fundamental classes are the a = {a,c}, d = {d,f}, e = {e,g} and the fundamental group is isomor- phic to (Z2 × Z2, +). In this Hv-group there is only one unit and every element has a unique double inverse. Only f has one more right inverse element, the d, since f ⊗d = {I,b}. Remark that if we need h/v-fields where the elements have at most one inverse element, then we must exclude the case of 2 ⊗ 5 = {1, 4} from (I), and the case 3 ⊗ 5 = {1, 3} from (II). Last decades Hv-structures have applications in other branches of mathematics and in other sciences. These applications range from biomathematics -conchology, inheritance- and hadronic physics or on leptons to mention but a few. The hyper- structure theory is related to fuzzy theory; consequently, hyperstructures can be widely applicable in industry and production, too [2], [5], [11], [12], [23], [25], [43], [47], [59]. The Lie-Santilli theory on isotopies was born in 1970’s to solve Hadronic Me- chanics problems. Santilli proposed a ’lifting’of the n-dimensional trivial unit ma- trix 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, correspond into the hyperstructures were introduced by Santilli & Vougiouklis in 1996 [25] and they are called e-hyperfields, [12], [24], [52], [56], [61]. 194 Hv-Fields, h/v-Fields Definition 3.2. 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. Definition 3.3. A hyperstructure (F, +, ·), where (+) is an operation and (·) is 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 3.4. 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). (G,⊗) is an Hv-group, in fact it is an Hb-group which contains the (G, ·). (G,⊗) is an e-hypergroup. Moreover, if for each x,y such that xy = e, then (G,⊗) becomes a strong e- hypergroup The main e-construction gives an extremely large number of e-hopes. Example. Consider the quaternions Q = {1,−1, i,−i,j,−j,k,−k}, with i2 = j2 = −1, ij = −ji = k, and denote i = {i,−i},j = {j,−j},k = {k,−k}. We define a lot of hopes (∗) by enlarging few products. For example, (−1) ∗k = k,k∗i = j and i∗j = k. Then the hyperstructure (Q,∗) is a strong e-hypergroup. The Lie-Santilli admissibility on non-quare matrices [12], [14], [24], [26], [57], [61]: Construction 3.1. Let L = (Mm×n, +) be an Hv-vector space of m × n hyper- matrices 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 canonical map 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 → P(L) : [x,y]RS = xRty −yStx. Notice that [x,y]RS = xRty −yStx = {xrty −ystx|r ∈ R and s ∈ S} 195 Thomas Vougiouklis An application, which combines the ∂-structures and fuzzy theory, is to re- place in questionnaires the scale of Likert by the bar of Vougiouklis & Vougiouklis [19]: Definition 3.5. 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