Ratio Mathematica Volume 38, 2020, pp. 313-328 Minimal Hv-fields Thomas Vougiouklis* Abstract Hyperstructures have applications in mathematics and in other sciences, which range from biology, hadronic physics, leptons, linguistics, sociology, to mention but a few. For this, the largest class of the hyperstructures, the Hv-structures, is used. They satisfy the weak axioms where the non-empty intersection replaces equality. The fundamental relations connect, by quo- tients, the Hv-structures with the classical ones. Hv-numbers are elements of Hv-field, and they are used in representation theory. We focus on minimal Hv-fields. Keywords: hyperstructure, Hv-structure, hope, hypernumbers, iso-numbers. 2010 AMS subject classifications: 20N20,16Y99.1 1 Introduction The class of hyperstructures called Hv-structures introduced in 1990 [Vou- giouklis, 1991a], [Vougiouklis, 1994] by Vougiouklis, satisfy the weak axioms where the non-empty intersection replaces equality. Algebraic hyperstructure (H, ·) is a set H equipped with a hyperoperation (ab- breviated: 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 commuta- tivity: xy∩yx 6= ∅,∀x,y ∈ H. (H, ·) is 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. The quotient of a group by a subgroup is a hypergroup, Marty 1934. The quotient of a group by any partition (equivalence) is an Hv-group, Vougiouklis 1990. *1Emeritus Professor (Democritus University of Thrace, Neapoli 14-6, Xanthi 67100, Greece; tvougiou@eled.duth.gr. 1Received on June 3rd, 2020. Accepted on June 23rd, 2020. Published on June 30th, 2020. doi: 10.23755/rm.v38i0.522. ISSN: 1592-7415. eISSN: 2282-8214. ©T. Vougiouklis This paper is published under the CC-BY licence agreement. 313 Thomas Vougiouklis In an Hv-semigroup the powers are: 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 (H, ·) is called cyclic of period s, if there exists an element h, called generator, and the minimum s, such that H = h1 ∪h2...∪hs. Analogously the cyclicity for the infinite period is defined. If thereare h and s, the minimum one, such that H = hs, then (H, ·) is a single-power cyclic of period s. (R,+, ·) is called Hv-ring if (+) and (·) are WASS, the reproduction 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. Let (R,+, ·) be an Hv-ring,a COW Hv-group (M,+) is called Hv-module over R, if there is 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= ∅, For more definitions and applications on Hv-structures one can see in books and papers as [Corsini, 1993],[Corsini and Leoreanu, 2003],[Davvaz, 2003],[Davvaz and Leoreanu, 2007],[Davvaz and Vougiouklis, 2018],[Vougiouklis, 1994],[Vou- giouklis, 1995],[Vougiouklis, 1999b]. Let (H, ·),(H,∗) Hv-semigroups, the hope (·) is smaller than (∗), and (∗) greater than (·), iff there exists an automorphism f ∈ Aut(H,∗) such that xy ⊂ f(x∗y), ∀x,y ∈ H. We write · ≤ ∗ and say that (H,∗) contains (H, ·). If (H, ·) is a structure then it is basic structure and (H,∗) is Hb −structure. Minimal is called an Hv-group which contains no other Hv-group defined on the same set. We extend this definition to any Hv-structures with any more properties. Theorem 1.1. (The Little Theorem). Greater hopes than the ones which are WASS or COW, are also WASS or COW, respectively. The little theorem leads to a partial order on Hv-structures and to posets. Let (H, ·) be hypergroupoid. We remove h ∈ H, if we take the restriction of (·) in H −{h}. h ∈ H absorbs h ∈ H if we replace h by h. 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 a class with representative h. 314 Minimal Hv-fields M. Koskas in 1970, introduced in hypergroups the relation β∗, which con- nects hypergroups with groups and it is defined in Hv-groups as well. Vougiouk- lis [Vougiouklis, 1985], [Vougiouklis, 1988], [Vougiouklis, 1991a], [Vougiouklis, 1994], [Vougiouklis, 1995], [Vougiouklis, 2016] introduced the γ* and �* rela- tions, which are defined, in Hv-rings and Hv-vector spaces, respectively. He also named all these relations, fundamental. Definition 1.1. The fundamental relations β*, γ* and �*, are defined, in Hv- groups, Hv-rings and Hv-vector space, respectively, as the smallest equivalences so that the quotient would be group, ring and vector spaces, respectively. Remark: Let (G, ·) be group and R a partition in G, then (G/R, ·) is an Hv- group, therefore the quotient (G/R, ·)/β* is a group, the fundamental one. Theorem 1.2. Let (H, ·) be an Hv-group and denote by U the set of all finite products of elements of H. Define the relation β in H by: xβy iff {x,y} ⊂ u where u ∈ U. Then β* is the transitive closure of β. Analogous theorems are for Hv-rings, Hv-vector spaces and so on [Vougiouk- lis, 1994]. Theorem 1.3. Let (R,+, ·) be Hv-ring. Denote U all finite polynomials of ele- ments of R. Define the relation γ in R by: xγy iff {x,y}⊂ u where u ∈ U. Then the relation γ* is the transitive closure of the relation γ. Proof. Let γ be the transitive closure of γ, and denote by γ(a) the class of a. First, we prove that the quotient set R/γ is a ring. In R/γ the sum (⊕) and the product (⊗) are defined in the usual manner: γ(a)⊕γ(b) = {γ(c) : c ∈ γ(a) + γ(b)}, γ∗(a)⊗γ(b) = {γ(d) : d ∈ γ∗(a) ·γ(b)}, ∀a,b ∈ R. Take a′ ∈ γ(a), b′ ∈ γ(b). Then we have a′γa iff ∃x1, ...,xm+1 with x1 = a′,xm+1 = a and u1, ...,um ∈ U such that {xi,xi+1}⊂ ui, i = 1, ...,m, and b′γb iff ∃y1, ...,yn+1 with y1 = b′,yn+1 = b and v1, ...,vn ∈ U such that {yj,yj+1}⊂ vj, i = 1, ...,n. 315 Thomas Vougiouklis From the above we obtain {xi,xi+1}+ y1 ⊂ ui + v1, i = 1, ...,m−1, xm+1 +{yj,yj+1}⊂ um + vj, j = 1, ...,n. The sums ui + v1 = ti, i = 1, ...m−1 and um + vj = tim+j−1, j = 1, ...,n are also polynomials, thus, tk ∈ U∀k ∈{1, ...,m + n−1}. Now, pick up z1, ...,zm+n such that zi ∈ xi + y1, i = 1, ...,n and zm+j ∈ xm+1 + yj+1, j = 1, ...,n, therefore, using the above relations we obtain {zk,zk+1}⊂ tk, k = 1, ...,m+n− 1. Thus, every z1 ∈ x1 + y1 = a′ + b′ is γ equivalent to every zm+n ∈ xm+1 + yn+1 = a + b. So γ(a)⊕γ(b) is a singleton so we can write γ(a)⊕γ(b) = γ(c),∀c ∈ γ(a) + γ(b) In a similar way we prove that γ(a)⊗γ(b) = γ(d),∀d ∈ γ(a) ·γ(b) The WASS and the weak distributivity on R guarantee that associativity and dis- tributivity are valid for R/γ*. Therefore R/γ* is a ring. Let σ be an equivalence relation in R such that R/σ is a ring and σ(a) the class of a. Then σ(a)⊕σ(b) and σ(a)⊗σ(b) are singletons ∀a,b ∈ R, i.e. σ(a)⊕σ(b) = σ(c),∀c ∈ σ(a) + σ(b), σ(a)⊗σ(b) = σ(d),∀d ∈ σ(a) ·σ(b). Therefore we write, for every a,b ∈ R and A ⊂ σ(a), B ⊂ σ(b), σ(a)⊕σ(b) = σ(a + b) = σ(A + B), σ(a)⊗σ(b) = σ(ab) = σ(A ·B) By induction, we extend these relations on finite sums and products. Thus, ∀u ∈ U, we have the relation σ(x) = σ(u) ∀x ∈ u. Consequently x ∈ γ(a) => x ∈ σ(a),∀x ∈ R. But σ is transitively closed, so we obtain: x ∈ γ(x) => x ∈ σ(a). That γ is the smallest equivalence relation in R such that R/γ is a ring, i.e. γ = γ*. An element is called single if its fundamental class is singleton [Vougiouklis, 1994]. General structures can be defined using fundamental structures. From 1990 there is the following [Vougiouklis, 1991a], [Vougiouklis, 1994]: 316 Minimal Hv-fields Definition 1.2. An Hv-ring (R,+, ·) is called Hv-field if R/γ* is a field. An Hv-module over an Hv-field F, it is called Hv-vector space. The analogous to Theorem 1.3, on Hv-vector spaces, can be proved: Theorem 1.4. Let (V,+) be Hv-vector space over the Hv-field F. Denote U the set of all expressions of finite hopes either on F and V or the external hope ap- plied on finite sets of elements of F and V. Define the relation � in V as follows: x�y iff {x,y}⊂ u where u ∈ U. Then �* is the transitive closure of the relation �. Definition 1.3. Let (L,+) be Hv-vector space over the Hv-field F, φ : F → F/γ* canonical; ωF = {x ∈ F : φ(x) = 0}, the core, 0 is the zero of F/γ. Let ωL be the core of φ′ : L → L/�* and denote by 0 the zero of L/�*, as well. Take 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,z ∈ L Definition 1.4. The Hv-semigroup (H·) is h/v-group if H/β* is a group [Vou- giouklis, 2003]. The h/v-group is a generalization of Hv-group where a reproductivity of classes is valid: if σ(x), ∀x ∈ H equivalence classes then xσ(y) = σ(xy) = σ(x)y,∀x,y ∈ H. Similarly h/v-rings, h/v-fields, h/v-vector spaces etc, are defined. The uniting elements method introduced by Corsini & Vougiouklis in 1989, is the following [Corsini and Vougiouklis, 1989]: Let G be a structure and a not valid property d, described by a set of equations. Take the partition in G for which put in the same class, all pairs of elements that causes the non-validity of d. The quotient by this partition G/d is an Hv-structure. Then, the quotient out G/d by β*, is a stricter structure (G/d)β* for which d is valid. Theorem 1.5. Let (R,+, ·) be a ring, and F = {f1, ...,fm,fm+1, ...,fm+n} be a system of equations on R consisting of subsystems Fm = {f1, ...,fm} and Fn = {fm+1, ...,fm+n}. Let σ, σm be the equivalence relations defined by the uniting elements procedure using F and Fm respectively, and σn the equivalence defined on Fn on the ring Rm = (R/σm)/γ*. Then (R/σ)/γ* ∼= (Rm/σn)/γ* 317 Thomas Vougiouklis 2 Large classes of hopes A class of Hv-structures, introduced in [Vougiouklis, 1991b], [Vougiouklis, 1994], [Vougiouklis, 2014b], is the following: Definition 2.1. An Hv-structure is called very thin if there exists a pair (a,b) ∈ H2 for which ab = A, with cardA > 1, and all the other products are singletons. From the very thin hopes the Attach Construction is obtained [Vougiouklis, 1999a], [Vougiouklis, 2014b], [Vougiouklis, 2017]: Let (H, ·) be an Hv-semigroup and v /∈ H. We extend the hope (·) 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. Let (H, ·) Hv-semigroup, and [x] the fundamental class of ∀x ∈ H. Unit class is [e] if ([e] · [x])∩ [x] 6= ∅ and ([x] · [e])∩ [x] 6= ∅,∀x ∈ H, and ∀x ∈ H, we call inverse class of [x], the class [x]−1, if ([x] · [x]−1)∩ [e] 6= ∅ and ([x]−1 · [x])∩ [e] 6= ∅. Enlarged hopes are the ones where a new element appears in one result. The useful cases are those h/v-structures with the same fundamental structure. Construction 2.1. (a) Let (H, ·) be an Hv-semigroup, v /∈ H. We extend (·) into H = H ∪{v} by: 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 single. Scalars and units of (H, ·) are scalars and units in (H, ·). If (H, ·) is COW then (H, ·) is COW. (b) (H, ·) Hv-semigroup, v /∈ H, (H, ·) its attached h/v-group. Take 0 /∈ H and define in H◦ = 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 318 Minimal Hv-fields Then (Ho,+, ·) is h/v-field with (Ho,+, ·)/γ*∼= Z3. (+) is associative, (·) is WASS and weak distributive to (+). 0 is zero absorbing in (+). (Ho,+, ·) is the attached h/v-field of (H, ·). Definition 2.2. [Vougiouklis, 2008], [Vougiouklis, 2016] Let (G, ·) be groupoid 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. If (·) is commutative then ∂ is commutative. If (·) is COW, then ∂ is COW. If (G, ·) is groupoid and f : G → P(G)−{∅} be multivalued map. We define the ∂-hope on G as follows x∂y = (f(x) ·y)∪ (x ·f(y)), ∀x,y ∈ G . Motivation for the ∂-hope is the derivative where only the product of functions is used. Basic property: if (G, ·) is semigroup then ∀f, the ∂-hope is WASS. Examples (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. Combining the uniting elements procedure with the enlarging theory or the ∂-theory, we can obtain analogous results [Vougiouklis, 1999a], [Vougiouklis, 2014b], [Vougiouklis, 2017]. Theorem 2.1. In the ring (Zn,+, ·), with n = ms we enlarge the multiplication only in the product of the 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 are scalars. 319 Thomas Vougiouklis Corolary 2.1. 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 results remain the same. Then (Zn,+,⊕) is a very thin Hv-field. Now we focus on Very Thin minimal Hv-fields obtained by a classical field. Theorem 2.2. In a field (F,+, ·), we enlarge only in the product of the special elements a and b, by setting a ⊗ b = {ab,c}, where c 6= ab, and the rest results remain the same. Then we obtain the degenerate, minimal very thin, Hv-field (F,+,⊗)/γ∗ ∼= {0} Thus, there is no non-degenerate Hv-field obtained by a field by enlarging any product. Proof. Take any x ∈ F −{0}, then from a⊗ b = {ab,c} we obtain (a⊗ b)−ab = {0,c−ab} and then (x(c−ab)−1)⊗ ((a⊗ b)−ab) = {0,x} thus, 0γx,x ∈ F −{0}. Which means that every x is in the same fundamental class with the element 0. Thus, (F,+,⊗)/γ∗ ∼= {0}. Theorem 2.3. In a field (F,+, ·), we enlarge only in the sum of the special el- ements a and b, by setting a ⊕ b = {a + b,c}, where c 6= a + b, and the rest results remain the same. Then we obtain the degenerate, minimal very thin, Hv- field (F,+,⊕)/γ∗ ∼= {0}. Thus, there is no non-degenerate Hv-field obtained by a field by enlarging any sum. Proof. Take any x ∈ F −{0}, then from a⊕ b = {a + b,c} we obtain (a⊕b)−a+b = {0,c−a+b} and then (x(c−a+b)−1)·((a⊕b)−a+b) = {0,x} thus, 0γx,x ∈ F −{0}. Which means that every x is in the same fundamental class with the element 0. Thus, (F,+,⊕)/γ∗ ∼= {0}. The above two theorems state that there is no non-degenerate Hv-field ob- tained by a field by enlarging any sum or product. Hopes defined on classical structures are the following [Corsini, 1993], [Corsini and Leoreanu, 2003], [Vougiouklis, 1987], [Vougiouklis, 1994] : 320 Minimal Hv-fields 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. If (G, ·) is semi- group, then xPy = (xP)y ∪x(Py) = xPy and (G,P) is a semihypergroup. 3 Representations and Applications Hv-structures used in Representation (abbr. rep) Theory of Hv-groups can be achieved by generalized permutations [Vougiouklis, 1992] or by Hv-matrices [Vougiouklis, 1985], [Vougiouklis, 1994], [Vougiouklis, 1999b]. Hv-matrix is called a matrix if has entries from an Hv-ring. The hyperproduct of 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 the n-ary circle hope on the hyper-sum. The problem of the Hv-matrix (or h/v-group) reps is the following: Definition 3.1. Let (H, ·) be Hv-group. Find an Hv-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 rep. If T(h1h2) ⊂ T(h1)T(h2),∀h1,h2 ∈ H, then T is an inclusion rep. If T(h1h2) = T(h1)T(h2),∀h1,h2 ∈ H, then T is a good rep. If T is a good rep and one to one then it is a faithful rep. The rep problem is simplified in cases such as if the h/v-rings have scalars 0 and 1. The main theorem of the theory of reps is the following: Theorem 3.1. A necessary condition in order to have an inclusion rep T of an h/v-group (H, ·) by n×n h/v-matrices over the h/v-ring (R,+, ·) is the following: ∀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}} The inclusion rep T : H → MR : a 7→ T(a) = (aij) induces an homomorphic T * of H/β* on R/γ* by 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 rep of T . 321 Thomas Vougiouklis In reps we need small Hv-fields with results of few elements. An important hope on non-square matrices is defined [Vougiouklis, 2009], [Vougiouklis and Vougiouklis, 2005]: Definition 3.2. Let A = (aij) ∈ Mm×n and s,t ∈ N, such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. Define a mod-like map st from Mm×n to Ms×t by corresponding to A the matrix Ast = (aij) with entries the sets aij = {ai+κs,j+λt|1 ≤ i ≤ s,1 ≤ j ≤ t and κ,λ ∈ N,i + κs ≤ m,j + λt ≤ n}. The map st : Mm×nMs×t : A → Ast(aij), is called helix-projection of type st. Ast is a set of s × t-matrices X = (xij) such that xij ∈ aij,∀i,j. Obviously Amn = A. LetA = (aij) ∈ Mm×n and s,t ∈ N, 1 ≤ s ≤ m, 1 ≤ t ≤ n. We apply the helix-projection first on the columns and then on the rows and the result is the same: (Asn)st = (Amt)st = Ast. Definition 3.3. Let A = (aij) ∈ Mm×n and B = (bij) ∈ Mu×v be matrices. Denote s=min(m,u), t=min(n,u), then we define the helix-sum by ⊕ : Mm×nMu×vP(Ms×t) : (A,B) → A⊕B = Ast + Bst = (aij) + (bij) ⊂ Ms×t, where (aij) + (bij) = {(cij) = (aij + bij)|aij ∈ aij and bij ∈ bij}. Denote s=min(n,u), then we define the helix-product by ⊗ : Mm×nMu×vP(Ms×t) : (A,B) → A⊗B = Ams + Bsv = (aij) + (bij) ⊂ Mm×v, where (aij) · (bij) = {(cij) = ∑ (aij + bij)|aij ∈ aij and bij ∈ bij}.. Remark. The definition of the Lie-bracket is immediate, therefore the helix- Lie Algebra is defined, as well. Last decades Hv-structures have applications in mathematics and in other sci- ences. Applications range from biology and hadronic physics or leptons to men- tion but a few. The hyperstructure theory is related to fuzzy one; consequently, can be widely applicable in industry and production, too [Corsini and Leore- anu, 2003], [Davvaz and Leoreanu, 2007], [Davvaz et al., 2015], [Davvaz and Vougiouklis, 2018], [Santilli and Vougiouklis, 1996], [Vougiouklis, 2014a], [Vou- giouklis, 2020], [Vougiouklis and Kambaki-Vougioukli, 2013]. An application, which combines Hv-hyperstructures and fuzzy theory, is to re- place in questionnaires the scale of Likert by the bar of Vougiouklis & Vougiouklis (V & V bar) [Vougiouklis and Kambaki-Vougioukli, 2013]. They suggest the fol- lowing: 322 Minimal Hv-fields Definition 3.4. In every question substitute the Likert scale with ’the bar’ whose poles are defined with ’0’ on the left end, and ’1’ on the right end: 0 1 The subjects/participants are asked instead of deciding and checking a specific grade on the scale, to cut the bar at any point s/he feels expresses her/his answer to the specific question. The use of V & V bar bar instead of a Likert scale has several advantages during both the filling-in and the research processing. The final suggested length of the bar, according to the Golden Ratio, is 6.2cm. The Lie-Santilli theory on isotopies was born to solve Hadronic Mechanics problems. Santilli proposed a ’lifting’ of the n-dimensional trivial unit matrix into an appropriate new matrix. The original theory is reconstructed to admit the new matrix as left and right unit. The isofields needed in this theory correspond into the hyperstructures called e-hyperfields, introduced by [Santilli and Vougiouklis, 1996, Davvaz et al., 2015]. Definition 3.5. 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. Definition 3.6. 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 exist a multiplicative scalar unit 1, i.e. 1 · x = x · 1 = x,∀x ∈ F , and for all 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 3.7. 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,⊗) becomes an Hv-group, actually is an Hb-group which contains the (G, ·). The Hv-group (G,⊗) is an e-hypergroup. 323 Thomas Vougiouklis 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 (Q,∗) is strong e-hypergroup. A generalization of P-hopes used in Santilli’s isotheory, is [Davvaz et al., 2015], [Vougiouklis, 2016] : Let (G, ·) be abelian group, P ⊂ 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 and y = e we call this hope Pe-hope. The hyperstructure (G,×p) is abelian Hv-group. 4 Small hypernumbers. Minimal h/v-fields The small non-degenerate h/v-fields on (Zn,+, ·) in iso-theory, satisfy the fol- lowing: 1. very thin minimal, 2. COW (non-commutative), 3. they have the elements 0 and 1, scalars, 4. if an element has inverse element, this is unique. Therefore, we cannot enlarge the result if it is 1 and we cannot put 1 in enlarge- ment. Theorem 4.1. [Vougiouklis, 2017] All multiplicative h/v-fields defined on (Z4,+, ·), with non-degenerate fundamental field, satisfying 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}. Fundamental classes: [0]=0,2, [1]=1,3 and we have (Z4,+,⊗)/γ∗ ∼= (Z2,+, ·) Example. Denote Eij the matrix with 1 in the ij-entry and zero in the rest en- tries. Take the 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, 324 Minimal Hv-fields then, we obtain for X={I,a,b,c,d,e,f,g}, that (X,⊗) is non-COW, Hv-group where the fundamental classes are a = {a,c},d = {d,f},e = {e,g} and the fundamen- tal group is isomorphic to (Z2 ×Z2,+). There is only one unit and every element has unique double inverse. Only f has one more right inverse element d, since f ⊗d = {I,b}. (X,⊗) is not cyclic. Theorem 4.2. All multiplicative h/v-fields on (Z6,+, ·), with non-degenerate fun- damental field, satisfying the above 4 conditions, are the following isomorphic cases: We have the only one hyperproduct, (I) 2⊗3 = {0,3},2⊗4 = {2,5},3⊗4 = {0,3},3⊗5 = {0,3},4⊗5 = {2,5}. The fundamental classes are [0]=0,3, [1]=1,4, [2]=2,5 and we have (Z6,+,⊗)/γ∗ ∼= (Z3,+, ·). (II) 2⊗3 = {0,2} or 2⊗3 = {0,4}, 2⊗4 = {0,2} or {2,4},2⊗5 = {0,4} or 2⊗5 = {2,4}, 3⊗4 = {0,2} or {0,4},3⊗5 = {3,5},4⊗5 = {0,2} or {2,4}. In all these cases the fundamental classes are [0]=0,2,4, [1]=1,3,5 and we have (Z6,+,⊗)/γ∗ ∼= (Z2,+, ·). Example. In the h/v-field (Z6,+,⊗) where only the hyperproduct is 2 ⊗ 4 = {2,5} take the h/v-matrices of type i = E11 +iE12 + 4E22, where i=0,1,...,5, then the multiplicative table of the hyperproduct of those h/v-matrices is ⊗ 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 The fundamental classes are (0) = 0,3,(1) = 1,4,(2) = 2,5 and the funda- mental group is isomorphic to (Z3,+). The (Z6,⊗) is h/v-group which is cyclic where 2 and 4 are generators of period 4. Example. Consider the h/v-field (Z10,+,⊗) where only 3 × 8 = {4,9} is a hyperproduct. Let us take the h/v-matrix A = 3E11 + E22 + 2E33 + 6E12 + 2E13 + 9E23 Then from the above formulas we obtain that the set of inverse h/v-matrices is A−1 = [2]E11 + [1]E22 + [3]E33 + [3]E12 + [2]E13 + [3]E23 325 Thomas Vougiouklis So, for example, if we take the h/v-matrix A−1 = 7E11 + 6E22 + 8E33 + 8E12 + 2E13 + 3E23 we obtain that A ·A−1 = E11 + E22 + E33 +{0,5}E12 + 5E23 therefore, it contains a unit h/v-matrix. Theorem 4.3. All multiplicative h/v-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. In all the above cases the fundamental classes are [0] = {0,3,6}, [1] = {1,4,7}, [2] = {2,5,8},andwehave(Z9,+,⊗)/γ∗ ∼= (Z3,+, ·). Theorem 4.4. All multiplicative h/v-fields on (Z10,+, ·), which have non-degenerate fundamental field, and satisfy the above 4 conditions, are the following isomor- phic cases: (I) We have the only one hyperproduct, 2⊗4 = {3,8},2⊗5 = {0,5},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}. In all the above cases the fundamental classes are [0] = {0,3,6}, [1] = {1,4,7}, [2] = {2,5,8}, and we have (Z9,+,⊗)/γ∗ ∼= (Z3,+, ·). (II) The cases with classes [0] = {0,2,4,6,8} and [1] = {1,3,5,7,9}, and fundamental field (Z10,+,⊗)/γ∗ ∼= (Z2,+, ·). are described as follows: 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 1, and we cannot enlarge only the 3⊗7 = 1. The number of those h/v-fields is 103. 326 Minimal Hv-fields References P. Corsini. Prolegomena of Hypergroup Theory. Aviani Editore, 1993. P. Corsini and V. Leoreanu. Application of Hyperstructure Theory. Klower Acad. Publ, 2003. P. Corsini and T. Vougiouklis. From groupoids to groups through hypergroups. Klower Rendiconti Mat. S., VII, 9,:173–181, 1989. B. Davvaz. A brief survey of the theory of hv-structures. 8th AHA, Greece,, pages 39–70, 2003. B. Davvaz and V. Leoreanu. Hyperring Theory and Applications. Int. Academic Press,, 2007. B. Davvaz and T. Vougiouklis. A Walk Through Weak Hyperstructures,Hv- Structures. World Scientific, 2018. B. Davvaz, R.M. Santilli, and T. Vougiouklis. Algebra, Hyperalgebra and Lie- Santilli Theory. J. Generalized Lie Theory and Appl., 9:2:1–5, 2015. R.M. Santilli and T. Vougiouklis. Isotopies, genotopies, hyperstructures and their applications. New Frontiers Hyperstr. Related Algebras, Hadronic, pages 177– 188, 1996. S. Vougiouklis. hv-vector spaces from helix hyperoperations. Int. J. Math. Anal. (New Series), 1(2):109–120, 2009. T. Vougiouklis. Representations of hypergroups, hypergroup algebra. Proc. Con- vegno: ipergrouppi, altre strutture multivoche appl. Udine, pages 59–73, 1985. T. Vougiouklis. Generalization of p-hypergroups. mRend. Circ. Mat. Palermo, S.II, 36:114–121, 1987. T. Vougiouklis. Groups in hypergroups. Annals Discrete Math., 37:459–468, 1988. T. Vougiouklis. The fundamental relation in hyperrings. the general hyperfield. 4th AHA, Xanthi 1990, World Scientific, pages 203–211, 1991a. T. Vougiouklis. The very thin hypergroups and the s-construction. Combinatorics ’88, Incidence Geom. Comb. Str., 2:471–477, 1991b. T. Vougiouklis. Representations of hypergroups by generalized permutations. Al- gebra Universalis, 29:172–183, 1992. 327 Thomas Vougiouklis T. Vougiouklis. Hyperstructures and their Representations. Monographs in Math., Hadronic,, 1994. T. Vougiouklis. Some remarks on hyperstructures. Contemporary Math., Amer. Math. Society, 184:427–431, 1995. T. Vougiouklis. Enlarging hv-structures. Algebras and Combinatorics, ICAC’97, Hong Kong,Springer Verlag, 184:455–463, 1999a. T. Vougiouklis. On hv-rings and hv-representations. Discrete Math., Elsevier, 208/209:615–620, 1999b. T. Vougiouklis. The h/v-structures. J. Discrete Math. Sciences and Cryptography, V.6,N.2-3:235–243, 2003. T. Vougiouklis. ∂-operations and hv-fields. Acta Math. Sinica, (Engl. Ser.), V.24, N.7:1067–1078, 2008. T. Vougiouklis. From hv-rings to hv-fields. Int. J. Algebraic Hyperstructures Appl., Vol.1, No.1:1–13, 2014a. T. Vougiouklis. Enlarged fundamentally very thin hv-structures. J. Algebraic Str. and Their Appl. (ASTA), Vol.1, No1:11–20, 2014b. T. Vougiouklis. On the hyperstructure theory. Southeast Asian Bull. Math., Vol. 40(4):603–620, 2016. T. Vougiouklis. hv-fields, h/v-fields. Ratio Mathematica, V.33:181–201, 2017. T. Vougiouklis. Fundamental relations in hv-structures. the ’judging from the results’ proof. J. Algebraic Hyperstrucures Logical Algebras, V.1, N.1:21–36, 2020. T. Vougiouklis and P. Kambaki-Vougioukli. Bar in questionnaires. Chinese Busi- ness Review, V.12, N.10:691–697, 2013. T. Vougiouklis and S. Vougiouklis. The helix hyperoperations. Italian J. Pure Appl. Math., 18:197–206, 2005. 328