Ratio Mathematica ISSN: 1592-7415 Vol. 31, 2016, pp. 65-78 eISSN: 2282-8214 65 Helix-Hopes on Finite Hyperfields Thomas Vougiouklis1, Souzana Vougiouklis2 1 Emeritus Professor, Democritus University of Thrace, Alexandroupolis, Greece tvougiou@eled.duth.gr 2 Researcher in Maths and Music, 17 Oikonomou, Exarheia, Athens 10683, Greece, elsouvou@gmail.com Received on: 03-12-2016. Accepted on: 14-01-2017. Published on: 28-02-2017 doi: 10.23755/rm.v31i0.321 © Thomas Vougiouklis and Souzana Vougiouklis Abstract Hyperstructure theory can overcome restrictions which ordinary algebraic structures have. A hyperproduct on non-square ordinary matrices can be defined by using the so called helix-hyperoperations. We study the helix- hyperstructures on the representations using ordinary fields. The related theory can be faced by defining the hyperproduct on the set of non square matrices. The main tools of the Hyperstructure Theory are the fundamental relations which connect the largest class of hyperstructures, the Hv- structures, with the corresponding classical ones. We focus on finite dimensional helix-hyperstructures and on small Hv-fields, as well. Keywords: hyperstructures, Hv-structures, h/v-structures, hope. 2010 AMS subject classification: 20N20, 16Y99. Thomas Vougiouklis, Souzana Vougiouklis 66 1 Introduction We deal with the largest class of hyperstructures called Hv-structures introduced in 1990 [10], [11], which satisfy the weak axioms where the non- empty intersection replaces the equality. Definitions 1.1 In a set H equipped with a hyperoperation (which we abbreviate it by hope) ∙ : HHP (H)-{}: (x,y) x∙yH we abbreviate by WASS the weak associativity: (xy)zx(yz), x,y,zH and by COW the weak commutativity: xyyx, x,yH. The hyperstructure (H,) is called Hv-semigroup if it is WASS and is called Hv- group if it is reproductive Hv-semigroup: xH=Hx=H, xH. (R,+,) is called Hv-ring if (+) and () are WASS, the reproduction axiom is valid for (+) and () is weak distributive with respect to (+): x(y+z)(xy+xz), (x+y)z(xz+yz), x,y,zR. For more definitions, results and applications on Hv-structures, see books and the survey papers as [2], [3], [11], [1], [6], [15], [16], [20]. An extreme class is the following: An Hv-structure is very thin iff all hopes are operations except one, with all hyperproducts singletons except only one, which is a subset of cardinality more than one. Thus, in a very thin Hv-structure in a set H there exists a hope () and a pair (a,b)H2 for which ab=A, with cardA>1, and all the other products, with respect to any other hopes (so they are operations), are singletons. The fundamental relations β* and γ* are defined, in Hv-groups and Hv-rings, respectively, as the smallest equivalences so that the quotient would be group and ring, respectively [9], [10], [11], [12], [13]. The main theorem is the following: Theorem 1.2 Let (H,) be an Hv-group and let us denote by U the set of all finite products of elements of H. We define the relation β in H as follows: xβy iff {x,y}u where uU. Then the fundamental relation β* is the transitive closure of the relation β. An element is called single if its fundamental class is a singleton. Motivation for Hv-structures: The quotient of a group with respect to an invariant subgroup is a group. Marty states that, the quotient of a group by any subgroup is a hypergroup. Helix-Hopes on Finite Hyperfields 67 Now, the quotient of a group with respect to any partition is an Hv-group. Definition 1.3 Let (H,), (H,) be Hv-semigroups defined on the same H. () is smaller than (), and () greater than (), iff there exists automorphism fAut(H,) such that xyf(xy), xH. Then (H,) contains (H,) and write  . If (H,) is structure, then it is basic and (H,) is an Hb-structure. The Little Theorem [11]. Greater hopes of the ones which are WASS or COW, are also WASS and COW, respectively. Fundamental relations are used for general definitions of hyperstructures. Thus, to define the general Hv-field one uses the fundamental relation γ*: Definition 1.4 [10], [11]. The Hv-ring (R,+,) is called Hv-field if the quotient R/γ* is a field. Let ω* be the kernel of the canonical map from R to R/γ*; then we call reproductive Hv-field any Hv-field (R,+,) if x(R-ω*)=(R-ω*)x=R-ω*,xR-ω*. From this definition, a new class is introduced [15]: Definition 1.5 The Hv-semigroup (H,) is h/v-group if the H/β* is a group. Similarly h/v-rings, h/v-fields, h/v-modulus, h/v-vector spaces, are defined. The h/v-group is a generalization of the Hv-group since the reproductivity is not necessarily valid. Sometimes a kind of reproductivity of classes is valid, i.e. if H is partitioned into equivalence classes σ(x), then the quotient is reproductive xσ(y)=σ(xy)=σ(x)y, xH. An Hv-group is called cyclic [11], if there is element, called generator, which the powers have union the underline set, the minimal power with this property is the period of the generator. If there exists an element and a special power, the minimum, is the underline set, then the Hv-group is called single-power cyclic. Definitions 1.6 [11], [14]. Let (R,+,) be an Hv-ring, (M,+) be 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), (a+b)x(ax+bx), (ab)xa(bx), then M is called an Hv-module over R. In the case of an Hv-field F instead of Hv- ring R, then the Hv-vector space is defined. Definition 1.7 [17]. Let (L,+) be Hv-vector space on (F,+,), φ:FF/γ*, the canonical map and ωF={xF:φ(x)=0}, where 0 is the zero of the fundamental Thomas Vougiouklis, Souzana Vougiouklis 68 field F/γ*. Similarly, let ωL be the core of the canonical map φ: LL/ε* and denote again 0 the zero of L/ε*. 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: [λ1x1+λ2x2,y](λ1[x1,y]+λ2[x2,y]) [x,λ1y1+λ2y](λ1[x,y1]+λ2[x,y2]), x,x1,x2,y,y1,y2L and λ1,λ2F (L2) [x,x]ωL, xL (L3) ([x,[y,z]]+[y,[z,x]]+[z,[x,y]])ωL, x,yL Two well known and large classes of hopes are given as follows [11], [16]: Definitions 1.8 Let (G,) be a groupoid, then for every subset PG, P, we define the following hopes, called P-hopes: x,yG P: xPy = (xP)yx(Py), Pr: xPry= (xy)Px(yP), Pl: xPly= (Px)yP(xy). The (G,P), (G,Pr) and (G,Pl) are called P-hyperstructures. The usual case is for semigroup (G,), then xPy=(xP)yx(Py)=xPy and (G,P) is a semihypergroup but we do not know about (G,Pr) and (G,Pl). In some cases, depending on the choice of P, the (G,Pr) and (G,Pl) can be associative or WASS. A generalization of P-hopes: Let (G,) be abelian group and P a subset of G with more than one elements. We define the hope P as follows: xPy = {xhy hP} if xe and ye xPy = xy if x=e or y=e we call this hope, Pe-hope. The hyperstructure (G,P) is an abelian Hv-group. Definition 1.9 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)} ( resp. xy = (f(x)y)(xf(y) ), x,yG. If () is commutative then  is commutative. If () is COW, then  is COW. Helix-Hopes on Finite Hyperfields 69 If (G,) is groupoid (or hypergroupoid) and f:GP(G)-{} 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 map derivative where only the product of functions can be used. Basic property: if (G,) is semigroup then f, the -hope is WASS. 2 Some Applications of Hv-Structures 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 hyperstructure theory is closely related to fuzzy theory; consequently, hyperstructures can be widely applicable in industry and production, too [2], [3], [7], [18]. 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. The isofields needed correspond into the hyperstructures introduced by Santilli & Vougiouklis in 1999 [7] 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. Definition 2.1 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. Definition 2.2 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 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. If the relation: 1=xx-1=x-1x, is valid, then we say that we have a strong e-hyperfield. Definition 2.3 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}, where g1, g2,…G-{e} Thomas Vougiouklis, Souzana Vougiouklis 70 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 main e-construction gives an extremely large number of e-hopes. Example 2.4 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. Mathematicalisation of a problem could make its results recognizable and comparable. This is because representing a research object or a phenomenon with numbers, figures or graphs might be simplest and in a recognizable way of reading the results. In questionnaires Vougiouklis & Vougiouklis proposed the substitution of Likert scales with the bar [5], [18].This substitution makes things simpler and easier for both the subjects of an empirical research and the researcher, either at the stage of designing or that of results processing, because it is really flexible. Moreover, the application of the bar opens a window towards the use of fuzzy sets in the whole procedure of empirical research, activating in this way more recent findings from different sciences, as well. The bar is closelly related with hyperstructure and fuzzy theories, as well. More specifically, the following was proposed: In every question, substitute the Likert scale with the ‘bar’ whose poles are defined with ‘0’ on the left and ‘1’ on the right: 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 they feel best expresses their answer to the specific question. The suggested length of the bar is approximately 6.18cm, or 6.2cm, following the golden ration on the well known length of 10cm. 3 Small Hv-Numbers. Hv-Matrix Representations In representations important role are playing the small hypernumbers. Construction 3.1 On the ring (Z4,+,∙) we will define all the multiplicative h/v- fields which have non-degenerate fundamental field and, moreover they are, (a) very thin minimal, Helix-Hopes on Finite Hyperfields 71 (b) COW (non-commutative), (c) they have the elements 0 and 1, scalars. Then, we have only the following isomorphic cases 23={0,2} or 32={0,2}. Fundamental classes: [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. Only f has one more right inverse element, the d, since fd={I,b}. Moreover, the (X,) is not cyclic. Construction 3.2 On (Z6,+,∙) we define, up to isomorphism, all multiplicative h/v-fields which have non-degenerate fundamental field and, moreover they are: (a) very thin minimal (b) COW (non-commutative) (c) they have the elements 0 and 1, scalars Then we have the following cases, by giving the only one hyperproduct, (i) 23={0,3} or 24={2,5} or 25={1,4} 34={0,3} or 35={0,3} or 45={2,5} In all 6 cases 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} 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={1,3} or 35={3,5} or 45={0,2} or 45={2,4}. In all 12 cases the fundamental classes are [0]={0,2,4}, [1]={1,3,5} and we have (Z6,+,)/γ*  (Z2,+,∙). 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). Hv-structures are used in Representation Theory of Hv-groups which can be achieved by generalized permutations or by Hv-matrices [11], [12], [13], [14]. 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 3.3 Let (H,) be 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 T(h) such that Thomas Vougiouklis, Souzana Vougiouklis 72 T(h1h2) T(h1)T(h2)  , h1,h2H. T is Hv-matrix (or h/v-matrix) representation. If T(h1h2)T(h1)T(h2), h1,h2H, then T is called inclusion. If T(h1h2)=T(h1)T(h2)= {T(h)hh1h2}, h1,h2H, then T is good and then an induced representation T* for the hypergroup algebra is obtained. If T is one to one and good then it is faithful. The main theorem on representations is [13]: Theorem 3.4 A necessary condition 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 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}} Inclusion T:HMR:a T(a)=(aij) induces homomorphic representation T* of H/β* on R/γ* by setting T*(β*(a))=[γ*(aij)], β*(a)H/β*, where γ*(aij)R/γ* is the ij entry of the matrix T*(β*(a)). T* is called fundamental induced of T. In representations, several new classes are used: Definition 3.5 Let M=Mmn be the module of mn matrices over R and P={Pi:iI}M. We define a P-hope P on M as follows P: MM  P(M): (A,B)  APB={APtiB: iI } M where Pt denotes the transpose of P. The hope P is bilinear map, is strong associative and inclusion distributive: AP(B+C)  APB+APC, A,B,CM Definition 3.6 Let M=Mmn the mn matrices over R and let us take sets S={sk:kK}R, Q={Qi:jJ}M, P={Pi:iI}M. Define three hopes as follows S: RMP(M): (r,A)rSA = {(rsk)A: kK} M Q+: MMP(M): (A,B)AQ+B = {A+Qj+B: jJ} M P: MMP(M): (A,B)APB = {APtiB: iI} M Then (M,S,Q+,P) is hyperalgebra on R called general matrix P-hyperalgebra. Helix-Hopes on Finite Hyperfields 73 4 Helix-Hopes and Applications Recall some definitions from [19], [8], [20], [4]: Definition 4.1 Let A=(aij)Mmn be mn matrix and s,tN be naturals such that 1sm, 1tn. We define the map cst from Mmn to Mst by corresponding to the matrix A, the matrix Acst=(aij) where 1is, 1jt. We call this map cut- projection of type st. Thus Acst is matrix obtained from A by cutting the lines, with index greater than s, and columns, with index greater than t. We use cut-projections on all types of matrices to define sums and products. Definitions 4.2 Let A=(aij)Mmn be an mn matrix and s,tN, 1sm, 1tn. We define the mod-like map st from Mmn to Mst by corresponding to A the matrix Ast=(aij) which has as entries the sets aij = {ai+κs,j+λt 1is, 1jt. and κ,λN, i+κsm, j+λtn}. Thus we have the map st: Mmn  Mst: A  Ast = (aij). We call this multivalued map helix-projection of type st. Ast is a set of st- matrices X=(xij) such that xijaij, i,j. Obviously Amn=A. Let A=(aij)Mmn be a matrix and s,tN such that 1sm, 1tn. Then it is clear that we can apply the helix-projection first on the rows and then on the columns, the result is the same if we apply the helix-progection on both, rows and columns. Therefore we have (Asn)st = (Amt)st = Ast. Let A=(aij)Mmn be matrix and s,tN such that 1sm, 1tn. Then if Ast is not a set but one single matrix then we call A cut-helix matrix of type st. In other words the matrix A is a helix matrix of type st, if Acst= Ast. Definitions 4.3 (a) Let A=(aij)Mmn , B=(bij)Muv be matrices and s=min(m,u), t=min(n,u). We define a hope, called helix-addition or helix-sum, as follows: : MmnMuvP(Mst): (A,B)AB=Ast+Bst=(aij)+(bij) Mst, where (aij)+( bij)= {(cij)= (aij+bij) aijaij and bijbij}. (b) Let A=(aij)Mmn and B=(bij)Muv be matrices and s=min(n,u). We define a hope, called helix-multiplication or helix-product, as follows: : MmnMuvP(Mmv):(A,B)AB=AmsBsv=(aij)(bij)Mmv, Thomas Vougiouklis, Souzana Vougiouklis 74 where (aij)(bij)= {( cij)=(aitbtj) aijaij and bijbij}. The helix-sum is external hope since it is defined on different sets and the result is also in different set. The commutativity is valid in the helix-sum. For the helix-product we remark that we have AB=AmsBsv so we have either Ams=A or Bsv=B, that means that the helix-projection was applied only in one matrix and only in the rows or in the columns. If the appropriate matrices in the helix-sum and in the helix-product are cut-helix, then the result is singleton. Remark. In Mmn the addition is ordinary operation, thus we are interested only in the ‘product’. From the fact that the helix-product on non square matrices is defined, the definition of the Lie-bracket is immediate, therefore the helix-Lie Algebra is defined [17], as well. This algebra is an Hv-Lie Algebra where the fundamental relation ε* gives, by a quotient, a Lie algebra, from which a classification is obtained. In the following we restrict ourselves on the matrices Mmn where mn and for m=n we have the classical theory. Notation. For given κℕ-{0}, we denote by κ the remainder resulting from its division by m if the remainder is non zero, and κ=m if the remainder is zero. Thus a matrix A=(aκλ)Mmn, m