Ratio Mathematica Vol. 33, 2017, pp. 167-179 ISSN: 1592-7415 eISSN: 2282-8214 Helix-Hopes on S-Helix Matrices Souzana Vougioukli∗, Thomas Vougiouklis† ‡doi:10.23755/rm.v33i0.385 Abstract A hyperproduct on non-square ordinary matrices can be defined by using the so called helix-hyperoperations. The main characteristic of the helix- hyperoperation is that all entries of the matrices are used. Such operations cannot be defined in the classical theory. Several classes of non-square ma- trices have results of the helix-product with small cardinality. We study the helix-hyperstructures on the representations and we extend our study up to Hv-Lie theory by using ordinary fields. We introduce and study the class of S-helix matrices. Keywords: hyperstructures; Hv-structures; h/v-structures; hope; helix-hopes. 2010 AMS subject classifications: 20N20, 16Y99. ∗17 Oikonomou str, Exarheia, 10683 Athens, Greece; elsouvou@gmail.com †Democritus University of Thrace, School of Education, 68100 Alexandroupolis, Greece; tvou- giou@eled.duth.gr ‡ c©Souzana Vougioukli and Thomas Vougiouklis. Received: 31-10-2017. Accepted: 26-12- 2017. Published: 31-12-2017. 167 Souzana Vougioukli and Thomas Vougiouklis 1 Introduction Our object is the largest class of hyperstructures, the Hv-structures, introduced in 1990 [10], satisfying the weak axioms where the non-empty intersection re- places the equality. Definition 1.1. In a set H equipped with a hyperoperation (abbreviate by hope) · : H ×H → P(H)−{∅} : (x,y) → x ·y ⊂ 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 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) 6= ∅, (x + y)z ∩ (xz + yz) 6= ∅, ∀x,y,z ∈ R. For more definitions, results and applications on Hv-structures, see [1], [2], [11], [12], [13], [17]. An interesting class is the following [8]: An Hv-structure is very thin, if and only if, all hopes are operations except one, with all hyperprod- ucts singletons except only one, which is a subset of cardinality more than one. Therefore, 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, 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 [8], [9], [11], [12], [13], [17]. The main theorem is the following: Theorem 1.1. 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: The quotient of a group with respect to any partition is an Hv- group. Definition 1.2. 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,y ∈ H. 168 Helix-Hopes on S-Helix Matrices 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 1.3. [10] The Hv-ring (R,+, ·) is called Hv-field if the quotient R/γ* is a field. This definition introduces a new class of which is the following [15]: Definition 1.4. The Hv-semigroup (H, ·) is called h/v-group if H/β* is a group. The class of h/v-groups is more general than the Hv-groups since in h/v-groups the reproductivity is not valid. The h/v-fields and the other related hyperstructures are defined in a similar way. An Hv-group is called cyclic [8], if there is an element, called generator, which the powers have union the underline set, the minimal power with this prop- erty is the period of the generator. Definition 1.5. [11], [14], [18]. 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) 6= ∅, (a + b)x∩ (ax + bx) 6= ∅, (ab)x∩a(bx) 6= ∅, then M is called an Hv-module over R. In the case of an Hv-field F instead of an Hv-ring R, then the Hv-vector space is defined. Definition 1.6. [16] 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 field F/γ*. Similarly, let ωL be the core of the canonical map φ′ : L → L/�* and denote again 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 169 Souzana Vougioukli and Thomas Vougiouklis (L2) [x,x]∩ωL 6= ∅, ∀x ∈ L (L3) ([x, [y,z]] + [y, [z,x]] + [z, [x,y]])∩ωL 6= ∅, ∀x,y,z ∈ L A well known and large class of hopes is given as follows [8], [9], [11]: Definition 1.7. Let (G, ·) be a groupoid, then for every subset 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), (G,P l) are called P-hyperstructures. The usual case is for semigroup (G, ·), then xPy = (xP)y ∪x(Py) = xPy, and (G,P) is a semihypergroup. A new important application of Hv-structures in Nuclear Physics is in the Santilli’s isotheory. In this theory a generalization of P-hopes is used, [4], [5], [22], which is defined as follows: Let (G,) be an abelian group and P a subset of G with more than one elements. We define the hyperoperation ×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 an abelian Hv-group. 2 Small hypernumbers and Hv-matrix representa- tions Several constructions of Hv-fields are uses in representation theory and ap- plications in applied sciences. We present some of them in the finite small case [18]. Construction 2.1. 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, (b) COW (non-commutative), (c) they have the elements 0 and 1, scalars. 170 Helix-Hopes on S-Helix Matrices 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. Construction 2.2. On the ring (Z6,+, ·) we define, up to isomorphism, all mul- tiplicative h/v-fields which have non-degenerate fundamental field and, moreover they are: (a) very thin minimal, i.e. only one product has exactly two elements (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,+, ·). Hv-structures are used in Representation Theory of Hv-groups which can be achieved by generalized permutations or by Hv-matrices [11], [14], [18]. Definition 2.1. Hv-matrix is a matrix with entries of an Hv-ring or Hv-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 hypersum. The hyperproduct of Hv-matrices is not necessarily WASS. The problem of the Hv-matrix representations is the following: Definition 2.2. Let (H, ·) be Hv-group (or h/v-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. 171 Souzana Vougioukli and Thomas Vougiouklis T is Hv -matrix (or h/v-matrix) representation. If T(h1h2) ⊂ T(h1)(h2) is called inclusion. If T(h1h2) = T(h1)(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 [11]: Theorem 2.1. A necessary condition 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 classes β∗(x),x ∈ H must exist elements aij ∈ H,i,j ∈ {1, . . . ,n} such that T(β∗(a)) ⊂{A = (a′ij)|aij ∈ γ ∗(aij), i,j ∈{1, . . . ,n}} Inclusion T : H → MR : a 7→ 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 2.3. 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 = {AP ti B : i ∈ I}⊆ M where P t denotes the transpose of P. The hope P is bilinear map, is strong associative and the inclusion distributive: AP(B + C) ⊆ APB + APC,∀A,B,C ∈ M Definition 2.4. Let M = Mm×n the m×n matrices over R and let us take sets S = {sk : k ∈ K}⊆ R, Q = {Qj : 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 = {AP ti B : i ∈ I}⊆ M Then (M,S,Q + ,P) is hyperalgebra on R called general matrix P-hyperalgebra. 172 Helix-Hopes on S-Helix Matrices 3 Helix-hopes Recall some definitions from [3], [4], [6], [7], [19], [20], [21]: Definition 3.1. 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 = (aij) 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. Definition 3.2. Let A = (aij) ∈ Mm×n be an m × n matrix and s,t ∈ N, such that 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-projection 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. Definition 3.3. 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} 173 Souzana Vougioukli and Thomas Vougiouklis b. Let A = (aij) ∈ Mm×n and B = (bij) ∈ Mu×v, be matrices and s=min(m,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, where (aij) · (bij) = {(cij = ( ∑ aitbtj)|aij ∈ aij and bij ∈ bij} The helix-sum is an external hope and the commutativity is valid. 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 [22], as well. This algebra is an Hv-Lie Algebra where the fundamental relation �∗ gives, by a quotient, a Lie algebra, from which a classifi- cation is obtained. In the following we restrict ourselves on the matrices Mm×n where m < n. We have analogous results if m > n 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 < n is a cut-helix matrix if we have aκλ = aκλ,∀κλ ∈ N−{0}. Moreover let us denote by Ic = (cκλ) the cut-helix unit matrix which the cut matrix is the unit matrix Im. Therefore, since Im = (δκλ), where δκλ is the Kronecker’s delta, we obtain that, ∀κ,λ, we have cκλ = δκλ. Proposition 3.1. For m < n in (Mm×n,⊗) the cut-helix unit matrix Ic = (cκλ), where cκλ = δκλ, is a left scalar unit and a right unit. It is the only one left scalar unit. Proof. Let A,B ∈ Mm×n then in the helix-multiplication, since m < n, we take helix projection of the matrix A, therefore, the result A⊗B is singleton if the matrix A is a cut-helix matrix of type m × m. Moreover, in order to have A⊗B = Amm·B = B, the matrix Amm must be the unit matrix. Consequently, Ic = (cκλ), where cκλ = δκλ,∀κ,λ ∈ N−{0}, is necessarily the left scalar unit. Let A = (auv) ∈ Mm×n and consider the hyperproduct A ⊗ Ic. In the entry κλ of this hyperproduct there are sets, for all 1 ≤ κ ≤ m, 1 ≤ λ ≤ n , of the form∑ aκscsλ = ∑ aκsδsλ = aκλ 3 aκλ. 174 Helix-Hopes on S-Helix Matrices Therefore A⊗ Ic 3 A,∀A ∈ Mm×n. 2 4 The S-helix matrices Definition 4.1. Let A = (aij) ∈ Mm×n be matrix and s,t ∈ N such that 1 ≤ s ≤ m, 1 ≤ t ≤ n. Then if Ast is a set of upper triangular matrices with the same diagonal, then we call A an S-helix matrix of type s× t. Therefore, in an S-helix matrix A of type s× t, the Ast has on the diagonal entries which are not sets but elements. In the following, we restrict our study on the case of A = (aij) ∈ Mm×n with m < n. Remark. According to the cut-helix notation, we have, aκλ = aκλ = 0, for all κ > λ and aκλ = aκλ, for κ = λ. Proposition 4.1. The set of S-helix matrices A = (aij) ∈ Mm×n with m < n, is closed under the helix product. Moreover, it has a unit the cut-helix unit matrix Ic, which is left scalar. Proof. It is clear that the helix product of two S-helix matrices, X = (xij),Y = (aij) ∈ Mm×n,X ⊗ Y , contain matrices Z = (zij), which are up- per diagonals. Moreover, for every zii, the entry ii is singleton since it is product of only z(i+km),(i+km) = zii, entries. The unit is, from Proposition 3.1, the matrix Ic = Im×n, where we have Im×n = Imm = Im. 2 An example of hyper-matrix representation, seven dimensional, with helix- hope is the following: Example 4.1. Consider the special case of the matrices of the type 3 × 5 on the field of real or complex. Then we have X =  x11 x12 x13 x11 x150 x22 x23 0 x22 0 0 x33 0 0   and Y =  y11 y12 y13 y11 y150 y22 y23 0 y22 0 0 y33 0 0   X ⊗Y =  x11 {x12, x15} x130 x22 x23 0 0 x33   ·  y11 y12 y13 y11 y150 y22 y23 0 y22 0 0 y33 0 0   = ( x11y11 x11y12 + {x12,x15}y22 x11y13 + {x12,x15}y23 + x13y33 x11y11 x11y15 + {x12,x15}y22 0 x22y22 x22y23 + x23y33 0 x22y22 0 0 x33y33 0 0 ) Therefore the helix product is a set with cardinality up to 8. The unit of this type is Ic =  1 0 0 1 00 1 0 0 1 0 0 1 0 0   175 Souzana Vougioukli and Thomas Vougiouklis Definition 4.2. We call a matrix A = (aij) ∈ Mm×n an S0-helix matrix if it is an S-helix matrix where the condition a11a22 . . .amm 6= 0, is valid. Therefore, an S0-helix matrix has no zero elements on the diagonal and the set S0 is a subset of the set S of all S-helix matrices. We notice that this set is closed under the helix product not in addition. Therefore it is interesting only when the product is used not the addition. Proposition 4.2. The set of S0-helix matrices A = (aij) ∈ Mm×n with m < n, is closed under the helix product, it has a unit the cut-helix unit matrix Ic, which is left scalar and S0-helix matrices X have inverses X−1, i.e. Ic ∈ X ⊗ X−1 ∩ X−1 ⊗X. Proof. First it is clear that on the helix product of two S0-helix matrices, the diagonal has not any zero since there is no zero on each of them. Therefore, the helix product is closed. The entries in the diagonal are inverses in the Hv-field. In the rest entries we have to collect equations from those which correspond to each element of the entry-set. 2 Example 4.2. Consider the special case of the above Example 4.1, of the matrices of the type 3 × 5. Suppose we want to find the inverse matrix Y = X−1, of the matrix X. Then we have Ic ∈ X ⊗Y ∩Y ⊗X. Therefore, we obtain x11y11 = x22y22 = x33y33 = 1 x11y12 +{x12,x15}y22 3 0,x11y13 +{x12,x15}y23 + x13y33 3 0, x11y15 +{x12,x15}y22 3 0,x23y22 + x33y23 3 0, Therefore a solution is y11 = 1 x11 ,y22 = 1 x22 ,y33 = 1 x33 y23 = −x23 x22x33 ,y12 = −x12 x11x22 ,y15 = −x15 x11x22 , and y13 = −x13 x11x33 + x23x12 x11x22x33 or y13 = −x13 x11x33 + x23x14 x11x22x33 Thus, a left and right inverse matrix of X is X−1 =   1x11 −x12x11x22 −x13x11x33 + x23x12x11x22x33 1x11 −x15x110 1 x22 −x23 x22x33 0 1 x22 0 0 1 x33 0 0   An interesting research field is the finite case on small finite Hv-fields. Impor- tant cases appear taking the generating sets by any S0-helix matrix. 176 Helix-Hopes on S-Helix Matrices Example 4.3. On the type 3 × 5 of matrices using the Construction 2.1, on (Z4,+, ·) we take the small Hv-field (Z4,+,⊗), where only 2 ⊗ 3 = {0,2} and fundamental classes {0,2},{1,3}. 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