Ratio Mathematica Vol. 33, 2017, pp. 89-102 ISSN: 1592-7415 eISSN: 2282-8214 Special Classes of Hb-Matrices Achilles Dramalidis∗ †doi:10.23755/rm.v33i0.382 Abstract In the present paper we deal with constructions of 2 × 2 diagonal or upper- triangular or lower-triangular Hb-matrices with entries either of an Hb-field on Z2 or on Z3. We study the kind of the hyperstructures that arise, their unit and inverse elements. Also, we focus our study on the cyclicity of these hyperstructures, their generators and the respective periods. Keywords: hope; Hv-structure; Hb-structure; Hv-matrix 2010 AMS subject classifications: 20N20. ∗Democritus University of Thrace, School of Education, 68100 Alexandroupolis, Greece; adra- mali@psed.duth.gr † c©Achilles Dramalidis. Received: 31-10-2017. Accepted: 26-12-2017. Published: 31-12- 2017. 89 Achilles Dramalidis 1 Introduction F. Marty, in 1934 [13], introduced the hypergroup as a set H equipped with a hyperoperation · : H × H → P(H) −{∅} which satisfies the associative law: (xy)z = x(yz), for all x,y,z ∈ H and the reproduction axiom: xH = Hx = H, for all x ∈ H. In that case, the reproduction axiom is not valid, the (H, ·) is called semihypergroup. In 1990, T. Vougiouklis [19] in the Fourth AHA Congress, introduced the Hv- structures, a larger class than the known hyperstructures, which satisfy the weak axioms where the non-empty intersection replaces the equality. Definition 1.1. [21], The (·) in H is called weak associative, we write WASS, if (xy)z ∩x(yz) 6= ∅,∀x,y,z ∈ H. The (·) is called weak commutative, we write COW, if xy ∩yx 6= ∅,∀x,y ∈ H. The hyperstructure (H, ·) is called Hv-semigroup if (·) is WASS. It is called Hv- group if it is Hv-semigroup and the reproduction axiom is valid. Further more, it is called Hv-commutative group if it is an Hv-group and a COW. If the commutativity is valid, then H is called commutative Hv-group. Analogous definitions for other Hv-structures, as Hv-rings, Hv-module, Hv-vector spaces and so on can be given. For more definitions and applications on hyperstructures one can see books [3], [4], [5], [6], [21] and papers as [2], [7], [9], [10], [12], [14], [20], [22], [23], [24], [26], [27]. An element e ∈ H is called left unit if x ∈ ex,∀x ∈ H and it is called right unit if x ∈ xe,∀x ∈ H. It is called unit if x ∈ ex∩xe,∀x ∈ H. The set of left units is denoted by E` [8]. The set of right units is denoted by Er and by E = E` ∩Er the set of units [8]. The element a′ ∈ H is called left inverse of the element a ∈ H if e ∈ a′a, where e unit element (left or right) and it is called right inverse if e ∈ aa′. If e ∈ a′a∩aa′ then it is called inverse element of a ∈ H. The set of the left inverses is denoted by I`(a,e) and the set of the right inverses is denoted by Ir(a,e)[8]. By I(a,e) = I`(a,e)∩ Ir(a,e), the set of inverses of the element a ∈ H, is denoted. 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 h, called generator and a natural s, the minimum: H = h1∪h2∪·· ·∪hs. Analogously the cyclicity for the infinite period 90 Special Classes of Hb-Matrices is defined [17],[21]. If there is an h and s, the minimum: H = hs, then (H, ·), is called single-power cyclic of period s. Definition 1.2. 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 [18],[19],[21],[22], (see also [1],[3],[4]). 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 [19] there is the following [20], [21]: Definition 1.3. An Hv-ring (R,+, ·) is called Hv-field if R/γ∗ is a field. Hv-matrix is a matrix with entries of an Hv-ring or Hv-field. The hyperprod- uct 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. Hv-matrices is a very useful tool in Representation Theory of Hv-groups [15],[16], [25],[28] (see also [11], [29]). 2 Constructions of 2×2 Hb-matrices with entries of an Hv-field on Z2 Consider the field (Z2,+, ·). On the set Z2 also consider the hyperoperation (�) defined by setting: 1�1 = {0,1} and x�y = x ·y for all (x,y) ∈ Z2 ×Z2 −{(0,1)}. Then (Z2,+,�) becomes an Hb-field. All the 2×2 Hb-matrices with entries of the Hb-field (Z2,+,�), are 24 = 16. Let us denote them by: 0 = ( 0 0 0 0 ) ,a1 = ( 1 0 0 0 ) ,a2 = ( 0 1 0 0 ) ,a3 = ( 0 0 1 0 ) , a4 = ( 0 0 0 1 ) ,a5 = ( 1 1 0 0 ) ,a6 = ( 1 0 1 0 ) ,a7 = ( 1 0 0 1 ) , a8 = ( 0 1 1 0 ) ,a9 = ( 0 1 0 1 ) ,a10 = ( 0 0 1 1 ) ,a11 = ( 1 1 1 0 ) , 91 Achilles Dramalidis a12 = ( 1 1 0 1 ) ,a13 = ( 1 0 1 1 ) ,a14 = ( 0 1 1 1 ) ,a15 = ( 1 1 1 1 ) . By taking a2i , i = 1, · · · ,15 there exist 15 closed sets, let us say Hi , i = 1, · · · ,15. Two of them are singletons, H2 = H3 = {0}. Also, H7 = H8 and H11 = H14 = H15. So, we shall study, according to the hyperproduct (·) of two Hb-matrices, the fol- lowing sets: H1 = {0,a1},H4 = {0,a4},H5 = {0,a1,a2,a5},H6 = {0,a1,a3,a6}, H7 = {0,a1,a4,a7},H9 = {0,a2,a4,a9},H10 = {0,a3,a4,a10}, H12 = {0,a1,a2,a4,a5,a7,a9,a12},H13 = {0,a1,a3,a4,a6,a7,a10,a13}, H15 = {0,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10a11,a12,a13,a14,a15}. 2.1 The case of diagonal 2×2 Hb-matrices Every set of H1,H4,H7 consists of diagonal 2 × 2 Hb-matrices. Then, the multiplicative tables of the hyperproduct, are the following: · 0 a1 0 0 0 a1 0 H1 , · 0 a4 0 0 0 a4 0 H4 · 0 a1 a4 a7 0 0 0 0 0 a1 0 0,a1 0 0,a1 a4 0 0 0,a4 0,a4 a7 0 0,a1 0,a4 H7 In all cases: x ·y = y ·x, ∀x,y ∈ Hi, i = 1,4,7 (x ·y) ·z = x · (y ·z), ∀x,y,z ∈ Hi, i = 1,4,7 So, we get the next propositions: Proposition 2.1. Every set H, consisting of diagonal 2 × 2 Hb-matrices with en- tries of the Hb-field (Z2,+,�), equipped with the usual hyperproduct (·) of ma- trices, is a commutative semihypergroup. 92 Special Classes of Hb-Matrices Notice that H1,H4 ⊂ H7 and since H1 ·H1 ⊆ H1 , H4 ·H4 ⊆ H4 then H1,H4 are sub-semihypergroups of (H7, ·). Proposition 2.2. For all commutative semihypergroups (H, ·), consisting of diag- onal 2×2 Hb-matrices with entries of the Hb-field (Z2,+,�): E = {ai}, I(ai,ai) = {ai}, where a2i = H. Remark 2.1. According to the above construction, the commutative semihyper- groups (H1, ·),(H4, ·) and (H7, ·), are single-power cyclic commutative semihy- pergroups with generators the elements a1,a4 and a7, respectively, with single- power period 2. 2.2 The case of upper- and lower- triangular 2×2 Hb-matrices Every set of H5,H9,H12 consists of upper-triangular 2 × 2 Hb-matrices and every set of H6,H10,H13 consists of lower-triangular 2 × 2 Hb-matrices. Then, the multiplicative tables of the hyperproduct, are the following: · 0 a1 a2 a5 0 0 0 0 0 a1 0 0,a1 0,a2 H5 a2 0 0 0 0 a5 0 0,a1 0,a2 H5 , · 0 a2 a4 a9 0 0 0 0 0 a2 0 0 0,a2 0,a2 a4 0 0 0,a4 0,a4 a9 0 0 H9 H9 · 0 a1 a2 a4 a5 a7 a9 a12 0 0 0 0 0 0 0 0 0 a1 0 0,a1 0,a2 0 0,a1, 0,a1 0,a2 0,a1, a2,a5 a2,a5 a2 0 0 0 0,a2 0 0,a2 0,a2 0,a2 a4 0 0 0 0,a4 0 0,a4 0,a4 0,a4 a5 0 0,a1 0,a2 0,a2 0,a1, 0,a1, 0,a2 0,a1, a2,a5 a2,a5 a2,a5 a7 0 0,a1 0,a2 0,a4 0,a1, 0,a1, 0,a2, H12 a2,a5 a4,a7 a4,a9 a9 0 0 0 0,a2, 0 0,a2, 0,a2, 0,a2, a4,a9 a4,a9 a4,a9 a4,a9 a12 0 0,a1 0,a2 0,a2, 0,a1, H12 0,a2, H12 a4,a9 a2,a5 a4,a9 93 Achilles Dramalidis · 0 a1 a3 a6 0 0 0 0 0 a1 0 0,a1 0 0,a1 a3 0 0,a3 0 0,a3 a6 0 H6 0 H6 , · 0 a3 a4 a10 0 0 0 0 0 a3 0 0 0 0 a4 0 0,a3 0,a4 H10 a10 0 0,a3 0,a4 H10 · 0 a1 a3 a4 a6 a7 a10 a13 0 0 0 0 0 0 0 0 0 a1 0 0,a1 0 0 0,a1 0,a1 0 0,a1 a3 0 0,a3 0 0 0,a3 0,a3 0 0,a3 a4 0 0 0,a3 0,a4 0,a3 0,a4 0,a3, 0,a3, a4,a10 a4,a10 a6 0 0,a1, 0 0 0,a1, 0,a1, 0 0,a1, a3,a6 a3,a6 a3,a6 a3,a6 a7 0 0,a1 0,a3 0,a4 0,a1, 0,a1, 0,a3, H13 a3,a6 a4,a7 a4,a10 a10 0 0,a3 0,a3 0,a4 0,a3 0,a3, 0,a3, 0,a3, a4,a10 a4,a10 a4,a10 a13 0 0,a1, 0,a3 0,a4 0,a1, H13 0,a3, H13 a3,a6 a3,a6 a4,a10 In all cases: (x ·y)∩ (y ·x) 6= ∅, ∀x,y ∈ Hi, i = 5,6,9,10,12,13 (x ·y) ·z = x · (y ·z), ∀x,y,z ∈ Hi, i = 5,6,9,10,12,13 So, we get the next proposition: Proposition 2.3. Every set H, consisting either of upper-triangular or lower- triangular 2 × 2 Hb-matrices with entries of the Hb-field (Z2,+,�), equipped with the usual hyperproduct (·) of matrices, is a weak commutative semihyper- group. Notice that H5,H9 ⊂ H12 and H6,H10 ⊂ H13. Since H5·H5 ⊆ H5 , H9·H9 ⊆ H9 , H6 ·H6 ⊆ H6 , H10 ·H10 ⊆ H10 , then H5,H9 are sub-semihypergroups of (H12, ·) and H6,H10 are sub-semihypergroups of (H13, ·). Proposition 2.4. For all weak commutative semihypergroups (H, ·), consisting either of upper-triangular or lower-triangular 2 × 2 Hb-matrices with entries of the Hb-field (Z2,+,�), the following assertions hold i) If ai,aj ∈ H : ai ·aj = H, ai ∈ a2i , a2j = H, ai ∈ aj ·ai , then a)E` = {ai,aj}, b)I(ai,ai) = I(aj,ai) = {ai,aj} 94 Special Classes of Hb-Matrices c)I(aj,aj) = I r(ai,aj) = {aj}, d)I`(ai,aj) = ∅ ii) If ai,aj ∈ H : aj ·ai = H, ai ∈ a2i , a2j = H, ai ∈ ai ·aj, then a)Er = {ai,aj}, b)I(ai,ai) = I(aj,ai) = {ai,aj} c)I(aj,aj) = I `(ai,aj) = {aj}, d)Ir(ai,aj) = ∅ iii) If ai,aj ∈ H : ai ·aj = aj ·ai = H, ai ∈ a2i , a2j = H, then a)E = {ai,aj}, b)I(ai,ai) = I(aj,ai) = I(aj,aj) = {ai,aj}, c)I(ai,aj) = {aj} Remark 2.2. According to the above construction, the weak commutative semi- hypergroups (Hi, ·), i=5,6,9,10,12,13 are single-power cyclic weak commutative semihypergroups with generators the elements a5,a6,a9,a10,a12,a13 respectively, with single-power period 2. 3 Constructions of 2×2 Hb-matrices with entries of an Hb-field on Z3 Consider the field (Z3,+, ·). On the set Z3, we consider four cases for the hyperoperation (�i), i = 1,2,3,4 defined, each time, by setting: 1) 1�1 2 = {1,2} and x�1 y = x ·y for all (x,y) ∈ Z3 ×Z3 −{(1,2)}. 2) 2�2 1 = {1,2} and x�2 y = x ·y for all (x,y) ∈ Z3 ×Z3 −{(1,2)}. 3) 1�3 1 = {1,2} and x�3 y = x ·y for all (x,y) ∈ Z3 ×Z3 −{(1,2)}. 4) 2�4 2 = {1,2} and x�4 y = x ·y for all (x,y) ∈ Z3 ×Z3 −{(1,2)}. Then, each time, (Z3,+,�i), i = 1,2,3,4 becomes an Hb-field. Now, consider the set H of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb- matrices, with entries of the Hb-field (Z3,+,�i). Let us denote them by: a11 = ( 1 0 0 1 ) ,a12 = ( 1 0 0 2 ) ,a21 = ( 2 0 0 1 ) ,a22 = ( 2 0 0 2 ) . So, H = {a11,a12,a21,a22}. 95 Achilles Dramalidis 3.1 The case of 1�1 2 = {1,2} The multiplicative table of the hyperproduct, is the following: · a11 a12 a21 a22 a11 a11 a11,a12 a11,a21 H a12 a12 a11 a12,a22 a11,a21 a21 a21 a21,a22 a11 a11,a12 a22 a22 a21 a12 a11 Notice that in the above multiplicative table: i) x ·H = H ·x = H,∀x ∈ H ii) (x ·y)∩ (y ·x) 6= ∅,∀x,y ∈ H iii) (x ·y) ·z ∩x · (y ·z) 6= ∅,∀x,y,z ∈ H So, we get the next proposition: Proposition 3.1. The set H, consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�1), equipped with the usual hyperproduct (·) of matrices, is an Hv-commutative group. Proposition 3.2. For the Hv-commutative group (H, ·), consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�1) : i) E = {a11} ii) Ir(x,a11) = {a22},∀x ∈ H iii) I`(x,a11) = {a11},∀x ∈ H Proposition 3.3. The Hv-commutative group (H, ·), consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�1), is a single-power cyclic Hv-commutative group with generator the element a22, with single-power period 3. 3.2 The case of 2�2 1 = {1,2} The multiplicative table of the hyperproduct, is the following: · a11 a12 a21 a22 a11 a11 a12 a21 a22 a12 a11,a12 a11 a21,a22 a21 a21 a11,a21 a12,a22 a11 a12 a22 H a11,a21 a11,a12 a11 As in the paragraph 3.1: Proposition 3.4. The set H, consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�2), equipped with the usual hyperproduct (·) of matrices, is an Hv-commutative group. 96 Special Classes of Hb-Matrices Now, take a map f onto and 1:1, f : H → H , such that f(a11) = a22, f(a12) = a21, f(a21) = a12, f(a22) = a11 Then, the successive transformations of the above multiplicative table are: · a22 a21 a12 a11 a22 a11 a12 a21 a22 a21 a11,a12 a11 a21,a22 a21 a12 a11,a21 a12,a22 a11 a12 a11 H a11,a21 a11,a12 a11 · a22 a21 a12 a11 a11 H a11,a21 a11,a12 a11 a12 a11,a21 a12,a22 a11 a12 a21 a11,a12 a11 a21,a22 a21 a22 a11 a12 a21 a22 · a11 a12 a21 a22 a11 a11 a11,a12 a11,a21 H a12 a12 a11 a12,a22 a11,a21 a21 a21 a21,a22 a11 a11,a12 a22 a22 a21 a12 a11 Then, the last multiplicative table is the table of the paragraph 3.1. So, we get: Proposition 3.5. The Hv-commutative group (H, ·) consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�2), is isomorphic to Hv-commutative group (H, ·) consisting of the diag(b11,b22) , b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�1). 3.3 The case of 1�3 1 = {1,2} The multiplicative table of the hyperproduct, is the following: · a11 a12 a21 a22 a11 H a12,a22 a21,a22 a22 a12 a12,a22 a11,a21 a22 a21 a21 a21,a22 a22 a11,a12 a12 a22 a22 a21 a12 a11 Notice that in the above multiplicative table: i) x ·H = H ·x = H, ∀x ∈ H 97 Achilles Dramalidis ii) x ·y = y ·x, ∀x,y ∈ H iii) (x ·y) ·z ∩x · (y ·z) 6= ∅, ∀x,y,z ∈ H So, we get the next proposition: Proposition 3.6. The set H, consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�3), equipped with the usual hyperproduct (·) of matrices, is a commutative Hv- group. Proposition 3.7. For the commutative Hv-group (H, ·), consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�3) : i) E = Er = E` = {a11} ii) I(x,a11) = Ir(x,a11) = I`(x,a11) = {x},∀x ∈ H Proposition 3.8. The commutative Hv-group (H, ·), consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�3) : i) is a single-power cyclic commutative Hv-group with generator the element a11, with single-power period 2. ii) is a single-power cyclic commutative Hv-group with generator the element a22, with single-power period 4. iii) is a cyclic commutative Hv-group of period 3 to each of the generators a12 and a21. 3.4 The case of 2�4 2 = {1,2} The multiplicative table of the hyperproduct, is the following: · a11 a12 a21 a22 a11 a11 a12 a21 a22 a12 a12 a11,a12 a22 a21,a22 a21 a21 a22 a11,a21 a12,a22 a22 a22 a21,a22 a12,a22 H Notice that in the above multiplicative table: i) x ·H = H ·x = H, ∀x ∈ H ii) x ·y = y ·x, ∀x,y ∈ H iii) (x ·y) ·z = x · (y ·z), ∀x,y,z ∈ H So, we get the next proposition: Proposition 3.9. The set H, consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�4), equipped with the usual hyperproduct (·) of matrices, is a commutative hypergroup. 98 Special Classes of Hb-Matrices Proposition 3.10. For the commutative hypergroup (H, ·), consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�4) : i) E = {a11} ii) I(x,a11) = {x},∀x ∈ H Proposition 3.11. The commutative hypergroup (H, ·), consisting of the diag(b11,b22), b11,b22 ∈ Z3 with b11b22 6= 0 Hb-matrices, with entries of the Hb-field (Z3,+,�4) is a single-power cyclic commutative hypergroup with generator the element a22, with single-power period 2. 4 Construction of 2×2 upper-triangular Hb-matrices with entries of an Hb-field on Z3 On the set Z3, consider the hyperoperation (�1) defined, by setting: 1�1 2 = {1,2} and x�1 y = x ·y for all (x,y) ∈ Z3 ×Z3 −{(1,2)} Now, consider the set H of the 2 × 2 upper-triangular Hb-matrices with b11,b22 ∈ Z3 and b11b22 6= 0, with entries of the Hb-field (Z3,+,�1). Let us denote the elements of H by: a1 = ( 1 0 0 1 ) ,a2 = ( 1 0 0 2 ) ,a3 = ( 1 1 0 1 ) ,a4 = ( 1 1 0 2 ) , a5 = ( 1 2 0 1 ) ,a6 = ( 1 2 0 2 ) ,a7 = ( 2 0 0 1 ) ,a8 = ( 2 0 0 2 ) , a9 = ( 2 1 0 1 ) ,a10 = ( 2 1 0 2 ) ,a11 = ( 2 2 0 1 ) ,a12 = ( 2 2 0 2 ) So, H = {a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12}. Since the multiplicative table is long enough, it is omitted. From this table we get: i) x ·H = H ·x = H, ∀x ∈ H ii) (·) is non-commutative iii) (x ·y) ·z ∩x · (y ·z) 6= ∅, ∀x,y,z ∈ H So, we get the next proposition: Proposition 4.1. The set H, consisting of the 2×2 upper-triangular Hb-matrices with b11,b22 ∈ Z3 and b11b22 6= 0, with entries of the Hb-field (Z3,+,�1), equipped with the usual hyperproduct (·) of matrices, is a non-commtative Hv- group. 99 Achilles Dramalidis Proposition 4.2. For the non-commtative Hv-group (H, ·), consisting of the 2×2 upper-triangular Hb-matrices with b11,b22 ∈ Z3 and b11b22 6= 0, with entries of the Hb-field (Z3,+,�1) : E = E` = Er = {a1},∀x ∈ H. Proposition 4.3. The non-commtative Hv-group (H, ·), consisting of the 2 × 2 upper-triangular Hb-matrices with b11,b22 ∈ Z3 and b11b22 6= 0, with entries of the Hb-field (Z3,+,�1) : i) is a single-power cyclic non-commutative Hv-group with generator the element a12, with single-power period 4. ii) is a single-power cyclic non-commutative Hv-group with generator the element a10, with single-power period 3. Now, take any Hb-field (Zp,+,�1) , p = prime 6= 2 and then consider a set H consisting of the 2 × 2 upper-triangular Hb-matrices with entries of this Hb-field, with b11b22 6= 0 , b11,b22 ∈ Zp. Then, for any such a set Zp, take for example the elements a3,a7 ∈ H, then: a7 ·a3 = a11 and a3 ·a7 = {a1,a7} So, we get the next general proposition: Proposition 4.4. Any set H, consisting of the 2×2 upper-triangular Hb-matrices with b11b22 6= 0 , b11,b22 ∈ Zp, p = prime 6= 2, with entries of the Hb- field (Zp,+,�1), equipped with the usual hyperproduct (·) of matrices, is a non- commutative hyperstructure. 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