Veblen and Bussey [ ], [ ] have defined a finite projective geometry, which is said to be a geometry of a k-dimensional space, in the following way Ratio Mathematica, 21, 2011, pp. 43-58 43 On geometrical hyperstructures of finite order Achilles Dramalidis School of Sciences of Education, Democritus University of Thrace, 681 00 Alexandroupolis, Greece adramali@psed.duth.gr Abstract It is known that a concrete representation of a finite k-dimensional Projective Geometry can be given by means of marks of a Galois Field GF [p n ], denoted by PG(k, p n ). In this geometry, we define hyperoperations, which create hyperstructures of finite order and we present results, propositions and examples on this topic. Additionally, we connect these hyperstructures to Join Spaces. AMS classification : 20N20, 16Y99, 51A45 Keywords : Hypergroups, Hv-groups, Hv-rings 1. Introduction The algebraic hyperstructures, which constitute a generalization of the ordinary algebraic structures, were introduced by Marty in 1934 [5]. Since then, many researchers worked on hyperstructures. The results of this work, as well as, applications of the hyperstructures theory can be found in the books [2] and [3]. Vougiouklis in 1991 introduced a larger class than the known hyperstructures, so called Hv- structures [8] and all about them can be found in his book [9]. Let us give some basic definitions, appearing in [3], [9]: mailto:adramali@psed.duth.gr 44 Let H be a set, P(H) the family of nonempty subsets of H and () a hyperoperation in H, that is  : HH  P(H) If (x,y)HH, its image under () is denoted by xy or xy. If A , B  H then AB or AB is given by AB =  {xy / xA , yB}. xA is used for {x}A and Ax for A{x}. Generally, the singleton {x} is identified with its member x. The hyperoperation () is called associative in H if (xy)z = x(yz) for all x,y,zH The hyperoperation () is called commutative in H if xy = yx for all x,yH A hypergroupoid (H,) that satisfies reproducibility, xH = Hx = H for all xH, and associativity, is called hypergroup. A join operation () [6] in a set J is a mapping of JJ into the family of subsets of J. A join space is defined as a system (J,), where () is a join operation in the arbitrary set J, which satisfies the postulates: i) ab≠ ii) ab = ba iii) (ab)c=a(bc) iv) a/b  c/d≠  ad  bc≠ v) a/b = {xJ / abx}≠ . The Hv-structures are hyperstructures satisfying the weak axioms, where the non-empty intersection replaces the equality. Let H≠ be a set equipped with the hyperoperations (+), (), then the weak associativity in () is given by the relation (xy) z  x(yz)  , x,y,zH. The (), is called weak commutative if xy  yx  , x,yH. The hyperstructure (H,) is called Hv-semigroup if () is weak associative and it is called Hv-quasigroup if the reproduction axiom is valid, i.e. xH=Hx=H, xH. The hyperstructure (H,) is called Hv-group if it is an Hv-quasigroup and an Hv-semigroup. It is called Hv-commutative group if it is an Hv- group and the weak commutativity is valid. 45 The weak distributivity of () with respect to (+) is given for all x,y,zH, by x(y+z)  (xy+xz)   , (x+y)z  (xz+yz)  . Using these axioms, the Hv-ring, which is the largest class of algebraic systems that satisfy ring-like axioms, is defined to be the triple (H,+,), where in both (+) and () the weak associativity is valid, the weak distributivity is also valid and (+) is reproductive, i.e x+H = H+x = H, xH. An Hv-ring (R,+,) is called dual Hv-ring if the hyperstructure (R,,+) is also an Hv-ring [4]. Let (H,) be a hypergroup or an Hv-group. The β* relation is defined as the smallest equivalence relation, one can say also congruence, such that, the quotient H/β* is a group. The β* is called fundamental equivalence relation. 2. Representation of the geometry of a k-dimensional space by means of Galois Fields Veblen and Bussey [7] have defined a finite projective geometry, which is said to be a geometry of a k-dimensional space, in the following way. It consists of a set of elements, called points for suggestiveness, which are subjected to the following five conditions or postulates: I. The set contains a finite number of points. It contains one or more subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains both A and B. III. If A, B, C are noncollinear points and if a line l contains a point D of the line AB and a point E of the line BC but does not contain A or B or C, then the line l contains a point F of the line CA. IV. If m is an integer less than k, not all the points considered are in the same m-space. 46 V. If (IV) is satisfied, there exists in the set of points considered no (k+1)-space. Furthermore, a point is called 0-space, a line is called 1-space and a plane is called 2-space. By means of marks of a Galois field, we shall now give a concrete representation of a finite k-dimensional projective geometry. We denote a point of the geometry by the ordered set of coordinates (μ0, μ1, μ2,…., μk), where μ0, μ1, μ2,…., μk are marks of the GF[p n ], at least one of which is different from zero. It is understood that the foregoing symbol (μ0, μ1, μ2,…., μk) denotes the same point as the symbol (μμ0, μμ1, μμ2,…., μμk), where μ is one of the p n -1 nonzero marks of the field. The ordered set of marks μ0, μ1, μ2,…., μk may be chosen in (p n ) κ+1 ways, but since the symbol (0, 0, 0,….,0) is excepted, then it may be chosen in (p n ) κ+1 -1 ways. So, there exists (p n ) κ+1 -1 points. In this totality, each point is represented in p n -1 ways (there are p n -1 nonzero marks in the field) and thus, it follows that the number of points defined is knn n kn pp p p     .....1 1 1)( 1 This representation of the finite κ-dimensional projective geometry by means of the marks of the GF[p n ] constitute the projective geometry PG(κ, p n ) [1]. Now, for the line containing the two distinct points (μ0, μ1, μ2,…., μk) and (ν0, ν1, ν2,…., νk) we consider the set of points : ( μμ0 + νν0 , μμ1 + νν1 , μμ2 + νν2 , ……, μμk + ννk ) where μ and ν run independently over the marks of the GF[p n ], subjected to the condition that μ and ν shall not be simultaneously zero. Then the number of possible combinations of μ and ν is (p n ) 2 – 1 and for each of these the corresponding symbol denotes a point, since not all the k+1 coordinates are zero. But the same point is 47 represented p n -1 times, due to the factor of proportionality involved in the definition of points. Therefore, a line so defined contains 1 1 1)( 2    n n n p p p points. It is obvious that any two points on the line may be used in this way to define the same line. The five postulates given above for the k-dimensional space are satisfied by the concrete elements thus introduced [1]. 3. On a hypergroup of finite order Let us denote by V the set of the elements of the PG(κ, p n ) and for x,yV let us denote by lxy the line which is defined by the points x and y. By lx- is denoted the line which is defined by the point x and any other point of V. We define the hyperoperation (∙) on V, as follows : Definition 1. For every x,yV , ∙ : VV  P(V) , such that x∙y =      yxifl yxifx xy Obviously, the above hyperoperation is a commutative one, since x∙y = lxy = lyx = y∙x for every x,yV and x ≠ y. One can compare the above defined hyperoperation with the join operation [6], when Euclidean Geometry is converted into Join Spaces by defining ab with a≠b, to be the open segment, whose endpoints are a and b. Moreover, aa is defined to be a. Proposition 2. For every noncollinear x,y,zV, ∣x∙(y∙z)∣=∣(x∙y)∙z∣ = p 2n +p n +1. 48 Proof. All the lines defined in V are having one point in common, at most. First, let us calculate the points of the set x∙(y∙z). For y ≠ z, the set y∙z = lyz consists of p n +1 points, including y and z. On the other hand, the point x (x ≠ y,z), with each of the p n +1 points of the line lyz, creates p n +1 lines of the type lx- which are having the point x in common. This means that the p n +1 lines of the type lx- are having no other point in common. So, each line lx- is having p n different points from the others. Then it follows that the set x∙(y∙z) consists of (p n +1)∙p n +1= p 2n +p n +1 different points. Similarly, it arises that ∣(x∙y)∙z∣ = p 2n +p n +1.■ Proposition 3. The hyperstructure (V, ∙) is a hypergroup. Proof. Easily follows, that for every xV x∙V =   )()( xvvx VvVv  V∙x = V Now, for every x, y, zV if x = y = z then x∙(y∙z) = (x∙y)∙z = x if x = y ≠ z then x∙(y∙z) = (x∙y)∙z = lxz if x = z ≠ y then x∙(y∙z) = (x∙y)∙z = lxy if y = z ≠ x then x∙(y∙z) = (x∙y)∙z = lxy if x ≠ y ≠ z and x, y, z collinear, then x∙(y∙z) = (x∙y)∙z = lxy if x, y, z noncollinear then for the line lyz containing the two distinct points y(y0, y1,..,yk) and z(z0, z1,…, zk) we take the set of points : ( μy0 + νz0 , μy1 + νz1 ,……, μyk + νzk ) where μ and ν run independently over the marks of the GF[p n ] subjected to the condition that μ and ν shall not be simultaneously zero. Let x(x0, x1,...,xk). Then for the set x∙(y∙z) we take the set of points : (ρx0 + λ(μy0 + νz0) , ρx1 + λ(μy1 + νz1) ,……, ρxk + λ(μyk + νzk)) (1) 49 where ρ and λ run independently over the marks of the GF[p n ] subjected to the condition that ρ and λ shall not be simultaneously zero. Let w x∙(y∙z) , then the coordinates of the point w is of the form of (1). For some i0,1,…..,k we have ρxi+λ(μyi+νzi) = ρxi+λμyi+λνzi = ρxi+λμyi+νzi, where νGF[p n ] If ρ=0 (2) then ρ=λ0 for every λGF[p n ] and then ρxi + λμyi + νzi = λ0xi + λμyi + νzi = λ(0xi + μyi ) + νzi If ρ≠0 (3) then ρ=λμ for every λ, μGF[p n ]-{0} and then ρxi + λμyi + νzi = λμxi + λμyi + νzi = λ(μxi + μyi ) + νzi The coordinates of the points of the set (x∙y)∙z are of the form κ(τxi + τyi ) + κzi where τ, τ, κ, κ run independently over the marks of the GF[p n ] subjected to the condition that τ, τ and κ, κ shall not be simultaneously zero. Due to the conditions (2) and (3) we get that w x∙(y∙z)  w (x∙y)∙z which means that x∙(y∙z)  (x∙y)∙z . In a similar way, it can be proven that for w(x∙y)∙z  wx∙(y∙z), which means that (x∙y)∙z  x∙(y∙z). So, x∙(y∙z) = (x∙y)∙z for every x,y,zV.■ Remark 4. For the hypergroup (V,∙), since {x,y}  x∙y x,yV, the Vβ* is a singleton. Proposition 5. The hypergroup (V, ∙) is a Join Space. Proof. Since the hyperoperation (∙) is commutative, the hyperstructure (V, ∙) is a commutative hypergroup. 50 Moreover, let a/b  c/d ≠  , a,b,c,dV . Then, there exists wV such that wa/b which implies that a lbw and wc/d which implies that c ldw. Since the lines of V are having one point in common at most, the lines lbw and ldw intersect at w. Let the ordered set of coordinates of the points w, a, d be (w0, w1,...,wk) , (a0, a1,...,ak), (d0, d1,...,dk) respectively. Then, the coordinates of the point b will be of the form ( λa0 + μw0 , λa1 + μw1 ,……, λak + μwk ), where λ,μ GF[p n ] and the coordinates of the point c will be of the form ( κd0 + ρw0 , κd1 + ρw1,… ……, κak + ρwk ), where κ,ρ GF[p n ]. Since the points w, a, d do not belong to the line lbc , the marks λ,μ,κ,ρ of the GF[p n ] are not zero. Now, lbc consists of the points of the form (νλa0+νμw0+ τκd0+τρw0 , νλa1+νμw1+ τκd1+τρw1 ,……, νλak+νμwk+ τκdk+τρwk ), where ν and τ run independently over the marks of the GF[p n ] and they are not simultaneously zero. It is known that for the nonzero marks μ and ρ, there exist nonzero marks ν and τ such that: νμ+τρ = 0 . Then, we get νμw0 + τρw0 = (νμ+τρ)w0 = 0, νμw1 + τρw1 = (νμ+τρ)w1 = 0 ,…….., νμwk + τρwk = (νμ+τρ)wk = 0 . In that case, the point (νλa0+τκd0 , νλa1+τκd1 ,……., νλak+τκdk) of the line lbc is additionally a point of the line lad. So, the lines lbc, lad intersect and then: a∙d  b∙c ≠  for all a,b,c,dV . ■ 4. On a Hv-group of finite order Now, we define a new hyperoperation (◦) on V as follows : Definition 6. For every x,yV , ◦ : VV  P(V) , such that x◦y =      yxifxl yxifx xy }{ 51 Every line of the set V contains pn+1 points. The hyperoperation (◦) is weak commutative, since the lines lxy-{x} and lyx-{y} are having p n +1-2 = p n -1 points in common, so (x◦y)  (y◦x) ≠  for every x,yV . Proposition 7. For every noncolliner x,y,zV, ∣x◦(y◦z)∣= p 2n > ∣(x◦y)◦z∣ = p 2n -p n +1 Proof. Since the line y◦z does not contain the point y, we get that ∣y◦z∣ = p n . The point x creates p n points with each of the p n points of the line y◦z (the point x is not participating according to the hyperoperation (◦)). So, ∣x◦(y◦z)∣= p n ‧p n = p 2n . On the other hand, the line x◦y consists of p n points. Each of these points creates p n points each time together with the point z, but since the point z appears p n times, we get ∣(x◦y)◦z∣ = p n ‧p n - p n +1 = p 2n -p n +1. Since p is prime and nIN, easily follows that ∣x◦(y◦z)∣ > ∣(x◦y)◦z∣.■ Proposition 8. The hyperstructure (V, ◦) is an Hv-group. Proof. Indeed, for every xV x◦V = )( vx Vv   = (x◦x)         )( }{ vx xVv  = x         }){( }{ xl xv xVv  = V since every line lxv always contains the point vV. On the other hand, for every xV V◦x = )( xv Vv   = (x◦x)         )( }{ xv xVv  = x         }){( }{ vl vx xVv  = V Indeed, having the fact that every line of V contains at least 3 points, for every line lvx–{v}=v◦x there exists at least one point vlvx –{v}, that v lvx– {v} = v◦x. So, x◦V = V◦x = V for every xV. 52 The hyperoperation (◦) is weak associative, since for every x,y,zV x◦(y◦z)  x◦z  z and (x◦y)◦z  y◦z  z But, we shall go further on, proving that the inclusion on the right parenthesis is valid, i.e (x◦y)◦z  x◦(y◦z). From the proposition 7 we get that ∣x◦(y◦z)∣= p 2n , since the p n +1 points of the line lxy are not contained into the set x◦(y◦z). Similarly, the set (x◦y)◦z does not contain the points of the line lxy, since (x◦y)◦z = (lxy – {x})◦z = (x1◦z ) (x2◦z ) …… ( 1 n p x ◦z ) (y◦z ), where x1, x2,……., 1 n p x lxy. Also, the set (x◦y)◦z does not contain the points of the line lxz, since x◦y=lxy– {x}. As the lines lxy and lxz intersect at the point x, they don’t have any other points in common. That means that the p 2n -p n +1 points of the set (x◦y)◦z (proposition 7), are also points of the set x◦(y◦z). So, we proved that (x◦y)◦z  x◦(y◦z), which, generally, means that (x◦y)◦z  x◦(y◦z) ≠  for every x,y,zV. ■ Remark 9. For the Hv-group (V,◦), since yx◦y x,yV, the Vβ* is a singleton. Since, yx◦y for every x,yV, we get the following proposition: Proposition 10. Every element of the Hv-commutative group (V,◦) is simultaneously a right zero and a left unit element. Example 11. By means of the marks of a Galois Field, we shall now give a concrete representation of a finite 2-dimensional projective geometry. We denote a point of the geometry by the ordered set of coordinates (μ0, μ1, μ2). The μ0, μ1, μ2 are marks of the GF[2 2 ] defined by means of the function x 2 +x+1. At least one of μ0, μ1, μ2 is different from 53 zero. Let us denote the marks of the GF[2 2 ] by 0, 1, a, b, then we have the following tables : + 0 1 a b ∙ 0 1 a b 0 0 1 a b 0 0 0 0 0 1 1 0 b a 1 0 1 a b a a b 0 1 a 0 a b 1 b b a 1 0 b 0 b 1 a The ordered set of marks μ0, μ1, μ2 may be chosen by (2 2 ) 2+1 = 64 ways, but since the symbol (0, 0, 0) is excepted, it may be chosen by 64-1=63 ways. So, there exist 63 points. In this totality, each point is represented by 3 ways (3 sets of symbols, since there are 3 nonzero marks in the field). Then the number of points defined is 63 ÷ 3 = 21. This representation of the finite 2-dimensional projective geometry by means of the marks of the GF[2 2 ], constitute the projective geometry PG(2, 2 2 ). The 21 points of PG(2, 2 2 ) = V will be denoted by letters in accordance with the following scheme : A(001) B(010) C(011) D(01a) E(01b) F(100) G(101) H(10a) I(10b) J(110) K(111) L(11a) M(11b) N(1a0) O(1a1) P(1aa) Q(1ab) R(1b0) S(1b1) T(1ba) U(1bb) Now, for the line containing the two distinct points A(001) and B(010) we take the set of points : ( μ0 + ν0 , μ0 + ν1 , μ1 + ν0 ) where μ and ν run independently over the marks of the GF[2 2 ] subjected to the condition that μ and ν shall not be simultaneously zero. Then the number of possible combinations of the μ and ν is (2 2 ) 2 – 1 = 15. For each of these combinations, the corresponding symbol 54 denotes a point. But the same point is represented by 3 of these combinations of μ and ν, due to the factor of proportionality involved in the definition of points. So, we get the following scheme: A : (001) , (00a) , (00b) B : (010) , (0a0) , (0b0) C : (011) , (0aa) , (0bb) D : (01a) , (0b1) , (0ab) E : (01b) , (0ba) , (0a1) Therefore, a line so defined, contains the 15 ÷ 3 = 5 points A,B,C,D,E. It is obvious that any two points on the line may be used in this way to define the same line. The 21 lines are those given in the following scheme and the letters in a given column denoting a line: A A A A A B B B B C C C C D D D D E E E E B F J N R F G H I F G H I F G H I F G H I C G K O S J K L M K J M L L M J K M L K J D H L P T N O P Q P Q N O Q P O N O N Q P E I M Q U R S T U U T S R S R U T T U R S Let us take the noncolliner points A, B, F, then A◦(B◦F) = A◦{F, J, N, R} = ={F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U}. (A◦B)◦F = {B, C, D, E}◦F = {F, J, N, R, K, P, U, L, Q, S, M, O, T}. Then, it follows that (A◦B)◦F  A◦(B◦F). Also, for the hyperoperation (∙) we proved that ∣x∙(y∙z)∣= ∣(x∙y)∙z∣ = p 2n +p n +1 and since the set V = PG(2, 22) consists of 21 points, it follows that x∙(y∙z)= (x∙y)∙z = V for every x,y,zV. 5. On a dual Hv-ring of finite order Working on dual Hv-rings (H,+,), one needs to prove not only the weak distributivity of () with respect to (+) but also the weak distributivity of (+) with respect to (). 55 Since the set V is now equipped with the hyperoperations () and (◦) mentioned above, the next propositions 12 to 19 serve the above purpose. Similarly, as in proposition 2, we can prove that: Proposition 12. For every noncollinear x,y,zV ∣(x∙y)∙(x∙z)∣ = ∣(x∙z)∙(y∙z)∣= p 2n +p n +1 Following a similar procedure, as in proposition 3 and according to the propositions 2 and 12, we get the next proposition: Proposition 13. For every x,y,zV x∙(y∙z) = (x∙y)∙(x∙z) and (x∙y)∙z = (x∙z)∙(y∙z) Proposition 14. For every noncolliner x,y,zV, ∣x∙(y◦z)∣= p 2n +1 and ∣(x∙y)◦(x∙z)∣= p 2n +p n +1 Proof. First we consider the set x∙(y◦z). Since the point y does not belong to the line y◦z, we get that ∣y◦z∣ = p n . Also, for every wy◦z we get that ∣x∙w∣=p n +1. Since the point x appears p n times in the set x∙(y◦z), we have ∣x∙(y◦z)∣=(p n +1) p n - p n +1 = p 2n +1. Consider now the set (x∙y)◦(x∙z). Each of the lines x∙y and x∙z contains p n +1 points, having in common only the point x, since the points x,y,z are noncollinear. Then, we get the following: i) x◦x = x, by definition. ii) Due to the hyperoperation (◦), the point xx∙y together with the rest p n points of the line x∙z create the p n points of the line x∙z. iii) Due to the hyperoperation (◦), the point yx∙y together with the p n +1 points of the line x∙z, create (p n +1)-2 = p n -1 points each time. Indeed, the point y does not participate (by definition) and the point wx∙z (which appears due to the hyperoperation y◦w), already exists due to the hyperoperation x◦w of the case (ii). 56 iv) Also, the point y(x∙y)◦(x∙z). Indeed, since there exists wx∙y such that yw◦x (where xx∙z). So, from the above 4 cases we get that: ∣(x∙y)◦(x∙z)∣= 1+ p n +( p n -1)( p n +1)+1 = p 2n +p n +1.■ In a similar way, we get the following proposition: Proposition 15. For every noncolliner x,y,zV ∣(x◦y)∙z∣= p 2n +1 and ∣(x∙z)◦(y∙z)∣= p 2n +p n +1 Proposition 16. For every noncolliner x,y,zV, ∣x◦(y∙z)∣=∣(x◦y)∙(x◦z)∣= p 2n +p n Proof. First, consider the set x◦(y∙z). The line y∙z consists of p n +1 points. The point x (due to the hyperoperation (◦)) together with the points of the line y∙z creates each time (p n +1) p n = p 2n +p n points, since x is not participating. Consider now, the set (x◦y)∙(x◦z). Each of the lines x◦y and x◦z contains p n different points, since the point x is not participating. Then, we get the following: i) The point yx◦y (due to the hyperoperation (∙)), together with every wx◦z creates p n +1 points each time, but since the point y appears p n times we get that the number of points in this case is (p n +1) p n - p n +1 = p 2n +1. ii) The point w(x◦y)-{y} (due to the hyperoperation (∙)), together with the point wx◦z creates each time the point w, since the point w already exists from case (i). The number of those w’s is p n -1. Then, the set (x◦y)∙(x◦z) consists of (p 2n +1)+(p n -1) = p 2n +p n points. As we mentioned, x is the only point which is not participating.■ Similarly, we get the following two propositions: Proposition 17. For every noncolliner x,y,zV 57 ∣(x∙y)◦z∣= p 2n and ∣(x◦z)∙(y◦z)∣= p 2n +p n +1 Proposition 18. For every noncollinear x,y,zV ∣(x◦y)◦(x◦z)∣ = p 2n and ∣(x◦z)◦(y◦z)∣= p 2n +p n +1 Following a similar procedure as above and according to the proposition 18 we get the next proposition: Proposition 19. For every x,y,zV x◦(y◦z) = (x◦y)◦(x◦z) and (x◦y)◦z  (x◦z)◦(y◦z) Proposition 20. The hyperstructure (V,▫,▪), where ▫,▪{∙,◦}, is a dual Hv-ring. Proof. There are four hyperstructures: (V,∙,∙) , (V,◦,◦) , (V,∙,◦) , (V,◦,∙). The hyperoperations (∙), (◦) (by propositions 3 and 8 respectively) are satisfying the reproduction axiom. The hyperoperation (∙) (by proposition 3) is associative and the hyperoperation (◦) (by proposition 8) is weak associative. Now, for the distributivity or the weak distributivity of (▪) with respect to (▫) we have the following cases: By proposition 13: x∙(y∙z) = (x∙y)∙(x∙z) and (x∙y)∙z = (x∙z)∙(y∙z) for every x,y,zV. By proposition 19: x◦(y◦z) = (x◦y)◦(x◦z) and (x◦y)◦z  (x◦z)◦(y◦z)≠  x,y,zV. Following a similar procedure as for the distributivity of the above hyperoperations and taking into account the propositions 16, 17, 18, 19 we get that: x◦(y∙z)=(x◦y)∙(x◦z) for every x,y,zV. On the right side, (x∙y)◦z  (x◦z)∙(y◦z) is valid, which means that (x∙y)◦z  (x◦z)∙(y◦z)≠ for every x,y,zV. Also, x∙(y◦z)  (x∙y)◦(x∙z), which means that 58 x∙(y◦z)  (x∙y)◦(x∙z) ≠ for every x,y,zV. Finally, on the right hand side (x◦y)∙z  (x∙z)◦(y∙z), which means that (x◦y)∙z  (x∙z)◦(y∙z) ≠ for every x,y,zV. So, the hyperstructure (V,▫,▪), where ▫,▪{∙,◦}, is a dual Hv-ring.■ References [1] Carmichael R.: Introduction to the theory of Groups of Finite Order, Dover Publications, Inc., New York. [2] Corsini P. : Prolegomena of hypergroup theory, Second edition, Aviani (1993). [3] Corsini P. – Leoreanu V. : Applications of Hyperstructures Theory, Kluwer Academic Publishers, Boston / Dordrecht / London. 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