Hyperhomographies on Krasner Hyperfields

In this paper, we introduce generalized homographic transformations as hyperhomographies over Krasner hyperfields.These particular algebraic hyperstructues are quotient structures of classical fields modulo normal groups. Besides, we define some hyperoperations and investigate the properties of the derived hypergroups and Hv-groups associated with the considered hyperhomographies. They are equipped hyperhomographies obtained as quotient sets of nondegenerate hyperhomographies modulo a special equivalence. Thus the symmetrical property of the equivalence relations plays a fundamental role in this constructions.


Introduction
In a recently published paper [1], the authors have initiated the study of elliptic hypercurves defined on Krasner hyperfields, generalizing the elliptic curves over fields. The main idea consists in substituting the field with a hyperfield, in particular with the associated quotient Krasner hyperfield. The power of this algebraic hyperstructure has been already used in solving different problems in affine algebraic schemes [2], theory of arithmetic functions [3], tropical geometry [4], algebraic geometry [5], etc. The quotient Krasner hyperfield is practically the quotientF = F/G of a classical field F by any normal subgroup G of the multiplicative part (F \ {0}, ·). It was introduced by Krasner in 1983 [6] and investigated from the hyperalgebraic point of view mostly by Massouros [7] around 1985. In this new environment, the definition of an elliptic curve over a field F can be naturally extended to the definition of an elliptic hypercurve over a quotient Krasner hyperfield. Besides the group operation on the set of elliptic curves is extended to a hyperoperation on a family of elliptic hypercurves. The properties of the associated hypergroup have been investigated also in relation with the Berardi's cryptographic system [8].
The study developed in this paper goes in the same direction as our recent study. This time we extend a particular quadratic equation in two variables from a field F to a Krasner hyperfield F/G. It is well known that a conic section, which is a curve obtained as the intersection between the surface of a cone and a plane, can be algebraically represented as a quadratic equation with coefficients in a filed, i.e., g(x, y) = ax 2 + bxy + cy 2 + dx + ey + f = 0. If a = c = 0 and b = 0, the equation g(x, y) = 0 models a homographic transformation. Generally, after a suitable change of variables, a homographic transformation y = ax+b cx+d , with ad − bc = 0, can be written in the form (X − A)(Y − B) = 1, where X = x, Y = 1 α y, where α = bc−ad c 2 = 0, A = − d c , and B = a αc are elements in the field F. Equivalently, a homography transformation is given by a function y = f a,b (x) = b + 1 x−a , with a, b elements in a field F. The aim of this paper is to generalize the homography transformation from the field F to the Krasner hyperfieldF. First, we generalize the reduced quadratic forms on Krasner hyperfields. This investigation leads us to introduce the notion of conic hypersection on a Krasner hyperfield. Secondly, using hyperconics, we define some hyperoperations and the associated hyperstructures give us the possibility of studying simultaneously some conics. As in our previous research paper [1], the results can be also applied in cryptography in relation with the Berardi's cryptographic system [8].

Preliminaries
We recall here some basic notions of conics and hyperstructures theory also we fix the notations used in this paper. We assign the readers to these topics in the following fundamental books [9][10][11].

Conic Sections
A conic is a plane affine curve of degree 2, defined by an irreducible polynomial g(x, y) = ax 2 + bxy + cy 2 + dx + ey + f = 0 with coefficients in a field F. Based on the number of the points at infinity (this number can be 2, 1, or 0), the irreducible conics are divided in three categories: hyperbola, parabola and ellipse. Certain sets of points on curves can form an algebraic structure, and till now it is very well known the group structure. Generally the group low on conics is defined over a field F, following the rule illustrated in Figure 1 or Figure 2. In particular, if we take O an arbitrary point on the conic, then for two arbitrary points p and q on the conic, their sum p + q is obtained as the second point of the intersection with the conic of the parallel line through O to the line joining p and q. In this case O is the identity element of the group. If we consider now that the identity element O is at infinity, then the sum p + q of two arbitrary points p and q on the conic is the image on the conic of the point obtained as intersection with the x-axis of the line passing through p and q, as shown in Figure 2.
If we take the identity element O at infinity, then the group operation is calculated as in the following Cayley table: The geometrical interpretation of the associativity of the group law is equivalent with a special case of Pascal's theorem, which is a very special case of Bezout's theorem. Theorem 1. For any conic and any six points p 1 , p 2 , ..., p 6 on it, the opposite sides of the resulting hexagram, extended if necessary, intersect at points lying on some straight line. More specifically, let L(p, q) denote the line through the points p and q. Then the points L(p 1 , p 2 ) ∩ L(p 4 , p 5 ), L(p 2 , p 3 ) ∩ L(p 5 , p 6 ), and L(p 3 , p 4 ) ∩ L(p 6 , p 1 ) lie on a straight line, called the Pascal line of the hexagon (see Figure 3).

Krasner Hyperrings and Hyperfields
In this section we briefly recall the main definitions and properties of hyperrings and hyperfields, focussing on the concept of Krasner hyperfield.
Let H be a non-empty set and P * (H) be the set of all non-empty subsets of H. Let • be a hyperoperation (or join operation) on H, that is, a function from the cartesian product H × H into P * (H).
An element e r (respectively e l ) of H is called a right identity (respectively left identity e l ) if for all a ∈ H, a ∈ a • e r (respectively a ∈ e l • a). An element e is called a two side identity, or for simplicity an identity if, for all a ∈ H, a ∈ a • e ∩ e • a. A right identity e r (resp. left identity e l ) of H is called a scalar right identity (respectively scalar left identity) if for all a ∈ H, a = a • e r (respectively a = e l • a). An element e is called a scalar identity if for all a ∈ H, a = a • e = e • a. An element a ∈ H is called a right inverse (respectively left inverse) of a in H if e r ∈ a • a , for some right identity e r in H (respectively e l ∈ a • a, for some left identity e l ). An element a ∈ H is called an inverse of a ∈ H if e ∈ a • a ∩ a • a , for some identity e in H. We denote the set of all right inverses, left inverses and inverses of a ∈ H by i r (a), i l (a), and i(a), respectively. In addition, if H has a scalar identity, and the inverse of a ∈ H exists, we indicate it by a −1 .

Definition 2.
A hypergroup H is called reversible, if the following conditions hold: (i) H has at least one identity e; (ii) every element x of H has at least one inverse, that is i(x) = ∅; (iii) x ∈ y • z implies that y ∈ x • z and z ∈ y • x, where z ∈ i(z) and y ∈ i(y). An exhaustive review for the theory of hypergroups appears in [9], while the book [12] contains a wealth of applications. The more general algebraic structure that satisfies the ring-like axioms is the hyperring. There are different kinds of hyperrings. The most general one, introduced by Vougiouklis [13], has both addition and multiplication defined as hyperoperations. If only the multiplication is a hyperoperation, then we talk about multiplicative hyperrings [14,15]. If only the addition + is a hyperoperation and the multiplication · is a usual operation, then we say that R is an additive hyperring. A special case of this type is the hyperring introduced by Krasner [6]. An exhaustive review for the theory of hyperrings appears in [16][17][18][19].

Definition 4 ([6]).
A Krasner hyperring is an algebraic structure (R, +, ·) which satisfies the following axioms: (1) (R, +) is a canonical hypergroup, i.e., (i) for every x, y, z ∈ R, x + (y + z) = (x + y) + z, (ii) for every x, y ∈ R, x + y = y + x, (iii) there exists 0 ∈ R such that 0 + x = {x} for every x ∈ R, (iv) for every x ∈ R there exists a unique element x ∈ R such that 0 ∈ x + x ; (we shall write −x for x and we call it the opposite of x.) (v) z ∈ x + y implies that y ∈ z − x and x ∈ z − y.
(3) The multiplication is distributive with respect to the hyperoperation +.
In the following we recall the first construction of a Krasner hyperfield, as a quotient structure of a classical field by a normal subgroup. Let (F, +, ·) be a field and G be a normal subgroup of with the hyperoperation and the multiplication defined by: From now on, we denoteā = aG, for all aG ∈ F G and the constructed hyperfield ( F G , ⊕, ) byF, and call it the Krasner hyperfield. Moreover, we denote the inverse ofā relative to ⊕ by ā and, forā =0, the multiplicative inverseā −1 by 1 a . Besides, we will

Hyperhomographies
In this section we define the notion of hyperhomography on a Krasner hyperfield, as a quotient structure of a classical field by a normal subgroup. Using it, we introduce some hyperoperations and investigate the properties of the associated hypergroups.

Definition 5.
LetF be the Krasner hyperfield associated with the field F and (Ā,B) ∈F 2 . Define the generalized homography transformation on F as1 ∈ (x Ā ) (ȳ B ) onF, and call it the hyperhomography relation. We call the set HĀ , x−a } is a homography, for all a ∈Ā and b ∈B.
Notice that the hyperhomography x−a , i.e., (x − a)(y − a) = 1. The classical operations on the field F have been extended to the hyperoperation ⊕ and operation onF, where bȳ x Ā we denote the hyperaddition betweenx and the opposite ofĀ with respect to the hyperoperation ⊕. Besides, since the result of a hyperoperation is a set, the equality relation in the definition of a homography is substitute by a "belongingness" relation in the definition of a hyperhomography.
Theorem 2. The relation between homographies and hyperhomographies is given by the following identity Then (xg 1 − a)(yg 2 − b) = 1 and the following implications hold: (⇒). Conversely, suppose that (x,ȳ) ∈ HĀ ,B (F), then the following implications hold, too: Thanks to Theorem 2, we call the set HĀ ,B (F) the hyperhomography on F, while the set H a,b (F) is a homography onF.
Consider now the hyperhomography and the homographies Then it follows that as stated by Theorem 2.

Definition 6.
A hyperhomography HĀ ,B (F) in F 2 is called nondegenerate, if the following conditions hold, respectively: By consequence, under the same conditions, also HĀ ,B (F) = HĀ ,B (F) is called a nondegenerate hyperhomography inF 2 .
For a nondegenerate hyperhomography in F 2 , we fix some new notations: Moreover set X = { x | x ∈ X}, where x = (x, f a,b (x)), for all (a, b) ∈Ā ×B and x ∈ X ⊆ F.
In the second case, solving the equation we get y = (ab−1)x+a(2−ab) bx+ (1−ab) . In the following, for a nondegenerate hyperhomography in F 2 , we define the lines passing through two points.
where f a,b means the formal derivative of f a,b . In addition we call L a,b ( x i , x j ) the line passing through the points x i and x j . Intuitively, for each a ∈ F, the line passing through (a, ∞) is a vertical line. In other words, (a, ∞) plays an asymptotic extension role for f a,b .
Taking two arbitrary points Using the definition of the lines L a,b ( x i , x j ) and L 0 , for x i = a = x j we have and hence x = x i • ab x j ∈ F. If x i or x j are equal to a, according to Definition 8, we have We will better illustrate the above defined notions in the following example.

Example 4.
Consider the field F = R and the homography transformation f 0,0 (x) = 1 x over F, so its graph is the hyperbola H 0,0 (R) represented below in Figure 4. Taking on H 0,0 (R) two arbitrary points x i = (x i , f 0,0 (x i )) and x j = (x j , f 0,0 (x j )), we draw the line L 0,0 ( x i , x j ) passing through x i and x j . Then x i • 00 x j = L 0 ∩ L 0,0 ( x i , x j ), where L 0 is the x-axis. Then we obtain the point x i • 00 x j = (x i • 00 x j , f 0,0 (x i • 00 x j )) on the hyperbola H 0,0 (R).
x j H 0,0 (R) Figure 4. Hyperbola H 0,0 (R) Proposition 1. Let HĀ ,B (F) be a nondegenerate hyperhomography in F 2 , and x i , x j ∈ H 2 a,b (F). Then, it follows that Proof. Based on Definition 8 and on the fact that f a,b (x) = b + 1 x−a and f a,b (x) = − 1 (x−a) 2 , by simple computations, we obtain

Remark 1. (H a,b (F),
• ab ) is a homography group, for all (a, b) ∈Ā ×B. Moreover, notice that "• ab " is the group operation on the homography H a,b (F).
On a nondegenerate hyperhomography HĀ ,B (F) in F 2 we introduce the equivalence relation "∼" by considering equip the quotient H a,b (F) with a group structure and gives us a group isomorphism (H a,b (F), • ab ) In addition, the concepts in Definition (8) can be similarly defined on H a,b (F), only by substituting a with O. Definition 9. Let HĀ ,B (F) be an equipped hyperhomography. We define the hyperoperation "•" on HĀ ,B (F) as follows.
If X = (a, ∞) or Y = (a, ∞) or Z = (a, ∞), then the associativity is obvious. If not, we have the following cases.
Case 1: |J| = 1. This means that H a,b (F) = H a ,b (F) = H a ,b (F) and we have Similarly, it holds that On the other hand we have In other words, for the six points (p 5 , p 6 ) ⊆ L 0 and therefore, by Pascal's theorem (see Theorem 1), it follows also that L a,b (p 3 , p 4 ) ∩ L a,b (p 6 , p 1 ) ⊆ L 0 , equivalently with By Definition 8 we know that where, by the associativity of the group operation "• ab ", it holds ( This leads to the equality On the other hand H a ,b (F), then the associativity holds, similarly as in the case (i).
On the other hand Therefore the hyperoperation "•" is associative.
In order to prove the reproduction axiom, we consider two cases as below: Case 1. If |Ā ×B| = 1, thenF = F and HĀ ,B (F) = H a,b (F), where a ∈Ā, b ∈B. It follows that (H a,b (F), •) is a homography group, so the reproduction axiom holds.
For a better understanding, we will explain all details in computing, for example, in the table of T the hyperproduct (1, 2) • (2, 4). For doing this, since T = H 0,1 (F), we use the function f 0,1 (x) = 1 + 1 x and the field F = Z 5 . Based on Corollary 1, we obtain and therefore, Based on Proposition 1, we have which imply that Then, for every b = 0, there is the homomorphism (H a,b (F), •) ∼ = (F * , ).
Proof. It is easy to see that (F * , ) is a hypergroup. Now, taking ν = a − b −1 , consider the bijective function ϕ : F \ {ν} −→ F * defined by ϕ(x) = bx + 1 − ab and the function ξ : is the graph of the function ϕ, thus it is the line passing through the points of (ν, 0) and (a, 1), while ξ is the map that projects the points of the hyperhomography H a,b (F) on the above mentioned line. Thus, using Proposition 1, for all Take now Π : F × F * −→ F * with Π((x, y)) = y as the projection map on the second component and define ψ : We have ψ((x, y)) = ϕ(x), for all (x, y) ∈ H a,b (F), thus ψ is a bijective map and also a homomorphism because Then, if b = 0, there is the homomorphism (H a,b (F), •) ∼ = (F, ).

Associated H v -Groups
Vougiouklis [13] introduced the notion of H v −group as a generalization of the notion of hypergroup, substituting the associativity of the hyperoperation with the weak associativity, i.e., The motivation of introducing this hyperstructure is the following one. We know that the quotient of a group with respect to a normal subgroup is a group, while the quotient of a group with respect to any subgroup is a hypergroup. Vougiouklis stated that the quotient of a group with respect to any partition of the group is an H v −group.

Conclusions
In the last few years, researchers in the hypercompositional structure theory have investigated, principally from a theoretical point of view, all types of hyperrings: general hyperrings [20], multiplicative hyperrings [15], additive hyperrings [21], superrings [22], but till now, only the Krasner hyperrings have found interesting and useful applications in number theory, algebraic geometry, scheme theory, as mentioned in the introductory part of this article. Here the authors continue the study on the research topic started in [1] about elliptic hypercurves defined on quotient Krasner hyperfield, with applications in cryptography [8]. In a similar way, the notion of a homography on a field is extended to hyperhomography over Krasner hyperfields. More exactly, considering an arbitrary field F and a normal subgroup G of its multiplicative group, we get a Krasner hyperfield F = F/G. Then the homography H a,b (F) = {(x, y) ∈ F 2 | y = f a,b (x) = b + 1 x−a }, where a, b ∈ F is naturally extended to the hyperhomography HĀ ,B (F) = {(x,ȳ) ∈F 2 |1 ∈ (x Ā ) (ȳ B )} over the hyperfield (F, ⊕, ). Besides, the group operation on a homography leads to a hyperoperation on the associated equipped hyperhomography HĀ ,B (F), that becomes a hypergroup. Then, all reversible subhypergroups of an equipped hyperhomography are characterized. In the last part of the paper, other hyperoperations are defined on hyperhomographies and their properties are investigated in connection with weak associativity.

Conflicts of Interest:
The authors declare no conflict of interest.