Hyperhomographies on Krasner Hyperfields

Abstract

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 H v -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.

Derived Hyperstructures from Hyperconics

“…Vougiouklis [39]. The hyperstructure (H, •) is called an H v −group if x • H = H = H • x, and also the weak associativity condition holds, that is [13,14] the authors have investigated some hyperoperations denoted by• and on some main classes of curves; elliptic curves and homographics over Krasner's hyperfields. In the following, we study them on hyperconic.…”

“…(i) For connecting it to number theory, incidence geometry, and geometry in characteristic one [8][9][10]. (ii) For connecting it to tropical geometry, quadratic forms [11,12] and real algebraic geometry [13,14]. (iii) For relating it to some other objects see [15][16][17][18][19].…”

“…Now let g(x, y) = ax 2 + bxy + cy 2 + dx + ey + f ∈ F[x, y] and g(x, y) = 0 be the quadratic equation of two variables in field of F, if a = c = 0 and b = 0 then the equation g(x, y) = 0 is called homographic transformation. In [14] Vahedi et. al extended this particular quadratic equation on Krasner's quotient hyperfield F G .…”

“…al extended this particular quadratic equation on Krasner's quotient hyperfield F G . The motivation of this paper goes in the same direction of [14]. If in the general form of the equation of quadratic form one suppose that ae = 0 and b = 0 then initiate an important quadratic equation which is called a conic.…”

“…Until now the study of conic curves has been on fields. At the recent works the authors have investigated some main classes of curves; elliptic curves and homographics over Krasner's hyperfields (see [13,14]). In the present work, we study the conic curves over some quotients of Krasner's hyperfields.…”