Derived Hyperstructures from Hyperconics

Abstract

In this paper, we introduce generalized quadratic forms and hyperconics over quotient hyperfields as a generalization of the notion of conics on fields. Conic curves utilized in cryptosystems; in fact the public key cryptosystem is based on the digital signature schemes (DLP) in conic curve groups. We associate some hyperoperations to hyperconics and investigate their properties. At the end, a collection of canonical hypergroups connected to hyperconics is proposed.

A Comparison of Complete Parts on m-Idempotent Hyperrings

“…In Reference [12], it is shown that the four fundamental relations defined on hyperrings are not equal in general, but for all m-idempotent Krasner hyperrings, it holds Γ = ε m = ξ m = α. Moreover, it is proved that ξ m = α on m-idempotent hyperrings satisfying relation (2), which states that the relation ξ * m is a new representation for the α * -relation on m-idempotent hyperrings satisfying relation (2).…”

“…The quotient R/α * is always a commutative ring [15], while the quotient R/ξ * m is not commutative in general [12]. Actually, if (R, +, •) is an m-idempotent hyperring satisfying relation (2), then ξ * m is the smallest strongly regular equivalence relation on R such that the quotient R/ξ * m is a commutative ring. In Reference [12], it is shown that the four fundamental relations defined on hyperrings are not equal in general, but for all m-idempotent Krasner hyperrings, it holds Γ = ε m = ξ m = α.…”

“…Theorem 1. Let R an m-idempotent hyperring (for 2 ≤ m ∈ N) satisfying condition (2). Then a nonempty subset M of R is a ξ m -part if and only if M is an α-part of R.…”

“…Besides, a Krasner hyperring is also known as an additive hyperring. There are also other types of hyperrings [2] and the most general one is the so called general hyperring, introduced by Vougiouklis [3], where both addition and multiplication are hyperoperations. A short review on the historical part, terminology and the importance of hyperrings is presented by Massouros [4] or Nakassis [5] in their expository papers.…”

Regular Parameter Elements and Regular Local Hyperrings

“…If both, the addition and the multiplication, are hyperoperations with the additive part being a canonical hypergroup, then we have superrings [5], which were introduced by Mittas in 1973 [6]. Until now, the most well known and studied type of hyperrings is the Krasner hyperring, that has a plentitude of applications in algebraic geometry [7,8], tropical geometry [9], theory of matroids [10], schemes theory [11], algebraic hypercurves [12,13], hypermomographies [14]. In addition, the theory of hypermodules was extensively investigated by Massouros [15].…”