We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is ...‘multi-valued’. This paper largely consists of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme (X,OX) and prove that Γ(X,OX)≃R for any integral affine hyperring scheme X=SpecR.
Generalized Centroid of Hyperrings Yazarli, Hasret; Davvaz, Bijan; Yilmaz, Damla
Discussiones mathematicae. General algebra and applications,
03/2022, Volume:
42, Issue:
1
Journal Article
Peer reviewed
Open access
In this paper, the notion of generalized centroid is applied to hyperrings. We show that the generalized centroid
of a semiprime hyperring
is a regular hyperring. Also, we show that if
is a ...hyperfield, then
is a prime hyperring.
On Refined Neutrosophic Hyperrings Ibrahim, M.A; Agboola, A.A.A; Ibrahim, Z.H ...
Neutrosophic sets and systems,
12/2021, Volume:
45
Journal Article
Peer reviewed
This paper presents the refinement of a type of neutrosophic hyperring in which +' and *' are hyperopraetions and studied some of its properties. Several interesting results and examples are ...presented. Keywords: Neutrosophic, neutrosophic hyperring, neutrosophic hypersubring, refined neutrosophic hyperring, refined neutrosophic hypersubhyperring, refined neutrosophic hyperring homomorphism.
Krasner hyperring is one of the generalizations of the classical ring in literature. In this paper, the notion of topological Krasner hyperring is introduced as a generalization of topological ring ...and variant of isomorphism theorems are studied
On Refined Neutrosophic Hyperrings M.A. Ibrahim; A.A.A. Agboola; Z.H. Ibrahim ...
Neutrosophic sets and systems,
08/2021, Volume:
45
Journal Article
Peer reviewed
Open access
This paper presents the refinement of a type of neutrosophic hyperring in which +’ and ·’ are hyperoperations and studied some of its properties. Several interesting results and examples are ...presented.
On a general hyperring, there is a fundamental relation, denoted
, such that the quotient set is a classical ring. In a previous paper, the authors defined the relation
on general hyperrings, proving ...that its transitive closure
is a strongly regular equivalence relation smaller than the
-relation on some classes of hyperrings, such that the associated quotient structure modulo
is an ordinary ring. Thus, on such hyperrings,
is a fundamental relation. In this paper, we discuss the transitivity conditions of the
-relation on hyperrings and
-idempotent hyperrings.
We introduce and study in detail the notion of compatibility between valuations and orderings in real hyperfields. We investigate their relation with valuations and orderings induced on factor and ...residue hyperfields. Much of the theory from real fields can be generalized to real hyperfields; we point out facts that cannot. We generalize the Baer-Krull theorem to real hyperfields.
The present study investigates the relation between derivations and hyperideals on ordered hyperrings with no zero divisors. Also, we identify some results for the ordered hyperrings induced by the ...homomorphism of the ordered hyperrings by derivations. The present work explores some aspects of derivations in ordered hyperrings. Also, we establish some results in connection with homomorphisms and hyperideals. Furthermore, we describe prime hyperideals associated to a derivation $d$ on an ordered hyperring $T$ and derive several results about homomorphisms and derivations on ordered hyperrings.
Totally ordered Krasner hyperrings are a special type of Krasner hyperrings that induce a total order. In this article, we use them to define (for the first time) meet plus hyperalgebra as an ...extension of max plus algebra. Moreover, we study matrices and systems of equations in meet plus hyperagebra.
Communicated by V. A. Artamonov
The α
-relation is a fundamental relation on hyperrings, being the smallest strongly regular relation on hyperrings such that the quotient structure
is a commutative ring. In this paper we introduce ...on hyperrings the relation ζ
, which is smaller than α
, and show that, on a particular class of
-idempotent hyperrings
, it is the smallest strongly regular relation such that the quotient ring
is commutative. Some properties of this new relation and its differences from the α
-relation are illustrated and discussed. Finally, we show that ζ
is a new representation for α
on this particular class of
-idempotent hyperrings.
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