Algebraic geometry over hyperrings

Abstract

Abstract. 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 consisits 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, O X ) and prove that Γ(X, O X ) ≃ R for any integral affine hyperring scheme X = Spec R.

Matroids over partial hyperstructures

“…, mu with |I| " r`1, |J| " r´1, and |IzJ| ě 3, the point p " pp I q lies on the "subvariety" of the projective space in the`m r˘h omogeneous variables x I for I P S defined by where signpi; I, Jq " p´1q s with s equal to the number of elements i 1 P I with i ă i 1 plus the number of elements j P J with i ă j. Although we will not explore this further in the present paper, when F is a hyperfield one can view the "equations" (3.23) as defining a hyperring scheme Gpr, mq in the sense of [Jun18], which we call the F -Grassmannian. In this geometric language, Theorem 3.17 says that a strong matroid of rank r on t1, .…”

An informal overview of triples and systems

“…A special case: The category of hyperfields as given in [45] can be embedded into the category of uniquely negated T -reversible systems ([64, Theorem 6.7]). Reversibility enables one to apply systems to matroid theory, although we have not yet embarked on that endeavor in earnest.…”

On the relation between hyperrings and fuzzy rings

“…In this section we briefly review the definition of a hyperring. For more details, we refer the reader to [Jun15] or [Bak16].…”