Algebraic Geometry Over Hyperrings

Valuations of semirings

“…5.3.2 Hyperfield valuations and analytic spaces. Our definition of hyperfield valuations provides new perspective on Berkovich's theory of analytic spaces through the theory of hyperring schemes which has been investigated by the author in [Jun15a].…”

“…By implementing the notion of hyperring schemes in [Jun15a], the above observation can be rephrased in the following way: the Berkovich spectrum M(A) of A can be identified with the set Hom(Spec T, Spec A) = Hom(A, T) of 'T-rational points' of an affine scheme Spec A, where Hom(Spec T, Spec A) is the set of morphisms of locally hyperringed spaces. In fact, one can globalize Observation 5.13.…”

On the relation between hyperrings and fuzzy rings

“…However, in the last decade that has changed. For example, Viro argues in [Vir10], [Vir11] that hyperrings are a natural structure deserving of study and in particular they should play a central role in tropical geometry, Gladki and Marshall relate hyperring theory to abstract real spectra and quadratic forms ([Mar06], [GM16]), Connes and Consani [CC11] use hyperrings in their work on the Adèle class space A K /K × over a global field K. The second author develops in [Jun15] a scheme theory based on hyperrings. Baker [Bak16] has shown that hyperfields provide an elegant tool for unifying several different flavors of matroid theory.…”

Association schemes and hypergroups

“…Note that linearly ordered hypergroups (or linearly ordered hyperfields) have been mainly used in tropical geometry for algebraic foundation of tropical geometry. See, [Vir10] or [Jun15] for more details.…”

“…In this subsection we consider congruence relations on hypergroups. The case when hypergroups are commutative was considered in [Jun15]. In general, a congruence relation is an equivalence relation which preserves algebraic structures.…”