[Paper Review] Classification of doubly distributive skew hyperfields and stringent hypergroups

Nathan Bowler, Ting Su|arXiv (Cornell University)|Mar 8, 2020

Rings, Modules, and Algebras9 references4 citations

TL;DR

This paper classifies doubly distributive skew hyperfields by first classifying stringent hypergroups—where addition yields a singleton unless elements are additive inverses—showing that every such hyperfield is a quotient of a skew field. The classification reveals that all doubly distributive skew hyperfields arise from a wedge sum construction over the Krasner hyperfield, sign hyperfield, or a skew field, with associated semirings corresponding to supertropical, symmetrized (max,+), or linearized (max,+) structures.

ABSTRACT

A hypergroup is stringent if $a \boxplus b$ is a singleton whenever $a eq -b$. A hyperfield is stringent if the underlying additive hypergroup is. Every doubly distributive skew hyperfield is stringent, but not vice versa. We present a classification of stringent hypergroups, from which a classification of doubly distributive skew hyperfields follows. It follows from our classification that every such hyperfield is a quotient of a skew field.

Motivation & Objective

• To classify all doubly distributive skew hyperfields by analyzing their underlying hypergroup structure.
• To establish that stringent hypergroups—where a⊕b is a singleton when a≠−b—form a foundational class for such hyperfields.
• To show that every doubly distributive skew hyperfield is a quotient of a skew field, extending known results in tropical geometry and matroid theory.
• To derive the associated semirings of these hyperfields and identify them as supertropical, symmetrized (max,+), or linearized (max,+) semirings.
• To provide a structural framework for matroids over hyperfields by characterizing the algebraic properties of doubly distributive skew hyperfields.

Proposed method

• Classify stringent hypergroups using a wedge sum construction over a totally ordered group G and hypergroups F_g indexed by G, with a projection ψ from F_g^× to G.
• Define the wedge sum F = ⋁_{g∈G} F_g with hyperaddition determined by the order on G and the hypergroup operations on each F_g.
• Prove that every doubly distributive skew hyperfield is stringent, using double distributivity to force singleton addition when elements are not additive inverses.
• Construct the associated semiring ⟨H⟩ from the power set of H, with operations ⊕ and ⊙ extended from hyperfield addition and multiplication.
• Analyze three cases based on the base hyperfield: Krasner, sign hyperfield, or a skew field—each yielding a distinct type of semiring (supertropical, symmetrized (max,+), or linearized (max,+)).
• Use valuation theory to show that ν⁻¹(1_G) is either the Krasner hyperfield, sign hyperfield, or a skew field, implying finite doubly distributive hyperfields are limited to these.

Experimental results

Research questions

• RQ1 What structural properties characterize doubly distributive skew hyperfields, and how do they relate to stringent hypergroups?
• RQ2 Can every doubly distributive skew hyperfield be realized as a quotient of a skew field?
• RQ3 What are the associated semirings of doubly distributive hyperfields, and how do they relate to known algebraic structures in tropical geometry?
• RQ4 How do the three canonical cases—Krasner, sign hyperfield, and skew field—determine the form of the resulting hyperfields and their semirings?
• RQ5 What role does stringency play in the classification, and why is it insufficient to characterize all doubly distributive hyperfields?

Key findings

• Every doubly distributive skew hyperfield is a quotient of a skew field, a key structural result with implications for matroid theory and tropical geometry.
• The only finite doubly distributive hyperfields are the Krasner hyperfield, the sign hyperfield, and finite fields, due to the infinite nature of nontrivial ordered groups.
• All doubly distributive skew hyperfields arise from a wedge sum construction F = ⋁_{g∈G} F_g, where each F_g is a Krasner hyperfield, sign hyperfield, or skew field, and G is a totally ordered group.
• The associated semiring ⟨H⟩ of any doubly distributive hyperfield H is isomorphic to one of three types: supertropical, symmetrized (max,+), or linearized (max,+), depending on the base hyperfield.
• For the Krasner base, ⟨H⟩ is the supertropical semiring; for the sign hyperfield, it is the symmetrized (max,+) semiring; and for a field base, it is the linearized (max,+) semiring.
• The valuation ν on any doubly distributive skew hyperfield satisfies ν⁻¹(1_G) ∈ {Krasner, sign hyperfield, skew field}, confirming a deep algebraic constraint on such structures.