[Paper Review] Ordered algebraic structures and classification of semifields

Guillaume Tahar|arXiv (Cornell University)|Sep 20, 2017

Advanced Algebra and Logic2 references3 citations

TL;DR

This paper classifies semifields by their characteristic, showing that all semifields of characteristic $p$ are fields, those of characteristic one correspond exactly to lattice-ordered groups (e.g., Max-Plus algebras), and strict semifields of characteristic zero are either cancellative (embeddable in partially ordered rings) or quotients of cancellative ones. The key contribution is a structural reduction of semifield classification to well-understood algebraic objects.

ABSTRACT

Semifields are semirings in which every nonzero element has a multiplicative inverse. A rough classification uses the characteristic of the semifield, that is the isomorphism type of the semifield generated by the two neutral elements. For every characteristic, we provide a structure theorem that reduces the classification of semifields to the classification of better-known algebraic structures. Every semifield of characteristic $p$ is actually a field. There is an equivalence between semifields of characteristic one and lattice-ordered groups. Strict semifields of characteristic zero are quotients of cancellative semifields and there is an equivalence between concellative strict semifields and a particular class of partially ordered rings.

Motivation & Objective

• To systematically classify semifields based on their characteristic, reducing complex algebraic structures to more familiar ones.
• To resolve the classification gap for strict semifields of characteristic zero by identifying cancellative and noncancellative cases.
• To establish a structural equivalence between semifields and known algebraic objects such as lattice-ordered groups and partially ordered rings.
• To clarify the role of idempotent and non-idempotent semifields in algebraic geometry and tropical mathematics.
• To show that all strict semifields of characteristic zero arise as quotients of cancellative semifields, thus unifying their construction.

Proposed method

• Define the characteristic of a semifield as the isomorphism type of the subsemifield generated by its additive and multiplicative identities.
• Prove that semifields of characteristic $p$ are fields via the existence of additive inverses in the multiplicative group.
• Establish a one-to-one correspondence between semifields of characteristic one and lattice-ordered groups using the positive cone defined by $1 + x^{-1} = 1$.
• Introduce cancellative semifields as those where $x + z = y + z$ implies $x = y$, and embed them into partially ordered rings via Grothendieck group construction.
• Construct the quotient semifield $Q(F)$ from formal sums and fractions of elements in $F$, showing that every strict semifield of characteristic zero is a quotient of a cancellative one.
• Use translation-invariance in lattice-ordered groups to prove distributivity of multiplication over supremum in characteristic one semifields.

Experimental results

Research questions

• RQ1 What is the complete classification of semifields based on their characteristic?
• RQ2 How do semifields of characteristic one relate to lattice-ordered groups?
• RQ3 What is the structural role of cancellative semifields in the classification of strict semifields of characteristic zero?
• RQ4 Can every strict semifield of characteristic zero be constructed as a quotient of a cancellative semifield?
• RQ5 What is the relationship between semifields and partially ordered rings in the context of characteristic zero?

Key findings

• Semifields of characteristic $p$ are necessarily fields, as the multiplicative group being torsion-free and finite implies triviality, forcing the existence of additive inverses.
• Semifields of characteristic one are in one-to-one correspondence with lattice-ordered groups, where addition is defined as supremum and multiplication as the group operation.
• Every cancellative strict semifield of characteristic zero embeds into a partially ordered ring where the positive elements are closed under inversion and generate the ring.
• Conversely, the set of strictly positive elements in any such ring forms a cancellative strict semifield of characteristic zero.
• All strict semifields of characteristic zero are quotients of cancellative strict semifields, constructed via formal sums and fractions modulo rational scaling relations.
• Noncancellative semifields, such as the semifield of positive rational functions under asymptotic equivalence, arise as quotients of cancellative ones, demonstrating a universal construction mechanism.