Which semifields are exact?

TL;DR

This paper characterizes semifields with a property similar to linear function extension in vector spaces, showing they are either skew fields or idempotent semirings, including tropical semirings.

Contribution

It provides a complete classification of left exact semifields, a result that is new even for tropical semirings, and addresses a problem posed by Wilding.

Findings

Semifields are left exact iff they are skew fields or idempotent semirings.

The result applies to tropical semirings, solving an open problem.

Identifies open problems for further research.

Abstract

Every (left) linear function on a subspace of a finite-dimensional vector space over a (skew) field can be extended to a (left) linear function on the whole space. This paper explores the extent to what this basic fact of linear algebra is applicable to more general structures. Semifields with a similar property imposed on linear functions are called (left) exact, and we present a complete description of such semifields. Namely, we show that a semifield $S$ is left exact if and only if $S$ is either a skew field or an idempotent semiring. In particular, our result is new even for the tropical semiring and gives a solution to the problem posed by Wilding. Also, we point out several problems that require further investigation.