Uniform extensions of layered semifields

TL;DR

This paper presents a canonical method for constructing simple uniform semifield extensions of layered semifields, decomposing them into bipotent and cancellative extensions, and unifying these into a comprehensive theory.

Contribution

It introduces a new canonical construction method for uniform semifield extensions, including a decomposition approach and a unifying theory for layered semifields.

Findings

Decomposition of uniform extensions into bipotent and cancellative parts

Characterization of the two types of semifield extensions

Unified theory for uniform extensions of layered semifields

Abstract

In this paper we introduce a canonical method of constructing simple uniform semifield extensions of uniform layered semifields introduced by Izhakian Knebusch and Rowen in the paper 'Layered tropical mathematics'. Our construction includes a decomposition of a uniform extension of a uniformly layered (uniform) semifield to the bipotent semifield extension of its $\nu$-values semifield and a cancellative semifields extension of its layers (sorting) semifield. We give a characterization of these two types of semifields extensions in the first two sections of the paper. The third section glues the pieces together to form a theory for a uniform extension of a uniformly layered semifield.