On the quotient class of non-archimedean fields

TL;DR

This paper investigates the algebraic structure of the quotient class of non-archimedean fields, focusing on Minkowski sum and product, revealing their properties as regular ordered semigroups and their interrelation through a modified distributivity law.

Contribution

It introduces a detailed algebraic analysis of the quotient class operations, highlighting their semigroup structures and the adapted distributivity law, which is a novel insight.

Findings

Addition and multiplication form regular ordered semigroups.

The two operations are connected by an adapted distributivity law.

The study enhances understanding of algebraic laws in non-archimedean contexts.

Abstract

The quotient class of a non-archimedean field is the set of cosets with respect to all of its additive convex subgroups. The algebraic operations on the quotient class are the Minkowski sum and product. We study the algebraic laws of these operations. Addition and multiplication have a common structure in terms of regular ordered semigroups. The two algebraic operations are related by an adapted distributivity law.