On Some Additive Properties of Multiplicative Subsemigroups of Semirings and Arithmetic Applications I

TL;DR

This paper explores additive properties of multiplicative subsemigroups in semirings, developing a process to analyze sum-keeping subsets, and applies these ideas to classical problems in additive number theory with new, simplified proofs.

Contribution

It introduces a novel elementary process for analyzing sum-keeping subsets in semirings and applies it to provide simpler proofs of classical number theory results.

Findings

Developed a collapse process for sum-keeping retractions

Provided new, simpler proofs for classical additive number theory results

Analyzed additive properties of multiplicative subsemigroups in semirings

Abstract

In this paper, we consider a question of sum-keeping about a multiplicative subsemigroup and its generator subsets in a semiring, and develop some elementary (collapse) process of the sum-keeping retraction through subsets until one minimal generators subset. As an application, we study and analyze several classical problems in additive number theory on the semiring of non-negative integers by this algebraic and combinatory idea, and provide new proofs in more simple and direct way for several classical results in number theory. Some further questions are also presented and discussed.