Addition theorems in partially ordered groups

TL;DR

This paper demonstrates that classical additive number theory results like Shnirel'man's inequality extend naturally to partially ordered groups, revealing their order-theoretic nature and broadening their applicability.

Contribution

It shows that fundamental additive theorems are inherently order-theoretic and extend to partially ordered abelian and nonabelian groups, including lattice point sets.

Findings

Shnirel'man's inequality extends to partially ordered groups

Addition theorems apply to sums of lattice points

Results reveal the order-theoretic nature of additive theorems

Abstract

Shnirel'man's inequality and Shnirel'man's basis theorem are fundamental results about sums of sets of positive integers in additive number theory. It is proved that these results are inherently order-theoretic and extend to partially ordered abelian and nonabelian groups. One abelian application is an addition theorem for sums of sets of $n$-dimensional lattice points.