Addition theorems in partially ordered groups

TL;DR

This work generalizes Shnirel'man density and addition theorems from integers to partially ordered abelian and nonabelian groups. It introduces a flexible density $\sigma_{\mathcal{J}}$ built from downward-closed finite subsets to capture order-theoretic structure, proving a nonabelian analogue of Shnirel'man's inequality $\sigma_{\mathcal{J}}(AB) \ge \alpha+\beta-\alpha\beta$ and deriving basis results in broad settings. The paper then specializes to lattice points, obtaining lattice-analogous addition theorems in $\mathbf{Z}^n$ with rectangular order and showing that $\sigma_{\mathcal{J}}(A)>0$ forces $A$ to be a basis for $\mathbf{N}_0^n\setminus\{\mathbf{0}\}$. These results unify and extend classical number-theoretic density arguments to new algebraic contexts and point to several open problems, including generalizations of Mann and Dyson-type theorems in partially ordered groups.

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.

Theorems & Definitions (25)

• Theorem 1
• proof
• Theorem 2: Shnirel'man's inequality
• proof
• Theorem 3: Shnirel'man's basis theorem
• Theorem 4
• proof
• Theorem 5: Szpilrajn szpi30
• Theorem 6
• proof