# nLab archimedean protoring

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A protoring $M$ is an archimedean protoring if for all terms $a, b, c, d \in M$, $c \lt d$ implies that there exists a positive natural number $n \in \mathbb{N}_+$ such that

$b + \sum_{i=1}^n c \lt a + \sum_{i=1}^n d$

where

$\sum_{i=1}^{(-)} (-):\mathbb{N}_+ \times M \to M$

is the canonical left non-unital $\mathbb{N}_+$-action for commutative semigroups? defined inductively by

$\sum_{i=1}^{1} c \coloneqq c$
$\sum_{i=1}^{n + 1} c \coloneqq c + \sum_{i=1}^{n}$