# nLab protoring

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A protoring is a commutative monoid $(M, 0, +)$ with a strict linear order $\lt$ such that

• $a \lt b$ if and only if $a + c \lt b + c$.

• The commutative subsemigroup of strictly positive elements $M_+ \coloneqq \{x \in M \vert 0 \lt x\}$ form a semiring $(M, +, 1, \cdot)$.

• $a \lt b$ if and only if $a \cdot c \lt b \cdot c$ and $c \cdot a \lt c \cdot b$.