#
nLab
totally ordered abelian group

Contents
### Context

#### Algebra

#### (0,1)-Category theory

# Contents

## Idea

A totally ordered abelian group is an ordered abelian group whose order forms a total order.

## Definition

The following definition is from Peter Freyd:

A **totally ordered abelian group** is a lattice-ordered abelian group $G$ such that for all elements $a$ in $G$, $a \leq 0$ or $-a \leq 0$.

In a totally ordered abelian group, the join is usually called the **maximum**, while the meet is usually called the **minimum**

## Examples

The integers, the rational numbers, and the real numbers are totally ordered abelian groups.

## References

- Peter Freyd,
*Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

Last revised on February 23, 2024 at 19:54:44.
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