nLab totally ordered abelian group

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.

Related concepts

• abelian group

• ordered abelian group

• total order

• lattice-ordered abelian group

• totally preordered abelian group?

• totally ordered ring

References

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