# nLab pseudolattice ordered abelian group

Contents

### Context

#### Algebra

higher algebra

universal algebra

(0,1)-category

(0,1)-topos

# Contents

## Idea

A pseudolattice ordered abelian group is an ordered abelian group whose order forms a pseudolattice.

## Definition

The following algebraic definition is from Peter Freyd:

A pseudolattice ordered abelian group is an abelian group $G$ with a function $ramp:G \to G$ such that for all $a$ and $b$ in $G$,

$a = ramp(a) - ramp(-a)$

and

$ramp(a - ramp(b)) = ramp(ramp(a) - ramp(b))$

The join $(-)\vee(-):G \times G \to G$ is defined as

$a \vee b \coloneqq a + ramp(b - a)$

and the meet $(-)\wedge(-):G \times G \to G$ is defined as

$a \wedge b \coloneqq a - ramp(a - b)$

The order relation is defined as in all pseudolattices: $a \leq b$ if $a = a \wedge b$.

## Examples

All totally ordered abelian groups, such as the integers, the rational numbers, and the real numbers, are pseudolattice ordered abelian groups.

An example of a pseudolattice ordered abelian group that is not totally ordered is the abelian group of Gaussian integers with $ramp(1) \coloneqq 1$ and $ramp(i) \coloneqq i$.

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