# 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 or l-group is an ordered abelian group whose order forms a pseudolattice.

## Definition

### With the join operation

A pseudolattice ordered abelian group or l-group is an abelian group $G$ with a binary join operation $(-)\vee(-):G \times G \to G$ such that $(G, \vee)$ is a commutative idempotent semigroup, and

• for all $a \in G$, $b \in G$, $c \in G$, $a \vee b = b$ implies that $(a + c) \vee (b + c) = b + c$ and $(c + a) \vee (c + b) = c + b$

The meet is defined as

$a \wedge b \coloneqq -(-a \vee -b),$

the ramp function is defined as

$ramp(a) \coloneqq a \vee 0,$

and the absolute value is defined as

$\vert a \vert \coloneqq a \vee -a$

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

### With the ramp function

The following algebraic definition is from Peter Freyd:

A pseudolattice ordered abelian group or l-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

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

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

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

and the absolute value is defined as

$\vert a \vert \coloneqq ramp(a) + ramp(-a)$

The order relation is defined as $a \leq b$ if $ramp(a - b) = 0$.

## 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 $ramp(i) \coloneqq i$.

## References

Last revised on July 23, 2022 at 15:04:39. See the history of this page for a list of all contributions to it.