#
nLab
pseudolattice ordered ring

Contents
### Context

#### Algebra

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

# Contents

## Idea

A pseudolattice ordered ring is an partially ordered ring whose partial order forms a pseudolattice.

## Definition

A **psuedolattice ordered ring** is a preordered ring where the partial order $\lt$ is a pseudolattice: it has binary joins and meets.

If the relation $\leq$ is only a preorder, then the preordered ring $R$ is said to be a **pseudolattice preordered ring**.

### Essentially algebraic definition

The following essentially algebraic definition is adapted from the algebraic definition of pseudolattice ordered abelian group by Peter Freyd:

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

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

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

and the following Horn clause:

$ramp(a) = a \wedge ramp(b) = b \vdash ramp(a b) = a b$

An element $a$ in $R$ is **non-negative** if $ramp(a) = a$. The Horn clause can then be stated as multiplication of non-negative elements is non-negative.

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

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

the meet $(-)\wedge(-):R \times R \to R$ 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 in all pseudolattices: $a \leq b$ if $a = a \wedge b$.

## Examples

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

## 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 December 7, 2022 at 15:58:47.
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