nLab lattice-ordered ring

Idea

A lattice-ordered ring is an partially ordered ring whose partial order forms a lattice. Lattices here are assumed not to have top or bottom elements, because otherwise the only such lattice-ordered ring is the trivial ring.

Definition

A lattice-ordered ring or l-ring is a preordered ring where the partial order $\lt$ is a lattice: it has binary joins and meets.

If the relation $\leq$ is only a preorder, then the preordered ring $R$ is said to be a prelattice-ordered ring.

Essentially algebraic definition

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

A lattice-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 lattices: $a \leq b$ if $a = a \wedge b$.

Examples

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

Related concepts

• ring

• ordered ring

• lattice

• totally ordered ring

• lattice-ordered abelian group

References

• Wikipedia, Lattice-ordered ring

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