nLab lattice-ordered ring




(0,1)-Category theory



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.


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 RR 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 RR with a function ramp:RRramp:R \to R such that for all aa and bb in GG,

a=ramp(a)ramp(a) a = ramp(a) - ramp(-a)
ramp(aramp(b))=ramp(ramp(a)ramp(b)) ramp(a - ramp(b)) = ramp(ramp(a) - ramp(b))

and the following Horn clause:

ramp(a)=aramp(b)=bramp(ab)=ab ramp(a) = a \wedge ramp(b) = b \vdash ramp(a b) = a b

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

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

aba+ramp(ba) a \vee b \coloneqq a + ramp(b - a)

the meet ()():R×RR(-)\wedge(-):R \times R \to R is defined as

abaramp(ab) a \wedge b \coloneqq a - ramp(a - b)

and the absolute value is defined as

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

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


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


Last revised on February 23, 2024 at 19:52:00. See the history of this page for a list of all contributions to it.