nLab
pseudolattice ordered ring

Contents

Context

Algebra

(0,1)-Category theory

Contents

Idea

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

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 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)

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

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

The order relation is defined as in all pseudolattices: aba \leq b if a=aba = 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 June 24, 2021 at 10:32:01. See the history of this page for a list of all contributions to it.