nLab
pseudolattice ordered abelian group

Contents

Context

Algebra

(0,1)-Category theory

Contents

Idea

A pseudolattice ordered abelian group is an ordered abelian group whose order forms a pseudolattice.

Definition

The following algebraic definition is from Peter Freyd:

A pseudolattice ordered abelian group is an abelian group GG with a function ramp:GGramp:G \to G such that for all aa and bb in GG,

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

and

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

The join ()():G×GG(-)\vee(-):G \times G \to G is defined as

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

and the meet ()():G×GG(-)\wedge(-):G \times G \to G 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 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)1ramp(1) \coloneqq 1 and ramp(i)iramp(i) \coloneqq i.

References

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

Created on June 18, 2021 at 16:38:23. See the history of this page for a list of all contributions to it.