nLab pseudolattice ordered abelian group

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Context

Algebra

(0,1)-Category theory

Contents

Idea

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

Definition

With the join operation

A pseudolattice ordered abelian group or l-group is an abelian group GG with a binary join operation ()():G×GG(-)\vee(-):G \times G \to G such that (G,)(G, \vee) is a commutative idempotent semigroup, and

  • for all aGa \in G, bGb \in G, cGc \in G, ab=ba \vee b = b implies that (a+c)(b+c)=b+c(a + c) \vee (b + c) = b + c and (c+a)(c+b)=c+b(c + a) \vee (c + b) = c + b

The meet is defined as

ab(ab), a \wedge b \coloneqq -(-a \vee -b),

the ramp function is defined as

ramp(a)a0, ramp(a) \coloneqq a \vee 0,

and the absolute value is defined as

|a|aa \vert a \vert \coloneqq a \vee -a

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

With the ramp function

The following algebraic definition is from Peter Freyd:

A pseudolattice ordered abelian group or l-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)

the meet ()():G×GG(-)\wedge(-):G \times G \to G 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 aba \leq b if ramp(ab)=0ramp(a - b) = 0.

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

Last revised on July 23, 2022 at 11:04:39. See the history of this page for a list of all contributions to it.