nLab pseudolattice ordered abelian group

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 $G$ with a binary join operation $(-)\vee(-):G \times G \to G$ such that $(G, \vee)$ is a commutative idempotent semigroup, and

• for all $a \in G$, $b \in G$, $c \in G$, $a \vee b = b$ implies that $(a + c) \vee (b + c) = b + c$ and $(c + a) \vee (c + b) = c + b$

The meet is defined as

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

the ramp function is defined as

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

and the absolute value is defined as

$$\vert a \vert \coloneqq a \vee -a$$

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

With the ramp function

The following algebraic definition is from Peter Freyd:

A pseudolattice ordered abelian group or l-group is an abelian group $G$ with a function $ramp:G \to G$ such that for all $a$ and $b$ in $G$,

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

and

$$ramp(a - ramp(b)) = ramp(ramp(a) - ramp(b))$$

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

$$a \vee b \coloneqq a + ramp(b - a)$$

the meet $(-)\wedge(-):G \times G \to G$ 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 $a \leq b$ if $ramp(a - b) = 0$.

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 $ramp(1) \coloneqq 1$ and $ramp(i) \coloneqq i$.

Related concepts

• abelian group

• ordered abelian group

• pseudolattice

• totally ordered abelian group

• pseudolattice ordered ring

References

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

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)