nLab lattice-ordered abelian group

Idea

A lattice-ordered abelian group or l-group is an ordered abelian group whose order forms a lattice. (Here we assume that lattices do not need to have top or bottom elements, because otherwise the only such object is the trivial group.)

Definition

Classical definition

A lattice-ordered abelian group or abelian l-group is an ordered abelian group $(G,\le)$ such that every pair of elements $a,b \in G$ admits a meet $a \wedge b$ in the underlying poset (Gilmer 1992, p. 158).

With the join operation

A lattice-ordered abelian group or abelian l-group is an abelian group $G$ with a binary join operation $(-)\vee(-)\colon G \times G \to G$ 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 lattice-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 lattice-ordered abelian groups.

• An example of a lattice-ordered abelian group that is not totally ordered is the abelian group of Gaussian integers with $ramp(1) \coloneqq 1$ and $ramp(i) \coloneqq i$.

• Let $R$ be an integral domain, $F(R)$ its field of fraction and $U(R)$ its group of units. The group of divisibility of $R$ (Gilmer 1992, p.174), denoted $D(R)$, is defined as

  $$D(R) = \frac{F(R) \backslash \{0\}}{U(R)}$$

  The abelian group $D(R)$ becomes an ordered abelian group if we define an order as follows: for every $rU(R),sU(R) \in D(R)$, we say that $rU(R) \le sU(R)$ iff $sr^{-1} \in R$.

Proposition

[Gilmer 1992, p.174] Let $R$ be an integral domain. Then $D(R)$ is a lattice-ordered abelian group iff $R$ is a GCD domain.

Jaffard-Ohm-Kaplansky theorem

Theorem

[Gilmer 1992, p.215] Every lattice-ordered abelian group $G$ is order-isomorphic to the group of divisibility of a Bézout domain i.e. there exists a Bézout domain $R$ and a group isomorphism $f:G \rightarrow D(R)$ such that $g \le h \Leftrightarrow f(g) \le f(h)$.

Related concepts

• abelian group

• ordered abelian group

• lattice

• totally ordered abelian group

• prelattice-ordered abelian group?

• lattice-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)

• Robert Gilmer: Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics (1992)