nLab lattice-ordered abelian group

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Context

Algebra

(0,1)-Category theory

Contents

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,)(G,\le) such that every pair of elements a,bGa,b \in G admits a meet aba \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 GG with a binary join operation ()():G×GG(-)\vee(-)\colon 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 lattice-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 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)1ramp(1) \coloneqq 1 and ramp(i)iramp(i) \coloneqq i.

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

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

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

Proposition

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

Jaffard-Ohm-Kaplansky theorem

Theorem

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

References

Last revised on June 21, 2024 at 03:59:23. See the history of this page for a list of all contributions to it.