Lattice-ordered abelian group

Definition

A lattice-ordered abelian group, or abelian $\ell$ -group, is an abelian group $(\Gamma,+,0)$ with a partial order $\leq$ such that:

$(\Gamma,\leq)$ is a lattice;
the order is translation-invariant: $a\leq b\quad\Longrightarrow\quad a+c\leq b+c.$

Equivalently, translation by $c$ preserves joins and meets:

(a\vee b)+c=(a+c)\vee(b+c),\qquad (a\wedge b)+c=(a+c)\wedge(b+c).

Morphisms

An $\ell$ -group homomorphism is a group homomorphism that preserves binary joins, equivalently binary meets. An order-preserving group homomorphism need not preserve joins when the order is not total, so this extra condition matters categorically.

Total and partial orders

Every totally ordered abelian group is an abelian $\ell$ -group, with $a\vee b=\max(a,b)$ and $a\wedge b=\min(a,b)$ . The converse fails: $\mathbb Z^2$ with the coordinatewise order is a lattice-ordered abelian group in which $(1,0)$ and $(0,1)$ are incomparable.

Positive and negative parts

Writing

x^+=x\vee0,\qquad x^-=(-x)\vee0,

one has $x=x^+-x^-$ and $x^+\wedge x^-=0$ . These lattice decompositions are useful when passing between $\ell$ -groups and idempotent semifields.

References

K. R. Goodearl, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, 1986. DOI record. Relevant: Chapter 1.
Guillaume Tahar, “Ordered algebraic structures and classification of semifields,” 2017. arXiv:1709.06923. Relevant: lattice-ordered groups and characteristic-one semifields.