## Abelian lattice-ordered groups

Abbreviation: AbLGrp

### Definition

An \emph{abelian lattice-ordered group} (or abelian $\ell$\emph{-group}) is a lattice-ordered group $\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ such that

$\cdot$ is commutative: $x\cdot y=y\cdot x$

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be $\ell$-groups. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a homomorphism: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\cdot y)=f(x)\cdot f(y)$.

Remark: It follows that $f(x\wedge y)=f(x)\wedge f(y)$, $f(x^{-1})=f(x)^{-1}$, and $f(e)=e$

### Definition

An \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a commutative residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle$ that satisfies the identity $x\cdot(x\to e)=e$.

Remark: $x^{-1}=x\to e$ and $x\to y=x^{-1}y$

### Examples

$\langle\mathbb{Z}, \mbox{max}, \mbox{min}, +, -, 0\rangle$, the integers with maximum, minimum, addition, unary subtraction and zero. The variety of abelian $\ell$-groups is generated by this algebra.

### Basic results

The lattice reducts of (abelian) $\ell$-groups are distributive lattices.

### Properties

Classtype variety decidable decidable hereditarily undecidable 1) 2) no yes (see lattices) yes yes, $n=2$ (see groups) yes, (see groups) yes, (see groups) yes no 3)

None

### References

1) Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups}, Algebra i Logika Sem., \textbf{6}, 1967, 45–62
2) Stanley Burris, \emph{A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups}, Algebra Universalis, \textbf{20}, 1985, 400–401, http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/HerUndecLOAG.pdf
3) Mona Cherri and Wayne B. Powell, \emph{Strong amalgamation of lattice ordered groups and modules}, International J. Math. & Math. Sci., Vol 16, No 1 (1993) 75–80, http://www.hindawi.com/journals/ijmms/1993/405126/abs/ doi:10.1155/S0161171293000080