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## Abelian lattice-ordered groups

Abbreviation: AbLGrp

### Definition

An abelian lattice-ordered group (or abelian $\ell$-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 abelian lattice-ordered group (or abelian $\ell$-group) is a

$\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle$ 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
Equational theory decidable
Quasiequational theory decidable
First-order theory hereditarily undecidable 1) 2)
Congruence distributive yes (see lattices)
Congruence n-permutable yes, $n=2$ (see groups)
Congruence regular yes, (see groups)
Congruence uniform yes, (see groups)

None

### References

1) Yuri Gurevic, Hereditary undecidability of a class of lattice-ordered Abelian groups, Algebra i Logika Sem., 6, 1967, 45–62 MRreview
2) Stanley Burris, A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups, Algebra Universalis, 20, 1985, 400–401 MRreview