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## Abelian lattice-ordered groups

Abbreviation: **AbLGrp**

### Definition

An ** abelian lattice-ordered group** (or abelian $\ell $

**) is a lattice-ordered group $\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ such that**

*-group*$\cdot$ 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

**) is a**

*abelian $\ell$-group*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 |
---|---|

Equational theory | decidable |

Quasiequational theory | decidable |

First-order theory | hereditarily undecidable ^{1)} ^{2)} |

Locally finite | no |

Residual size | |

Congruence distributive | yes (see lattices) |

Congruence modular | yes |

Congruence n-permutable | yes, $n=2$ (see groups) |

Congruence regular | yes, (see groups) |

Congruence uniform | yes, (see groups) |

Congruence extension property | |

Definable principal congruences | |

Equationally def. pr. cong. | |

Amalgamation property | yes |

Strong amalgamation property | |

Epimorphisms are surjective |

### Finite members

None

### Subclasses

### Superclasses

### References

^{1)}Yuri Gurevic,

**, Algebra i Logika Sem.,**

*Hereditary undecidability of a class of lattice-ordered Abelian groups***6**, 1967, 45–62 MRreview

^{2)}Stanley Burris,

**, Algebra Universalis,**

*A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups***20**, 1985, 400–401 MRreview

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