### Table of Contents

## Normal valued lattice-ordered groups

Abbreviation: **NVLGrp**

### Definition

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

$(x\vee x^{-1})(y\vee y^{-1}) \le (y\vee y^{-1})^2(x\vee x^{-1})^2$

##### 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 $f(e)=e$

### Examples

### Basic results

The variety of normal valued $\ell$-groups is the largest proper subvariety of lattice-ordered groups ^{1)}.

### Properties

Classtype | variety |
---|---|

Equational theory | |

Quasiequational theory | |

First-order theory | hereditarily undecidable ^{2)} ^{3)} |

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 | |

Strong amalgamation property | |

Epimorphisms are surjective |

### Finite nontrivial members

None

### Subclasses

### Superclasses

### References

^{1)}W. Charles Holland, \emph{The largest proper variety of lattice-ordered groups}, Proceedings of the AMS, \textbf{57}(1), 1976, 25–28

^{2)}Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups}, Algebra i Logika Sem., \textbf{6}, 1967, 45–62

^{3)}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