## Normal valued lattice-ordered groups

Abbreviation: NVLGrp

### Definition

A normal valued lattice-ordered group (or normal valued $\ell$-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 $f(e)=e$

### Basic results

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

### Properties

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

None

### References

1) W. Charles Holland, The largest proper variety of lattice-ordered groups, Proceedings of the AMS, 57(1), 1976, 25–28
2) Yuri Gurevic, Hereditary undecidability of a class of lattice-ordered Abelian groups, Algebra i Logika Sem., 6, 1967, 45–62
3) Stanley Burris, A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups, Algebra Universalis, 20, 1985, 400–401, http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/HerUndecLOAG.pdf