## Lattice-ordered groups

Abbreviation: LGrp

### Definition

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

$\langle L, \vee, \wedge\rangle$ is a lattice

$\langle L, \cdot, ^{-1}, e\rangle$ is a group

$\cdot$ is order-preserving: $x\leq y\Longrightarrow uxv\leq uyv$

Remark: $xy=x\cdot y$, $x\leq y\Longleftrightarrow x\wedge y=x$ and $x\leq y\Longleftrightarrow x\vee y=y$

### Definition

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

$\langle L,\vee\rangle$ is a semilattice

$\langle L,\cdot,^{-1},e\rangle$ is a group

$\cdot$ is join-preserving: $u(x\vee y)v=uxv\vee uyv$

Remark: $x\wedge y=( x^{-1}\vee y^{-1}) ^{-1}$

### Definition

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

Remark: $x^{-1}=e/x=x\backslash e$, $x/y=xy^{-1}$ and $x\backslash y=x^{-1}y$

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be $\ell$-groups. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\to M$ that is a homomorphism: $f(x\vee y)=f(x)\vee f(y)$, $f(x\wedge y)=f(x)\wedge f(y)$, $f(x\cdot y)=f(x)\cdot f(y)$, $f(x^{-1})=f(x)^{-1}$, and $f(e)=e$.

### Examples

$\langle Aut(\mathbf{C}),\mbox{max},\mbox{min},\circ,^{-1},id_{\mathbf{C}}\rangle$, the group of order-automorphisms of a Chains $\mathbf{C}$, with $\mbox{max}$ and $\mbox{min}$ (applied pointwise), composition, inverse, and identity automorphism.

### Basic results

The lattice reducts of lattice-ordered groups are distributive lattices.

### Properties

Classtype variety decidable1) undecidable2) hereditarily undecidable3) 4) yes, see lattices yes, $n=2$, see groups yes, see groups yes, see groups no no

None

### References

1) W. Charles Holland, Stephen H. McCleary, \emph{Solvability of the word problem in free lattice-ordered groups}, Houston J. Math., \textbf{5} 1979, 99–105 MRreviewZMATH [http://www.chapman.edu/~jipsen/lgroups/lgroupDecisionProc.html implementation]
2) A. M. W. Glass, Yuri Gurevich, \emph{The word problem for lattice-ordered groups}, Trans. Amer. Math. Soc., \textbf{280} 1983, 127–138 MRreviewZMATH
3) Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups}, Algebra i Logika Sem., \textbf{6}, 1967, 45–62
4) 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