Table of Contents
Abelian lattice-ordered groups
Abbreviation: AbLGrp
Definition
An \emph{abelian lattice-ordered group} (or abelian $\ell $\emph{-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 \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a 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 | no 3) |
| Epimorphisms are surjective |
Finite members
None
