Abbreviation: RLGrp
A \emph{representable lattice-ordered group} (or \emph{representable} $\ell$\emph{-group}) is a lattice-ordered group $\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ that satisfies the identity
$(x\wedge y)^2 = x^2\wedge y^2$
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 $f(e)=e$
Every representable $\ell$-group is a subdirect product of totally ordered groups.
| Classtype | variety |
|---|---|
| Equational theory | |
| Quasiequational theory | |
| 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 | no 3) |
| Strong amalgamation property | no 4) |
| Epimorphisms are surjective |
None