Table of Contents
Representable lattice-ordered groups
Abbreviation: RLGrp
Definition
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
$(x\wedge y)^2 = x^2\wedge y^2$
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
Every representable $\ell$-group is a subdirect product of totally ordered groups.
Properties
| 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 |
Finite members
None
Subclasses
Superclasses
References
1)
Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups},
Algebra i Logika Sem.,
\textbf{6}, 1967, 45–62
2)
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
3)
A. M. W. Glass, D. Saracino and C. Wood,
\emph{Non-amalgamation of ordered groups},
Math. Proc. Camb. Phil. Soc. 95 (1984), 191–195
4)
Mona Cherri and Wayne B. Powell,
\emph{Strong amalgamation of lattice ordered groups and modules},
International J. Math. & Math. Sci., Vol 16, No 1 (1993) 75–80, http://www.hindawi.com/journals/ijmms/1993/405126/abs/ doi:10.1155/S0161171293000080
