−Table of Contents
Normal valued lattice-ordered groups
Abbreviation: NVLGrp
Definition
A \emph{normal valued lattice-ordered group} (or \emph{normal valued} ℓ\emph{-group}) is a lattice-ordered group L=⟨L,∨,∧,⋅,−1,e⟩ that satisfies
(x∨x−1)(y∨y−1)≤(y∨y−1)2(x∨x−1)2
Morphisms
Let L and M be ℓ-groups. A morphism from L to M is a function f:L→M that is a homomorphism: f(x∨y)=f(x)∨f(y) and f(x⋅y)=f(x)⋅f(y).
Remark: It follows that f(x∧y)=f(x)∧f(y), f(x−1)=f(x)−1, and f(e)=e
Examples
Basic results
The variety of normal valued ℓ-groups is the largest proper subvariety of lattice-ordered groups 1).
Properties
| Classtype | variety |
|---|---|
| Equational theory | |
| Quasiequational theory | |
| First-order theory | hereditarily undecidable 2) 3) |
| 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 | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
Finite nontrivial members
None
Subclasses
Superclasses
References
1)
W. Charles Holland, \emph{The largest proper variety of lattice-ordered groups},
Proceedings of the AMS, \textbf{57}(1), 1976, 25–28
2)
Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups},
Algebra i Logika Sem.,
\textbf{6}, 1967, 45–62
3)
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
