Abbreviation: NVLGrp
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
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
The variety of normal valued ℓ-groups is the largest proper subvariety of lattice-ordered groups 1).
| 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 |
None