Table of Contents
Representable lattice-ordered groups
Abbreviation: RLGrp
Definition
A representable lattice-ordered group (or representable $\ell$-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, Hereditary undecidability of a class of lattice-ordered Abelian groups,
Algebra i Logika Sem.,
6, 1967, 45–62
2)
Stanley Burris, A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups,
Algebra Universalis,
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,
Non-amalgamation of ordered groups,
Math. Proc. Camb. Phil. Soc. 95 (1984), 191–195
4)
Mona Cherri and Wayne B. Powell,
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
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