−Table of Contents
Representable lattice-ordered groups
Abbreviation: RLGrp
Definition
A \emph{representable lattice-ordered group} (or \emph{representable} \emph{-group}) is a lattice-ordered group that satisfies the identity
Morphisms
Let and be -groups. A morphism from to is a function that is a homomorphism: and .
Remark: It follows that , , and
Examples
Basic results
Every representable -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, (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