Table of Contents
Abelian ordered groups
Abbreviation: AoGrp
Definition
An \emph{abelian ordered group} is an ordered group $\mathbf{A}=\langle A,+,-,0,\le\rangle$ such that
$+$ is commutative: $x+y=y+x$
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be abelian ordered groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an orderpreserving homomorphism: $h(x + y)=h(x) + h(y)$ and $x\le y\Longrightarrow h(x)\le h(y)$.
Examples
Example 1: $\langle\mathbb Z,+,-,0,\le\rangle$, the integers with the usual ordering.
Basic results
Every ordered group with more than one element is infinite.
Properties
| Classtype | universal |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | decidable |
| First-order theory | |
| Locally finite | no |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
Finite members
None
