### 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