Table of Contents

Ordered abelian groups

Abbreviation: OGrp

Definition

An \emph{ordered abelian group} is an ordered group $\mathbf{G}=\langle G,+,-,0,\le\rangle$ such that

$+$ is \emph{commutative}: $x+y=y+x$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be ordered groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism: $h(x + y)=h(x) + h(y)$, $x\le y\Longrightarrow h(x)\le h(y)$.

Definition

A \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Finite members

one-element group

Subclasses

[[Abelian ordered groups]]

Superclasses

[[Partially ordered groups]]
[[Ordered monoids]] reduced type

References