Ordered abelian groups

Abbreviation: OGrp

Definition

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

$+$ is 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 is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Finite members

one-element group

Subclasses

[[Abelian ordered groups]]

Superclasses

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

References