Abelian partially ordered groups

Abbreviation: APoGrp

Definition

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

$\cdot$ is \emph{commutative}: $xy=yx$

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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 … . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \ldots y)=h(x) \ldots h(y)$

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

Finite members

$\begin{array}{lr}

f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\

\end{array}$ $\begin{array}{lr}

f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

Subclasses

[[Abelian lattice-ordered groups]] expanded type

Superclasses

[[Partially ordered groups]]
[[Abelian groups]] reduced type

References


1) F. Lastname, \emph{Title}, Journal, \textbf{1}, 23–45 MRreview

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