Abbreviation: PoGrp

A \emph{partially ordered group} is a structure $\mathbf{G}=\langle G,\cdot,^{-1},1,\le\rangle$ such that

$\langle G,\cdot,^{-1},1\rangle$ is a group

$\langle G,\le\rangle$ is a partially ordered set

$\cdot$ is \emph{orderpreserving}: $x\le y\Longrightarrow wxz\le wyz$

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

Example 1: The integers, the rationals and the reals with the usual order.

Any group is a partially ordered group with equality as partial order.

Any finite partially ordered group has only the equality relation as partial order.

$\begin{array}{lr}

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

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

f(6)= &2\\
f(7)= &1\\
f(8)= &5\\
f(9)= &2\\
f(10)= &2\\

\end{array}$

Abelian partially ordered groups

Lattice-ordered groups expanded type

Partially ordered monoids reduced type