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partially_ordered_groups [2010/07/29 15:46] external edit
partially_ordered_groups [2016/11/26 16:12] (current)
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$\cdot$ is \emph{orderpreserving}:  $x\le y\Longrightarrow wxz\le wyz$ $\cdot$ is \emph{orderpreserving}:  $x\le y\Longrightarrow wxz\le wyz$
-Remark: This is a template. 
-If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page. 
-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== ==Morphisms==
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$x\le y\Longrightarrow h(x)\le h(y)$ $x\le y\Longrightarrow h(x)\le h(y)$
-====Definition==== +====Examples==== 
-A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +Example 1: The integers, the rationals and the reals with the usual order.
-...\rangle$ such that+
-$...$ is ...:  $axiom$ +====Basic results====
-   +
-$...$ is ...:  $axiom$+
-====Examples==== +Any [[group]] is a partially ordered group with equality as partial order.
-Example 1: +
-====Basic results====+Any finite partially ordered group has only the equality relation as partial order.
====Properties==== ====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. 
^[[Classtype]]                        |quasivariety  | ^[[Classtype]]                        |quasivariety  |
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$\begin{array}{lr} $\begin{array}{lr}
  f(1)= &1\\   f(1)= &1\\
-  f(2)= &\\ +  f(2)= &1\\ 
-  f(3)= &\\ +  f(3)= &1\\ 
-  f(4)= &\\ +  f(4)= &2\\ 
-  f(5)= &\\+  f(5)= &1\\
\end{array}$     \end{array}$    
$\begin{array}{lr} $\begin{array}{lr}
-  f(6)= &\\ +  f(6)= &2\\ 
-  f(7)= &\\ +  f(7)= &1\\ 
-  f(8)= &\\ +  f(8)= &5\\ 
-  f(9)= &\\ +  f(9)= &2\\ 
-  f(10)= &\\+  f(10)= &2\\
\end{array}$ \end{array}$
====Subclasses==== ====Subclasses====
-  [[Abelian partially ordered groups]]+[[Abelian partially ordered groups]]
-  [[Lattice-ordered groups]] expanded type+[[Lattice-ordered groups]] expanded type
====Superclasses==== ====Superclasses====
-  [[Partially ordered monoids]] reduced type+[[Partially ordered monoids]] reduced type