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partially_ordered_monoids [2010/07/29 15:46] external edit
partially_ordered_monoids [2016/11/26 15:55] (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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$h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$,
$x\le y\Longrightarrow h(x)\le h(y)$ $x\le y\Longrightarrow h(x)\le h(y)$
-A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle 
-...\rangle$ such that 
-$...$ is ...:  $axiom$ 
-$...$ is ...:  $axiom$ 
====Examples==== ====Examples====
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====Basic results==== ====Basic results====
 +Every monoid with the discrete partial order is a po-monoid.
====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} $\begin{array}{lr}
  f(1)= &1\\   f(1)= &1\\
-  f(2)= &\\ +  f(2)= &4\\ 
-  f(3)= &\\ +  f(3)= &37\\ 
-  f(4)= &\\+  f(4)= &549\\
  f(5)= &\\   f(5)= &\\
-  f(6)= &\\ 
-  f(7)= &\\ 
-  f(8)= &\\ 
-  f(9)= &\\ 
-  f(10)= &\\ 
\end{array}$ \end{array}$
====Subclasses==== ====Subclasses====
-  [[Commutative partially ordered monoids]]+[[Commutative partially ordered monoids]]
-  [[Lattice-ordered monoids]] expanded type+[[Lattice-ordered monoids]] expanded type
====Superclasses==== ====Superclasses====
-  [[Partially ordered semigroups]] reduced type+[[Partially ordered semigroups]] reduced type