# Differences

This shows you the differences between two versions of the page.

partially_ordered_monoids [2010/07/29 15:46]
127.0.0.1 external edit
partially_ordered_monoids [2016/11/26 15:55] (current)
jipsen
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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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$h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$,
$x\le y\Longrightarrow h(x)\le h(y)$ $x\le y\Longrightarrow h(x)\le h(y)$
-
-====Definition====
-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)= &\\
-\end{array}$-$\begin{array}{lr}
-  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

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