Differences

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}:  $x\le y\Longrightarrow wxz\le wyz$
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-	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.
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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==		==Morphisms==
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	$h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$,		$h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$,
	$x\le y\Longrightarrow h(x)\le h(y)$		$x\le y\Longrightarrow h(x)\le h(y)$
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-	====Definition====
-	A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
-	...\rangle$ such that
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-	$...$ 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