# 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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