# Differences

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

partially_ordered_groups [2010/07/29 15:46] 127.0.0.1 external edit |
partially_ordered_groups [2016/11/26 16:12] (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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$x\le y\Longrightarrow h(x)\le h(y)$ | $x\le y\Longrightarrow h(x)\le h(y)$ | ||

- | ====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 |

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