# Differences

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

residuated_partially_ordered_monoids [2010/07/29 15:46] 127.0.0.1 external edit |
residuated_partially_ordered_monoids [2019/02/24 14:15] (current) jipsen |
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- | =====Name of class===== | + | =====Residuated partially ordered monoids===== |

- | Abbreviation: **Abbr** | + | Abbreviation: **RpoMon** |

====Definition==== | ====Definition==== | ||

- | A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle | + | A \emph{residuated partially ordered monoid} (or \emph{rpo-monoid}) is a structure $\mathbf{A}=\langle A,\le,\cdot,1,\backslash,/\rangle$ such that |

- | ...\rangle$ such that | + | |

- | $\langle A,...\rangle$ is a [[name of class]] | + | $\langle A,\le\rangle$ is a [[partially ordered set]], |

- | $op_1$ is (name of property): $axiom_1$ | + | $\langle A,\cdot,1\rangle$ is a [[monoid]] and |

- | $op_2$ is ...: $...$ | + | $\backslash$ is the left residual of $\cdot$: $x\cdot y\le z\iff y\le x\backslash z$ |

- | Remark: This is a template. | + | $/$ is the right residual of $\cdot$: $x\cdot y\le z\iff x\le z/y$. |

- | 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== | ||

- | Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: | + | Let $\mathbf{A}$ and $\mathbf{B}$ be residuated po-monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an order-preserving homomorphism: |

- | $h(x ... y)=h(x) ... h(y)$ | + | $x\le y\implies h(x)\le h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \backslash y)=h(x) \backslash h(y)$, $h(x / y)=h(x) / h(y)$. |

- | | + | |

- | ====Definition==== | + | |

- | A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle | + | |

- | ...\rangle$ such that | + | |

- | | + | |

- | $...$ is ...: $axiom$ | + | |

- | | + | |

- | $...$ is ...: $axiom$ | + | |

====Examples==== | ====Examples==== | ||

- | Example 1: | ||

====Basic results==== | ====Basic results==== | ||

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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. | 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]] |(value, see description) [(Ln19xx)] | | + | ^[[Classtype]] |order variety [(Ln19xx)] | |

^[[Equational theory]] | | | ^[[Equational theory]] | | | ||

^[[Quasiequational theory]] | | | ^[[Quasiequational theory]] | | | ||

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====Subclasses==== | ====Subclasses==== | ||

- | [[...]] subvariety | ||

- | [[...]] expansion | + | [[Commutative residuated partially ordered monoids]] |

+ | | ||

+ | [[Involutive residuated partially ordered monoids]] | ||

+ | | ||

+ | [[Residuated lattices]] | ||

====Superclasses==== | ====Superclasses==== | ||

- | [[...]] supervariety | ||

- | [[...]] subreduct | + | [[Partially ordered monoids]] |

+ | | ||

+ | [[Residuated partially ordered semigroups]] | ||

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