# Differences

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commutative_residuated_partially_ordered_monoids [2010/07/29 15:46] 127.0.0.1 external edit |
commutative_residuated_partially_ordered_monoids [2019/12/12 08:00] (current) pnotthesamejipsen |
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$\cdot$ is \emph{commutative}: $xy=yx$ | $\cdot$ is \emph{commutative}: $xy=yx$ | ||

- | Remark: This is a template. | + | Remark: These algebras are also known as \emph{lineales}.[(dePaiva2005)] |

- | 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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$h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, | $h(x \cdot y)=h(x) \cdot h(y)$, $h(1)=1$, | ||

$h(x \to y)=h(x) \to h(y)$, and $x\le y\Longrightarrow h(x)\le h(y)$. | $h(x \to y)=h(x) \to h(y)$, and $x\le y\Longrightarrow h(x)\le h(y)$. | ||

- | |||

- | ====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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$\begin{array}{lr} | $\begin{array}{lr} | ||

f(1)= &1\\ | f(1)= &1\\ | ||

- | f(2)= &\\ | + | f(2)= &2\\ |

- | f(3)= &\\ | + | f(3)= &5\\ |

- | f(4)= &\\ | + | f(4)= &24\\ |

- | f(5)= &\\ | + | f(5)= &131\\ |

- | \end{array}$ | + | f(6)= &1001\\ |

- | $\begin{array}{lr} | + | |

- | f(6)= &\\ | + | |

f(7)= &\\ | f(7)= &\\ | ||

f(8)= &\\ | f(8)= &\\ | ||

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

- | [[Commutative residuated lattices]] expansion | + | [[Commutative residuated lattices]] expansion |

- | [[Pocrims]] same type | + | [[Pocrims]] same type |

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

- | [[Residuated partially ordered monoids]] supervariety | + | [[Residuated partially ordered monoids]] supervariety |

- | [[Commutative partially ordered monoids]] subreduct | + | [[Commutative partially ordered monoids]] subreduct |

====References==== | ====References==== | ||

- | [(Lastname19xx> | + | [(dePaiva2005> |

- | F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] | + | V. de Paiva, \emph{Lineales: Algebras and Categories in the Semantics of Linear Logic}, Proofs and Diagrams, CSLI Publications, Stanford, 123-142, 2005, [[https://research.nuance.com/wp-content/uploads/2014/10/Lineales-algebras-and-categories-in-the-semantics-of-Linear-Logic.pdf]])] |

- | )] | + | |

Trace: