# Differences

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tarski_algebras [2010/07/29 15:46] 127.0.0.1 external edit |
tarski_algebras [2015/06/06 11:50] (current) jipsen |
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=====Name of class===== | =====Name of class===== | ||

- | % Note: replace "Template" with Name_of_class in previous line | ||

- | Abbreviation: **Abbr** | + | Abbreviation: **TarskiA** |

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

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

- | ...\rangle$ such that | + | 2\rangle$ such that $\to$ satisfies the following identities: |

- | $\langle A,...\rangle$ is a [[name of class]] | + | $(x\to y)\to x=x$ |

- | $op_1$ is (name of property): $axiom_1$ | + | $(x\to y)\to y=(y\to x)\to x$ |

- | $op_2$ is ...: $...$ | + | $x\to(y\to z)=y\to(x\to z)$ |

- | | + | |

- | 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== | ||

- | 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 Tarski algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: |

- | $h(x ... y)=h(x) ... h(y)$ | + | $h(x \to y)=h(x) \to h(y)$ |

- | | + | |

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

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

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

- | | + | |

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

- | | + | |

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

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

- | Example 1: | + | Example 1: $\langle\{0,1\},\to\rangle$ where $x\to y=0$ iff $x=1$ and $y=0$. |

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

+ | Tarski algebras are the implication subreducts of Boolean algebras. | ||

====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. | 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]] | variety | |

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

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

Trace: