# consequence operator is determined by its fixed points

###### Theorem 1

Suppose that ${C}_{\mathrm{1}}$ and ${C}_{\mathrm{2}}$ are consequence operators on a set $L$ and that, for every $X\mathrm{\subseteq}L$, it happens that ${C}_{\mathrm{1}}\mathit{}\mathrm{(}X\mathrm{)}\mathrm{=}X$ if and only if ${C}_{\mathrm{2}}\mathit{}\mathrm{(}X\mathrm{)}\mathrm{=}X$. Then ${C}_{\mathrm{1}}\mathrm{=}{C}_{\mathrm{2}}$.

###### Theorem 2

Suppose that $C$ is a consequence operators on a set $L$. Define $K\mathrm{=}\mathrm{\{}X\mathrm{\subseteq}L\mathrm{\mid}C\mathit{}\mathrm{(}X\mathrm{)}\mathrm{=}X\mathrm{\}}$. Then, for every $X\mathrm{\in}L$, there exists a $Y\mathrm{\in}K$ such that $X\mathrm{\subseteq}Y$ and, for every $Z\mathrm{\in}K$ such that $X\mathrm{\subseteq}Z$, one has $Y\mathrm{\subseteq}Z$.

###### Theorem 3

Given a set $L$, suppose that $K$ is a subset of $L$ such that, for every $X\mathrm{\in}L$, there exists a $Y\mathrm{\in}K$ such that $X\mathrm{\subseteq}Y$ and, for every $Z\mathrm{\in}K$ such that $X\mathrm{\subseteq}Z$, one has $Y\mathrm{\subseteq}Z$. Then there exists a consequence operator $C\mathrm{:}\mathrm{P}\mathit{}\mathrm{(}L\mathrm{)}\mathrm{\to}\mathrm{P}\mathit{}\mathrm{(}L\mathrm{)}$ such that $C\mathit{}\mathrm{(}X\mathrm{)}\mathrm{=}X$ if and only if $X\mathrm{\in}K$.

Title | consequence operator is determined by its fixed points |
---|---|

Canonical name | ConsequenceOperatorIsDeterminedByItsFixedPoints |

Date of creation | 2013-03-22 16:29:48 |

Last modified on | 2013-03-22 16:29:48 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 03G10 |

Classification | msc 03B22 |

Classification | msc 03G25 |