bottom

In a poset $P$, a **bottom** is a **least** element: an element $\bot$ of $P$ such that $\bot \leq a$ for every element $a$. Such a bottom may not exist; if it does, then it is unique.

In a proset, a bottom may be defined similarly, but it need not be unique. (However, it is still unique up the natural equivalence in the proset.)

A bottom of $P$ can also be understood as a join of zero elements in $P$.

A poset that has both top and bottom is called *bounded*.

As a poset is a special kind of category, a bottom is simply an initial object in that category.

The bottom of the poset of subsets or subobjects of a given set or object $A$ is called the empty subset or subobject. In a category (such as Set) with a strict initial object $\varnothing$, this will always serve as the bottom of any subobject poset.

**basic symbols used in logic**

$\phantom{A}$symbol$\phantom{A}$ | $\phantom{A}$meaning$\phantom{A}$ |
---|---|

$\phantom{A}$$\in$ | $\phantom{A}$element relation |

$\phantom{A}$$\,:$ | $\phantom{A}$typing relation |

$\phantom{A}$$=$ | $\phantom{A}$equality |

$\phantom{A}$$\vdash$$\phantom{A}$ | $\phantom{A}$entailment / sequent$\phantom{A}$ |

$\phantom{A}$$\top$$\phantom{A}$ | $\phantom{A}$true / top$\phantom{A}$ |

$\phantom{A}$$\bot$$\phantom{A}$ | $\phantom{A}$false / bottom$\phantom{A}$ |

$\phantom{A}$$\Rightarrow$ | $\phantom{A}$implication |

$\phantom{A}$$\Leftrightarrow$ | $\phantom{A}$logical equivalence |

$\phantom{A}$$\not$ | $\phantom{A}$negation |

$\phantom{A}$$\neq$ | $\phantom{A}$negation of equality / apartness$\phantom{A}$ |

$\phantom{A}$$\notin$ | $\phantom{A}$negation of element relation $\phantom{A}$ |

$\phantom{A}$$\not \not$ | $\phantom{A}$negation of negation$\phantom{A}$ |

$\phantom{A}$$\exists$ | $\phantom{A}$existential quantification$\phantom{A}$ |

$\phantom{A}$$\forall$ | $\phantom{A}$universal quantification$\phantom{A}$ |

$\phantom{A}$$\wedge$ | $\phantom{A}$logical conjunction |

$\phantom{A}$$\vee$ | $\phantom{A}$logical disjunction |

$\phantom{A}$$\otimes$$\phantom{A}$ | $\phantom{A}$multiplicative conjunction$\phantom{A}$ |

$\phantom{A}$$\oplus$$\phantom{A}$ | $\phantom{A}$multiplicative disjunction$\phantom{A}$ |

Last revised on July 3, 2018 at 03:05:31. See the history of this page for a list of all contributions to it.