# lowest upper bound

Let $S$ be a set with a partial ordering $\le $, and let $T$ be a subset of $S$. A *lowest upper bound*, or *supremum ^{}*, of $T$ is an upper bound

^{}$x$ of $T$ with the property that $x\le y$ for every upper bound $y$ of $T$. The lowest upper bound of $T$, when it exists, is denoted $\mathrm{sup}(T)$.

A lowest upper bound of $T$, when it exists, is unique.

Greatest lower bound is defined similarly: a *greatest lower bound*, or *infimum ^{}*, of $T$ is a lower bound $x$ of $T$ with the property that $x\ge y$ for every lower bound $y$ of $T$. The greatest lower bound of $T$, when it exists, is denoted $\mathrm{inf}(T)$.

If $A=\{{a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}\}$ is a finite set^{}, then the supremum of $A$ is simply $\mathrm{max}(A)$, and the infimum of $A$ is equal to $\mathrm{min}(A)$.

Title | lowest upper bound |
---|---|

Canonical name | LowestUpperBound |

Date of creation | 2013-03-22 11:52:18 |

Last modified on | 2013-03-22 11:52:18 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 13 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 06A05 |

Defines | least upper bound |

Defines | greatest lower bound |

Defines | supremum |

Defines | infimum |