# complete lattice

## Complete lattices

A *complete lattice ^{}* is a poset $P$
such that every subset of $P$ has both a supremum

^{}and an infimum

^{}in $P$.

For a complete lattice $L$,
the supremum of $L$ is denoted by $1$,
and the infimum of $L$ is denoted by $0$.
Thus $L$ is a bounded lattice^{},
with $1$ as its greatest element and $0$ as its least element.
Moreover, $1$ is the infimum of the empty set^{},
and $0$ is the supremum of the empty set.

## Generalizations

A *countably complete lattice* is a poset $P$
such that every countable^{} subset of $P$
has both a supremum and an infimum in $P$.

Let $\kappa $ be an infinite^{} cardinal.
A $\kappa $-complete lattice is a lattice^{} $L$
such that for every subset $A\subseteq L$
with $|A|\le \kappa $, both $\bigvee A$ and $\bigwedge A$ exist.
(Note that an ${\mathrm{\aleph}}_{0}$-complete lattice
is the same as a countably complete lattice.)

Every complete lattice is a for every infinite cardinal $\kappa $, and in particular is a countably complete lattice. Every countably complete lattice is a bounded lattice.

Title | complete lattice |
---|---|

Canonical name | CompleteLattice |

Date of creation | 2013-03-22 12:56:44 |

Last modified on | 2013-03-22 12:56:44 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 06B23 |

Classification | msc 03G10 |

Related topic | TarskiKnasterTheorem |

Related topic | CompleteLatticeHomomorphism |

Related topic | Domain6 |

Related topic | CompleteSemilattice |

Related topic | InfiniteAssociativityOfSupremumAndInfimumRegardingItself |

Related topic | CompleteBooleanAlgebra |

Related topic | ArbitraryJoin |

Defines | countably complete lattice |

Defines | countably-complete lattice |

Defines |
$\kappa $-complete^{} |

Defines | $\kappa $-complete lattice |