# semilattice

A *lower semilattice ^{}* is a partially ordered set

^{}S in which each pair of elements has a greatest lower bound

^{}.

A *upper semilattice ^{}* is a partially ordered set S in which each pair of elements has a least upper bound.

Note that it is not normally necessary to distinguish lower from upper semilattices, because one may be converted to the other by reversing the partial order^{}. It is normal practise to refer to either structure^{} as a *semilattice* and it should be clear from the context whether greatest lower bounds or least upper bounds exist.

Alternatively, a semilattice can be considered to be a commutative^{} band, that is a semigroup^{} which is commutative, and in which every element is idempotent^{}. In this context, semilattices are important elements of semigroup theory and play a key role in the structure theory of commutative semigroups.

A partially ordered set which is both a lower semilattice and an upper semilattice is a lattice^{}.

Title | semilattice |

Canonical name | Semilattice |

Date of creation | 2013-03-22 12:57:23 |

Last modified on | 2013-03-22 12:57:23 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20M99 |

Classification | msc 06A12 |

Related topic | Lattice |

Related topic | Poset |

Related topic | Idempotent2 |

Related topic | Join |

Related topic | Meet |

Related topic | CompleteSemilattice |

Defines | lower semilattice |

Defines | upper semilattice |