# a semilattice is a commutative band

This note explains how a semilattice is the same as a commutative^{} band.

Let $S$ be a semilattice, with partial order^{} $$ and each pair of elements $x$ and $y$ having a greatest lower bound^{} $x\wedge y$.
Then it is easy to see that the operation^{} $\wedge $ defines a binary operation^{} on $S$ which makes it a commutative semigroup, and that every element is idempotent^{} since $x\wedge x=x$.

Conversely, if $S$ is such a semigroup^{}, define $x\le y$ iff $x=xy$. Again, it is easy to see that this defines a partial order on $S$, and that greatest lower bounds exist with respect to this partial order, and that in fact $x\wedge y=xy$.

Title | a semilattice is a commutative band |
---|---|

Canonical name | ASemilatticeIsACommutativeBand |

Date of creation | 2013-03-22 12:57:28 |

Last modified on | 2013-03-22 12:57:28 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Proof |

Classification | msc 20M99 |

Classification | msc 06A12 |

Related topic | Lattice^{} |