# example of a non-lattice homomorphism

Consider the Hasse diagram of the lattice of subgroups of the quaternion group^{}
of order $8$, ${Q}_{8}$. [The use of ${Q}_{8}$ is only for a concrete realization of the lattice^{}.]

$$ |

To establish an order-preserving map which is not a lattice isomorphism^{}
one can simply “skip” $\u27e8-1\u27e9$, which we display graphically as:

$$ |

Since containment is still preserved the map is order-preserving. However, the intersection^{} (meet) of
$\u27e8i\u27e9$ and $\u27e8j\u27e9$, which is $\u27e8-1\u27e9$, is not perserved under this
map. Thus it is not a lattice homomorphism.

Title | example of a non-lattice homomorphism^{} |
---|---|

Canonical name | ExampleOfANonlatticeHomomorphism |

Date of creation | 2013-03-22 16:58:31 |

Last modified on | 2013-03-22 16:58:31 |

Owner | Algeboy (12884) |

Last modified by | Algeboy (12884) |

Numerical id | 8 |

Author | Algeboy (12884) |

Entry type | Example |

Classification | msc 06B23 |