# ultrametric

Any metric $d:X\times X\to \mathbb{R}$ on a set $X$ must satisfy the triangle inequality:

$$(\forall x,y,z)\mathit{\hspace{1em}}d(x,z)\le d(x,y)+d(y,z)$$ |

An ultrametric must additionally satisfy a stronger version of the triangle inequality:

$$(\forall x,y,z)\mathit{\hspace{1em}}d(x,z)\le \mathrm{max}\{d(x,y),d(y,z)\}$$ |

Here is an example of an ultrametric on a space with 5 points, labelled $a,b,c,d,e$:

$$\begin{array}{cccccc}& \hfill a\hfill & \hfill b\hfill & \hfill c\hfill & \hfill d\hfill & \hfill e\hfill \\ \hfill a\hfill & \hfill 0\hfill & \hfill 12\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 12\hfill \\ \hfill b\hfill & & \hfill 0\hfill & \hfill 12\hfill & \hfill 12\hfill & \hfill 5\hfill \\ \hfill c\hfill & & & \hfill 0\hfill & \hfill 6\hfill & \hfill 12\hfill \\ \hfill d\hfill & & & & \hfill 0\hfill & \hfill 12\hfill \\ \hfill e\hfill & & & & & \hfill 0\hfill \end{array}$$ |

In the table above, an entry $n$ in the for element $x$ and the for element $y$ indicates that $d(x,y)=n$, where $d$ is the ultrametric. By symmetry of the ultrametric ($d(x,y)=d(y,x)$), the above table yields all values of $d(x,y)$ for all $x,y\in \{a,b,c,d,e\}$.

The ultrametric condition is equivalent^{} to the ultrametric three point condition:

$$(\forall x,y,z)\mathit{\hspace{1em}}x,y,z\text{can be renamed such that}d(x,z)\le d(x,y)=d(y,z)$$ |

Ultrametrics can be used to model bifurcating hierarchical systems. The distance^{} between nodes in a weight-balanced binary tree is an ultrametric. Similarly, an ultrametric can be modelled by a weight-balanced binary tree, although the choice of tree is not necessarily unique. Tree models of ultrametrics are sometimes called *ultrametric trees*.

The metrics induced by non-Archimedean valuations are ultrametrics.

Title | ultrametric |
---|---|

Canonical name | Ultrametric |

Date of creation | 2013-03-22 13:28:28 |

Last modified on | 2013-03-22 13:28:28 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 21 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 54E35 |

Related topic | MetricSpace |

Related topic | Valuation^{} |

Related topic | UltrametricSpace |