# weakest extension of a partial ordering

Let $A$ be a commutative ring with partial ordering ${A}^{+}$ and suppose that $B$ is a ring that admits a partial ordering. If $f:A\to B$ is a ring monomorphism (thus we regard $B$ as an over-ring of $A$), then any partial ordering of $B$ that can contain $f({A}^{+})$ will also contain the set ${B}^{+}\subset B$ defined by

$${B}^{+}:=\{\sum _{i=1}^{n}f({a}_{i}){b}_{i}^{2}:n\in \mathbb{N},{a}_{1},\mathrm{\dots},{a}_{n}\in {A}^{+}\}$$ |

${B}^{+}$ is itself a partial ordering and it is called the
*weakest partial ordering of $B$ that extends ${A}^{\mathrm{+}}$ (through $f$)*. It is called ”weakest” because this is the smallest partial ordering ${B}^{+}$ of $B$ that will transform $f$ into a poring monomorphism^{} (i.e. a monomorphism in the category^{} of partially ordered rings) $f:(A,{A}^{+})\to (B,{B}^{+})$ (for simplicity, we abuse the symbol $f$ here).

Title | weakest extension^{} of a partial ordering |
---|---|

Canonical name | WeakestExtensionOfAPartialOrdering |

Date of creation | 2013-03-22 18:51:40 |

Last modified on | 2013-03-22 18:51:40 |

Owner | jocaps (12118) |

Last modified by | jocaps (12118) |

Numerical id | 9 |

Author | jocaps (12118) |

Entry type | Definition |

Classification | msc 13J30 |

Classification | msc 13J25 |

Defines | weakest extension of a partial ordering |