# R-minimal element

Let $A$ be a set and $R$ be a relation on $A$. Suppose that $B$ is a subset of $A$. An element $a\in B$ is said to be $R$-minimal^{} in $B$ if and only if there is no $x\in B$ such that $xRa$. An $R$-minimal element in $A$ is simply called $R$-minimal.

From this definition, it is evident that if $A$ has an $R$-minimal element, then $R$ is not reflexive^{}. However, the definition of $R$-minimality is sometimes adjusted slightly so as to allow reflexivity: $a\in B$ is $R$-minimal (in $B$) iff the only $x\in B$ such that $xRa$ is when $x=a$.

Remark. Using the second definition, it is easy to see that when $R$ is a partial order^{}, then an element $a$ is $R$-minimal iff it is minimal.

Title | R-minimal element |
---|---|

Canonical name | RminimalElement |

Date of creation | 2013-03-22 12:42:43 |

Last modified on | 2013-03-22 12:42:43 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 11 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 03B10 |

Synonym | R-minimal |

Synonym | $R$-minimal |

Related topic | WellFoundedRelation |