Let $(V,\preceq )$ be a poset or a strictly ordered set and let $S\subseteq V$.

An element $m\in S$ is called:

maximal in $S$ |
no $x\in S$ is greater, formally $\not\exists x\in S\; x\succ m$ |

minimal in $S$ |
no $x\in S$ is smaller, formally $\not\exists x\in S\; x\prec m$ |

maximum of (greatest in) $S$ |
all $x\in S$ are smaller or equal, formally $\forall x\in S\; x\preceq m$ |

minimum of (smallest in) $S$ |
all $x\in S$ are greater or equal, formally $\forall x\in S\; x\succeq m$ |

An element $m\in V$ is called:

upper bound of $S$ |
all $x\in S$ are smaller or equal, formally $\forall x\in S\; x\preceq m$ |

lower bound of $S$ |
all $x\in S$ are greater or equal, formally $\forall x\in S\; x\succeq m$ |

supremum of $S$ |
$m$ is minimum of all upper bounds of $S$, formally $m=\sup(S):=\min(\{n\in V\mid \forall x\in S\; x\preceq n\})$ |

infimum of $S$ |
$m$ is maximum of all lower bounds of $S$, formally $m=\inf(S):=\max(\{n\in V\mid \forall x\in S\; x\succeq n\})$ |

| | | | | created: 2018-12-16 23:14:40 | modified: 2020-07-06 20:53:35 | by: *bookofproofs* | references: [577], [979]

[577] **Knauer Ulrich**: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001

[979] **Reinhardt F., Soeder H.**: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10