# Definition: Special Elements of Ordered Sets

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 $m=\inf(S):=\max(\{n\in V\mid \forall x\in S\; x\succeq n\})$

Definitions: 1 2 3 4 5
Examples: 6 7
Explanations: 8 9 10
Lemmas: 11
Proofs: 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Propositions: 29 30 31 32 33

Github: ### References

#### Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001
2. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition