**Definition**: Special Elements of Posets

Let $(V,\prec)$ be a poset and let $S\subseteq V$ be ordered by $”\prec”$.

An element $m\in V$ is called:

- an
**upper bound**of $S$, if $x\prec m$ for all $x\in S.$ If $m\in S$, then $m$ is called the**maximum**(**greatest element**) of $S$ and denote it by $\max(S).$ - a
**lower bound**of $S$, if $m\prec x$ for all $x\in S.$ If $m\in S$, then $m$ is called the**minimum**(**least element**) of $S$ and denote it by $\min(S).$ - Let $S^u$ be the set of all upper bounds of $S$. If the maximum of $S^u$ exists, then it is called the
**supremum**of $S$ and is defined by $\sup(S):=\min(S^u).$ - Let $S^l$ be the set of all lower bounds of $S$. If the minimum of $S^l$ exists, then it is called the
**infimum**of $S$ and is defined by $\inf(S):=\max(S^l).$

An element $m\in S$ is called:

- a
**maximal element**of $S$, if from $m\prec x$ and $x\in S$ it follows that $m=x.$ - a
**minimal element**of $S$, if from $x\prec m$ and $x\in S$ it follows that $x=m.$

| | | | | Contributors: *bookofproofs* | References: [577], [979]

(none)

[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

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