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## 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]

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