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

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