Absolute Minimal and Maximal Elements

Topics: Discrete Mathematics - Relation

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(definition)

Let $(X,⪯)$ be a poset.

• An element $m\in X$ is an absolute maximal element if $x⪯m$ for all $x\in X$
• An element $s\in X$ is an absolute minimal element if $s⪯x$ for all $x\in X$

Examples

With $(K,\subseteq )$ given:

$$K=\{\{x\},\{x,y\},\{w,x,y,z\},\{y,z\}\}$$

…notice that:

• $\{x\}$ is a minimal element, but it’s not an absolute minimal element since $\{x\}⊈\{y,z\}$
• $\{w,x,y,z\}$ is a maximal element and an absolute maximal element