Maximal extension in a set

Set

 context $A$ … set context $a\in A$ definiendum $\mathrm{max}(a,A)\equiv\bigcup\{b\mid b\in A\land a\subseteq b\}$

Discussion

Idea

Given $a\in A$, the maximal extension $\mathrm{max}(a,A)$ is the largest set in $A$ which encompasses $a$.

Predicate

 predicate $x$ maximal in $X \equiv \mathrm{max}(x,X)=x$