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maximal_extension_in_a_set [2014/12/04 13:59] nikolaj |
maximal_extension_in_a_set [2014/12/04 14:21] nikolaj |
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| ===== Maximal extension in a set ===== | ===== Maximal extension in a set ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#55CCEE: context | @#55CCEE: $X$ ... set | | + | | @#55CCEE: context | @#55CCEE: $A$ ... set | |
| - | | @#55CCEE: context | @#55CCEE: $a\in X$ | | + | | @#55CCEE: context | @#55CCEE: $a\in A$ | |
| - | | @#FFBB00: definiendum | @#FFBB00: $\mathrm{max}(a,A)$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\mathrm{max}(a,A)\equiv\bigcup\{b\mid b\in A\land a\subseteq b\}$ | |
| - | | @#55EE55: postulate | @#55EE55: $\mathrm{max}(a,A)\in X$ | | + | |
| - | | @#55EE55: postulate | @#55EE55: $a\subseteq\mathrm{max}(a,A)$ | | + | |
| - | | @#FFFDDD: forall | @#FFFDDD: $b\in X$ | | + | |
| - | | @#55EE55: postulate | @#55EE55: $a\subseteq b\implies b\subseteq \mathrm{max}(a,A)$ | | + | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | >todo: write this down order theoretical (define it in terms of an ordering defined via $\subseteq$) | ||
| === Idea === | === Idea === | ||
| - | Given $a\in X$, the maximal extension $a'$ is the largest amongs $X$ which does contain $a$. | + | Given $a\in A$, the maximal extension $\mathrm{max}(a,A)$ is the largest set in $A$ which encompasses $a$. |
| === Predicate === | === Predicate === | ||
| - | | @#EEEE55: predicate | @#EEEE55: $x$ maximal in $X \equiv \mathrm{max}(x,A)=x$ | | + | | @#EEEE55: predicate | @#EEEE55: $x$ maximal in $X \equiv \mathrm{max}(x,X)=x$ | |
| === Reference === | === Reference === | ||
