Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision | Next revision Both sides next revision | ||
partial_order [2008/04/28 16:08] vkuncak |
partial_order [2008/04/28 16:14] vkuncak |
||
---|---|---|---|
Line 38: | Line 38: | ||
Duality minimal/maximal, least/greatest, supremum/infimum | Duality minimal/maximal, least/greatest, supremum/infimum | ||
- | Note: | + | Note |
- | * minimal element need not exist | + | * minimal element need not exist: $(0,1)$ interval of rationals |
- | * there may be multiple minimal elements | + | * there may be multiple minimal elements: $\{\{a\},\{b\},\{a,b\}\}$ |
- | * if minimal element exists, it need not be least | + | * if minimal element exists, it need not be least: above example |
- | * there are no two distinct minimal elements for the same set | + | * there are no two distinct least elements for the same set |
* least element is always glb and minimal | * least element is always glb and minimal | ||
* if glb belongs to the set, then it is always least and minimal | * if glb belongs to the set, then it is always least and minimal | ||
Line 52: | Line 52: | ||
x \leq y\ \rightarrow\ \alpha (x) \sqsubseteq \alpha (y) | x \leq y\ \rightarrow\ \alpha (x) \sqsubseteq \alpha (y) | ||
\end{equation*} | \end{equation*} | ||
- | |||
- |