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partial_order [2008/04/28 16:08] vkuncak |
partial_order [2008/04/28 16:14] vkuncak |
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| 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 | ||
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| x \leq y\ \rightarrow\ \alpha (x) \sqsubseteq \alpha (y) | x \leq y\ \rightarrow\ \alpha (x) \sqsubseteq \alpha (y) | ||
| \end{equation*} | \end{equation*} | ||
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