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partial_order [2008/04/28 16:14] vkuncak |
partial_order [2008/04/28 16:33] vkuncak |
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| Duality minimal/maximal, least/greatest, supremum/infimum | Duality minimal/maximal, least/greatest, supremum/infimum | ||
| - | Note | + | Notes |
| * minimal element need not exist: $(0,1)$ interval of rationals | * minimal element need not exist: $(0,1)$ interval of rationals | ||
| * there may be multiple minimal elements: $\{\{a\},\{b\},\{a,b\}\}$ | * there may be multiple minimal elements: $\{\{a\},\{b\},\{a,b\}\}$ | ||
| Line 45: | Line 45: | ||
| * 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 | ||
| + | * for relation $\subseteq$ on sets, $glb$ is intersection, $lub$ is union (not all families of sets are closed under $\cap$, $\cup$) | ||
| ===== Monotonic functions ===== | ===== Monotonic functions ===== | ||