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partial_order [2008/04/29 22:29] vkuncak |
partial_order [2008/04/30 10:00] vkuncak |
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| **Example:** Let $U = \{ 1,2,3\}$ and let $A = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{2,3\} \}$. Then $(A, \le)$ is a partial order. We can draw it as a Hasse diagram. | **Example:** Let $U = \{ 1,2,3\}$ and let $A = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{2,3\} \}$. Then $(A, \le)$ is a partial order. We can draw it as a Hasse diagram. | ||
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| + | ===== Hasse diagram ===== | ||
| + | |||
| + | [[wk>Hasse diagram]] presents the relation as a directed graph in a plane, such that | ||
| + | * the direction of edge is given by which nodes is drawn above | ||
| + | * transitive and reflexive edges are not represented (they can be derived) | ||
| ===== Extreme Elements in Partial Orders ===== | ===== Extreme Elements in Partial Orders ===== | ||
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| * **greatest element** of $S$ if $a \in S$ and for all $a' \in S$ we have $a' \le a$ | * **greatest element** of $S$ if $a \in S$ and for all $a' \in S$ we have $a' \le a$ | ||
| * **least element** of $S$ if $a \in S$ and for all $a' \in S$ we have $a \le a'$ | * **least element** of $S$ if $a \in S$ and for all $a' \in S$ we have $a \le a'$ | ||
| - | * **least upper bound** (lub, supremum, $\sqcup$) of $S$ if $a$ is the least element in the set of all upper bounds of $S$ | + | * **least upper bound** (lub, supremum, meet, $\sqcup$) of $S$ if $a$ is the least element in the set of all upper bounds of $S$ |
| - | * **greatest lower bound** (glb, infimum, $\sqcap$) of $S$ if $a$ is the least element in the set of all upper bounds of $S$ | + | * **greatest lower bound** (glb, infimum, join, $\sqcap$) of $S$ if $a$ is the least element in the set of all upper bounds of $S$ |
| Taking $S=A$ we obtain minimal, maximal, greatest, least elements for the entire partial order. | Taking $S=A$ we obtain minimal, maximal, greatest, least elements for the entire partial order. | ||