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partial_order [2008/05/07 16:26]
pedagand
partial_order [2009/03/26 17:12] (current)
vkuncak
Line 27: Line 27:
* the direction of edge is given by which nodes is drawn above   * the direction of edge is given by which nodes is drawn above
* transitive and reflexive edges are not represented (they can be derived)   * 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, ​meet, $\sqcup$) of $S$ if $a$ is the least element in the set of all upper bounds of $S$ +  * **least upper bound** (lub, supremum, ​join, $\sqcup$) 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 greatest element in the set of all lower bounds of $S$+  * **greatest lower bound** (glb, infimum, ​meet, $\sqcap$) of $S$ if $a$ is the greatest element in the set of all lower 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.

partial_order.txt · Last modified: 2009/03/26 17:12 by vkuncak

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