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 partial_order [2008/05/07 16:26]pedagand partial_order [2009/03/26 17:12] (current)vkuncak Both sides previous revision Previous revision 2009/03/26 17:12 vkuncak 2008/05/07 16:26 pedagand 2008/04/30 15:21 vkuncak 2008/04/30 15:20 vkuncak 2008/04/30 10:00 vkuncak 2008/04/30 09:15 vkuncak 2008/04/29 22:29 vkuncak 2008/04/28 16:33 vkuncak 2008/04/28 16:14 vkuncak 2008/04/28 16:08 vkuncak 2008/04/28 16:05 vkuncak 2008/04/28 16:02 vkuncak 2008/04/28 15:53 vkuncak 2007/04/05 01:53 gizil.oguz 2007/03/30 21:01 vkuncak 2007/03/30 21:00 vkuncak 2007/03/30 21:00 vkuncak 2007/03/30 20:32 vkuncak 2007/03/30 20:32 vkuncak 2007/03/30 20:31 vkuncak created 2009/03/26 17:12 vkuncak 2008/05/07 16:26 pedagand 2008/04/30 15:21 vkuncak 2008/04/30 15:20 vkuncak 2008/04/30 10:00 vkuncak 2008/04/30 09:15 vkuncak 2008/04/29 22:29 vkuncak 2008/04/28 16:33 vkuncak 2008/04/28 16:14 vkuncak 2008/04/28 16:08 vkuncak 2008/04/28 16:05 vkuncak 2008/04/28 16:02 vkuncak 2008/04/28 15:53 vkuncak 2007/04/05 01:53 gizil.oguz 2007/03/30 21:01 vkuncak 2007/03/30 21:00 vkuncak 2007/03/30 21:00 vkuncak 2007/03/30 20:32 vkuncak 2007/03/30 20:32 vkuncak 2007/03/30 20:31 vkuncak created 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 ===== Line 37: Line 38: * **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.