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# Lab for Automated Reasoning and Analysis LARA

# Differences

This shows you the differences between two versions of the page.

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partial_order [2008/05/07 16:26] pedagand |
partial_order [2009/03/26 17:12] (current) vkuncak |
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* 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. |