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partial_order [2008/04/30 15:20] vkuncak |
partial_order [2009/03/26 17:12] vkuncak |
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| Given a partial ordering relation $\le$, the corresponding **strict ordering relation** $x < y$ is defined by $x \le y \land x \neq y$ and can be viewed as a shorthand for this conjunction. | Given a partial ordering relation $\le$, the corresponding **strict ordering relation** $x < y$ is defined by $x \le y \land x \neq y$ and can be viewed as a shorthand for this conjunction. | ||
| - | We can view partial order $(A,r)$ as a first-order interpretation $I=(A,\alpha)$ of language ${\cal L}={\le\}$ where $\alpha({\le\})=r$. | + | We can view partial order $(A,r)$ as a first-order interpretation $I=(A,\alpha)$ of language ${\cal L}=\{\le\}$ where $\alpha({\le})=r$. |
| ===== Example Partial Orders ===== | ===== Example Partial Orders ===== | ||
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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) | ||
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| ===== 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 least element in the set of all upper 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. | ||