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partial_order [2008/04/30 10:00] vkuncak |
partial_order [2008/04/30 15:21] 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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| Given a partial order $(A,\le)$ and a set $S \subseteq A$, we call an element $a \in A$ | Given a partial order $(A,\le)$ and a set $S \subseteq A$, we call an element $a \in A$ | ||
| - | * **upper bound** of $S$ if for all $a' \in S$ we have $a \le a'$ | + | * **upper bound** of $S$ if for all $a' \in S$ we have $a' \le a$ |
| * **lower bound** of $S$ if for all $a' \in S$ we have $a \le a'$ | * **lower bound** of $S$ if for all $a' \in S$ we have $a \le a'$ | ||
| * **minimal element** of $S$ if $a \in S$ and there is no element $a' \in S$ such that $a' < a$ | * **minimal element** of $S$ if $a \in S$ and there is no element $a' \in S$ such that $a' < a$ | ||