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partial_order [2008/04/30 09:15] vkuncak |
partial_order [2008/05/07 16:26] pedagand |
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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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**Example:** Let $U = \{ 1,2,3\}$ and let $A = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{2,3\} \}$. Then $(A, \le)$ is a partial order. We can draw it as a Hasse diagram. | **Example:** Let $U = \{ 1,2,3\}$ and let $A = \{ \emptyset, \{1\}, \{2\}, \{3\}, \{2,3\} \}$. Then $(A, \le)$ is a partial order. We can draw it as a Hasse diagram. | ||
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+ | ===== Hasse diagram ===== | ||
+ | |||
+ | [[wk>Hasse diagram]] presents the relation as a directed graph in a plane, such that | ||
+ | * the direction of edge is given by which nodes is drawn above | ||
+ | * transitive and reflexive edges are not represented (they can be derived) | ||
===== Extreme Elements in Partial Orders ===== | ===== Extreme Elements in Partial Orders ===== | ||
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 $a' < a$ | ||
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* **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, meet, $\sqcup$) 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, join, $\sqcap$) 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. |