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preorder [2007/03/30 20:40] vkuncak |
preorder [2007/03/30 20:52] vkuncak |
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A (reflexive) preorder relation $\rho$ on set $A$ is a binary relation $r \subseteq A^2$ that is reflexive and transitive, that is, these two properties hold: | A (reflexive) preorder relation $\rho$ on set $A$ is a binary relation $r \subseteq A^2$ that is reflexive and transitive, that is, these two properties hold: | ||
+ | |||
* $x \mathop{\rho} x$ | * $x \mathop{\rho} x$ | ||
- | * $x \mathop{\rho} y\ \land\ y \mathop{\rho} z \ \rightarrow\ x \mathop{\rho} z$ | ||
+ | * $x \mathop{\rho} y\ \land\ y \mathop{\rho} z \ \rightarrow\ x \mathop{\rho} z$ | ||
+ | ===== Constructing a partial order from a preorder ===== | ||
- | ===== Constructing a [[partial order]] from a [[preorder]] ===== | + | Intuitively, preorder differs from partial order in that there are distinct elements that have same ordering properties with respect to other elements. For such elements we therefore have $x \rho y$ and $y \rho x$. By identifying these elements we obtain a partial order. |
- | Let $\rho$ be a preorder. Define relation $\sim$ by | + | More precisely, let $\rho$ be a preorder. Define relation $\sim$ by |
\begin{equation*} | \begin{equation*} | ||
x \sim y \ \iff\ x \mathop{\rho} y\ \land\ y \mathop{\rho} x | x \sim y \ \iff\ x \mathop{\rho} y\ \land\ y \mathop{\rho} x |