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preorder [2007/03/30 20:40] vkuncak |
preorder [2007/03/30 20:52] vkuncak |
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===== Constructing a partial order from a preorder ===== | ===== Constructing a partial order from a preorder ===== | ||
- | Let $\rho$ be a preorder. Define relation $\sim$ by | + | 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. |
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+ | 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 |