I am trying to understand directed preorders, a.k.a. directed sets. Are they analogous to connected DAGs?

  • $\begingroup$ One may visualize any preorder as follow. A preorder is a certain type of category. If the category is small, we may take it's nerve. One might also might be able to visualize this as a graph of some sort. $\endgroup$ – Baby Dragon May 22 '13 at 3:52

Not quite. A DAG cannot have edges $x\to y$ and $y\to x$, while a directed preorder can have $x\leq y$ and $y\leq x$.

In general, you can visualize a preorder as the reachability relation on the vertices of a directed graph. Then a directed preorder corresponds to a graph where any two vertices are reachable from some third vertex.

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  • 1
    $\begingroup$ So it is necessary that the directed graph be at least weakly connected. $\endgroup$ – ThomasMcLeod Aug 7 '13 at 22:14

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