If I have a set $S$ and a relation $\rho$ on $S$, such that $\rho$ is reflexive and antisymmetric, and I draw the arrow diagram of this, and this diagram is connected, with no cycles, is transitivity induced?
2026-04-13 06:04:39.1776060279
Almost a partial order, is a partial order induced anyway?
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You can always find the transitive closure of any such relation by drawing a link between $A$ and $B$ if there is a path from $A$ to $B$. If the original graph contains no cycles (ignoring the reflexive ones), then its transitive closure will contain no cycles either, and it will still be anti-symmetric. It will also still be reflexive of course and, since it is the transitive closure, transitive. Thus, the transitive closure of such a relation will indeed be a partial order.