Grothendieck topologies on a poset

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We investigate Grothendieck topologies (in the sense of sheaf theory) on a poset ¶ that are generated by some subset of ¶. We show that such Grothendieck topologies exhaust all possibilities if and only if ¶ is Artinian. If ¶ is not Artinian, other families of Grothendieck topologies on ¶ exist that are not generated by some subset of ¶, but even those are related to the Grothendieck topologies generated by subsets. Furthermore, we investigate several notions of equivalences of Grothendieck topologies, and using a posetal version of the Comparison Lemma, a sheaf-theoretic result known as the Comparison Lemma, going back to Grothendieck et al SGA4, we calculate the sheaves with respect to most of the Grothendieck topologies we have found.