Grothendieck topologies on a poset

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 \cite{SGA4}, we calculate the sheaves with respect to most of the Grothendieck topologies we have found.

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