Ideals of Priestley powers of semilattices (Q5932482)

Let \(P,Q\) be ordered topological spaces, let \(P^Q_\tau \) be the poset of continuous order-preserving maps from \(Q\) to \(P\) (the latter being endowed with the topology \(\tau \)) and let \(P^Q\) be \(P^Q_\tau \) where \(\tau \) is the discrete topology on \(P\). Suppose that \(\Sigma \) and \(\Lambda \) stand for the Scott and the Lawson topology, respectively. If \(S\) is a \(\vee \)-semilattice, \(S^\sigma \) its ideal semilattice, and \(T\) a bounded distributive lattice with Priestley dual space \(P(T)\), it is shown that the following isomorphisms hold: \[ (S^{P(T)})^\sigma \cong (S^\sigma)^{P(T)}_\Sigma \cong (S^\sigma)^{P(T^\sigma)}_\Lambda . \] Moreover, \[ (S^\sigma)^{P(T^\sigma)}_\Lambda \cong (S^\sigma)^{P(T^\sigma)} \text{ if and only if }(S^\sigma)^{P(T^\sigma)}_\Lambda =(S^\sigma)^{P(T^\sigma)}, \] and sufficient and necessary conditions for the isomorphism to hold are obtained (both necessary and sufficient if \(S\) is a distributive \(\vee \)-semilattice).