## Abstract.

Let *X* be a poset and *Y* an ordered space; *X* ^{ Y } \((X^Y_\Sigma, X^Y_\Lambda)\) denotes the poset of continuous order-preserving maps from *Y* to *X* with the discrete (respectively, Scott, Lawson) topology. If *S* is a \(\lor\)-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: \((S^{P(T)})^\sigma \cong (S^\sigma) ^{P(T^\sigma)}_\Lambda.\) Moreover, \($(S^\sigma)^{P(T^\sigma)}_\Lambda \cong (S^\sigma) ^{P(T^\sigma)}$ if and only if $(S^\sigma)^{P(T^\sigma)}_\Lambda = (S^\sigma)^{P(T^\sigma)},\) and sufficient conditions and necessary conditions for the isomorphism to hold are obtained (both necessary and sufficient if *S* is a distributive \(\lor\)-semilattice).

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