Abstract. Let $L$ be a lattice and $M$ a bounded distributive lattice. Let $\mathop{\rm Con}L$ denote the congruence lattice of $L$, $P(M)$ the Priestley dual space of $M$, and $L^{P(M)}$ the lattice of continuous order-preserving maps from $P(M)$ to $L$ with the discrete topology. It is shown that $\mathop{\rm Con}(L^{P(M)})\cong(\mathop{\rm Con}L)_{\Lambda}^{P(\mathop{\rm Con}M)}$, the lattice of continuous order-preserving maps from $P(\mathop{\rm Con}M)$ to $\mathop{\rm Con}L$ with the Lawson topology. Various other ways of expressing $\mathop{\rm Con}(L^P)$ as a lattice of continuous functions or semilattice homomorphisms are presented. Indeed, a link is established between semilattice homomorphisms from a semilattice $S$ into a bounded distributive lattice $T$ (or its ideal lattice) and continuous order-preserving maps from $P(T)$ into the ideal lattice of $S$ with the Scott, Lawson, or discrete topology. It is also shown that, in general, $\mathop{\rm Con}(L^{P(M)})\not\cong(\mathop{\rm Con}L)^{P(\mathop{\rm Con}M)}$, solving a problem of E. T. Schmidt (independently solved by Grätzer and Schmidt).

AMS Subject Classification (1991): 06B10, 08A30, 08B26, 06B30, 06F30, 22A26, 06E15, 06B35, 06A12, 08B25, 18A30, 18A35

Keyword(s): Priestley power, Boolean power, (generalized) function lattice, congruence lattice, algebraic lattice, semilattice, (bounded) distributive lattice, Priestley space, Scott topology, Lawson topology, exponentiation

Received September 4, 1995 and in revised form February 7, 1996. (Registered under 5707/2009.)