Congruence lattices of function lattices (Q1344243)

Let \(D\) be a bounded, distributive lattice, \(L\) a lattice. The generalized function lattice \(L[D]\) has as elements the continuous isotone maps of \(X\) into \(L\), where \(X\) is the poset of all prime-filters of \(D\) with the usual topology. The authors show for congruence lattices the isomorphism \(\text{Con }L[D]\cong (\text{Con } L)[\text{Con }D]\) iff \(\text{Con }L\) or \(D\) is finite. \(E[A]\) is complete (or algebraic) for every algebraic lattice \(A\) iff \(E\) is finite.