Priestley powers of lattices and their congruences . A problem of E . T .

Abstract. Let L be a lattice and M a bounded distributive lattice. Let ConL denote the congruence lattice of L, P (M) the Priestley dual space of M , and L (M) the lattice of continuous order-preserving maps from P (M) to L with the discrete topology. It is shown that Con(L ) ∼= (ConL) P (ConM) Λ , the lattice of continuous order-preserving maps from P (ConM) to ConL with the Lawson topology. Various other ways of expressing Con(L ) 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, Con(L ) 6∼= (ConL) , solving a problem of E. T. Schmidt (independently solved by Grätzer and Schmidt).