Functions on distributive lattices with the congruence substitution property: Some problems of Gratzer from 1964

Let L be a bounded distributive lattice. For k greater than or equal to 1, let S-k(L) be the lattice of k-ary functions on L with the congruence subst itution property (Boolean functions); let S(L) be the lattice of all Boolea n functions. The lattices that can arise as S-k(L) or S(L) for some bounded distributive lattice L are characterized in terms of their Priestley space s of prime ideals. For bounded distributive lattices L and M, it is shown t hat S-1(L) congruent to S-1 (M) implies S-k(L) congruent to S-k(M). If L an d M are finite, then Sk(L) congruent to Sk(M) implies L congruent to M. Som e problems of Gratzer dating to 1964 are thus solved. (C) 2000 Academic Pre ss.