FUNCTIONS ON DISTRIBUTIVE LATTICES WITH THE CONGRUENCE SUBSTITUTION PROPERTY: SOME PROBLEMS

OF GRÄTZER

JONATHAN DAVID FARLEY

Abstract. Let L be a bounded distributive lattice. For k ≥ 1, let S_{k}(L) be the lattice of k-ary functions on L with the congruence substitution property (Boolean functions); let S(L) be the lattice of all Boolean 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 spaces of prime ideals. For bounded distributive lattices L and M, it is shown that S_{1}(L) ≅ S_{1}(M) implies S_{k}(L) ≅ S_{k}(M). If L and M are finite, then S_{k}(L) ≅ S_{k}(M) implies L ≅ M.

Some problems of Grätzer dating to 1964 are thus solved.