Functions on Distributive Lattices with the Congruence Substitution Property: Some Problems of Grätzer

Jonathan David Farley

\define\iso{\cong} Let $L$ be a bounded distributive lattice. For $k\ge1$, 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 $S_1(L)\iso S_1(M)$ implies $S_k(L)\iso S_k(M)$. If $L$ and $M$ are finite, then $S_k(L)\iso S_k(M)$ implies $L\iso M$.

Some problems of Gr\"atzer dating to 1964 are thus solved.