Functions on Distributive Lattices with the Congruence Substitution Property: Some Problems of Grätzer
Jonathan David Farley
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 $S_1(L)\cong S_1(M)$ implies $S_k(L)\cong S_k(M)$. If $L$ and $M$ are finite, then $S_k(L)\cong S_k(M)$ implies $L\cong M$.
Some problems of Grätzer dating to 1964 are thus solved.