Functions on distributive lattices with the congruence substitution property: Some problems of Grätzer from 1964
The author solves two problems raised by G. Grätzer in 1964, using Priestley duality. For a bounded distributive lattice \(L\), let \(S_k(L)\) (resp. \(S(L))\) denote the lattice of all \(k\)-ary (resp. finitary) ``Boolean functions'' on \(L\), that is, functions \(f: L^k\to L\) such that \(\vec a \theta \vec b\) implies \(f(\vec a) \theta f(\vec b)\) for each congruence \(\theta\) on \(L\). Such functions are determined by their restriction on \(\{0,1\}^k\) -- a well-known result which receives a new proof in this paper. Examples of computations of \(S_1(L)\) and \(S_2(L)\) are given in full detail for some small lattices \(L\). It is shown that \(S_{k+ 1}(L)\cong S_1(S_k(L))\), whence \(S_1(L)\cong S_1(M)\) implies \(S_k(L)\cong S_k(M)\) for all \(k\), solving Problem II.14 of \textit{G. Grätzer's} General lattice theory (1978; Zbl 0385.06015). Also the Priestley dual of \(S_k(L)\) (resp. \(S(L)\)) is characterized as a certain product of the Priestley dual of \(L\) with the \(2^k\)-element discrete space (resp. with the Cantor space). It follows that for each \(k\), \(S_k(L)\) characterizes \(L\) up to isomorphism, but \(S(L)\) does not.
Boolean functions on a distribute lattice
scientific article; zbMATH DE number 1422995