On isotone functions with the substitution property in distributive lattices (Q1904383)

Let \(\text{Id } L\) denote the lattice of all ideals of a lattice \(L\). One may regard \(L\) as a sublattice of \(\text{Id } L\), with the embedding \(x\mapsto (x]\). Let \(p\) be an \(n\)-ary polynomial over \(\text{Id } L\) with the property that it maps \(L^n\) into \(L\). The corresponding \(n\)-ary function \(p_L\) on \(L\) is called an \(n\)-ary \(I\)-polynomial. A \(D\)-polynomial is defined dually. An \(ID\)-polynomial is a composition of \(I\)- and \(D\)-polynomials. The following propositions are the main results of the paper. Theorem 3. Let \(L\) be a distributive lattice. Then the set of translations on \(L\) is the same as the set of all unary \(ID\)-polynomials. Theorem 4. Let \(L\) be a distributive lattice. All unary functions with the Substitution Property are \(ID\)-polynomials if and only if \(L\) contains no proper Boolean interval.