Representing an isotone map between two bounded ordered sets by principal lattice congruences (Q1669662)

For a lattice \(L\), \(\operatorname{Princ}(L\)) denotes the ordered set of principal congruences of \(L\) (congruences generated by a pair of elements). Let \(P_1\) and \(P_2\) be ordered sets with 0 and \(f\,: P_1\to P_2\) a 0-preserving isotone map. The map \(f\) is said to be \textit{representable by principal congruences of bounded lattices} it there exist bounded lattices \(L_1\) and \(L_2\), order isomorphisms \(h_i\,: P_i\to\operatorname{Princ}(L_i\)), \(i\in \{1,2\}\), and a lattice homomorphism \(g\,: L_1 \to L_2\) such that \(f=h_2^{-1}\circ \operatorname{Princ}(g)\circ h_1\), where \(\operatorname{Princ}(g)\,: \operatorname{Princ}(L_1)\to \operatorname{Princ}(L_2)\) is defined by \(\operatorname{con}(x, y) \mapsto \operatorname{con}((g(x), g(y))\). If \(L_1\) and \(L_2\) are of lengths at most \(m\) and \(n\), respectively, then \(f\) is said to be \textit{representable by principal congruences of lattices of lengths at most \(m\) and \(n\)}. It is proved that if \(f\,: P_1\to P_2\) is a 0-preserving isotone map between bounded ordered sets, then \(f\) is representable by principal congruences of bounded lattices of lengths at most 5 and 7.