Isotone maps on lattices (Q1762477)

The main result of the present paper is the following theorem: Let \textbf{V} be a nontrivial variety of lattices, \({\mathcal L}=(L_i\,|\,i\in I)\) a family of lattices in \textbf{V} and Free(\({\mathcal L}\),\textbf{V}) the free product of the \(L_i\) in \textbf{V}. Furthermore, let \(M\) be a lattice \text{(}not necessarily in \textbf{V}\text{)} and \((\varphi_i:L_i\to M\,|\,i\in I)\) a family of isotone maps from the \(L_i\) to \(M\). Then there exists an isotone map \(\varphi:\;\)Free(\({\mathcal L}\),\textbf{V})\(\;\to M\) whose restriction to each \(L_i\) is \(\varphi_i\). This result was already known for \textbf{V} = \textbf{L}, the variety of all lattices; see [\textit{Yu.~I.~Sorkin}, Mat. Sb., N. Ser. 30(72), 677--694 (1952; Zbl 0047.02903); \textit{G.~Grätzer} et al., Fundam.\ Math.\ 69, 233--240 (1970; Zbl 0206.29703)]. Free(\({\mathcal L}\),\textbf{V}) may be regarded as the free lattice (in \textbf{V}) on the partial lattice given by the disjoint union of the \(L_i\). So there arises the natural question: When does the analog of the above theorem hold for the inclusion of a general partial lattice \(P\) in its free lattice Free \(P\)? For this case, the authors show the following result: Let \(M\) be a lattice. Then the following conditions are equivalent: (1) \(M\) is complete. (2) Any isotone map from a partial lattice \(P\) to \(M\) can be extended to an isotone map from Free \(P\) to \(M\). Furthermore, for lattices \(L_i\) amalgamated over a convex retract, an analog of the main result is proved. Finally, several ``semilattice variants'' are given. The paper is well organized and written in a very clear style. The proofs are partly rather tricky and hard.