Canonical extensions, free completely distributive lattices, and complete retracts (Q2057106)

\textit{W. Morton} and \textit{C. J. van Alten} [Int. J. Algebra Comput. 28, No. 3, 521--541 (2018; Zbl 06902200)] proved the following theorem. Theorem 1. The canonical extension \(D^{\sigma}\) of a bounded distributive lattice \(D\) is the free completely distributive lattice generated by \(D\). The authors give a simple proof of this result. In addition the following results are proved. Theorem 2. If \(A\) is a bounded sublattice of \(C\), both are completely distributive, and \(A\) completely generates \(C\), then \(A\) is a complete retract of \(C\). Proposition. Suppose \(B\) is a complete sublattice of a Raney lattice \(C\). If \(B\) and \(C\) are Boolean, or are chains, then \(B\) is a complete retract of \(C\). The following open problem is posed. Problem: Characterize when a complete lattice \(A\) that is a bounded sublattice of a completely distributive lattice \(C\) is a complete retract.