On the lattice of convex sublattices of a lattice (Q5906527)

Let \(L\) be a lattice and \(CS(L)\) be the family of all convex sublattices of \(L\) including the empty set. Then \(CS(L)\), partially ordered by inclusion, forms an atomic algebraic lattice. \textit{K. M. Koh} investigated some basic concepts and results of \(CS(L)\), and raised the following problem [Nanta Math. 5, No. 3, 18-37 (1972; Zbl 0284.06006)]. Problem 1. Is \(L\) a chain if \(CS(L)\) is the lower semimodular lattice \((\text{LSM}_ 2)\)? \textit{G. Grätzer} posed the following open problem [General lattice theory (1978; Zbl 0385.06015), p. 56]. Problem 2. For what equational classes \(K\) of lattices do \(L\in K\) and \(CS(L) \cong CS(L_ 1)\) imply \(L_ 1\in K\)? The aim of this note is to discuss the two problems.