On the lattice of convex sublattices of a lattice (Q5917473)

In Algebra Univers. 35, 63-71 (1996; Zbl 0842.06003), \textit{S. Lavanya} and \textit{S. Parameshwara Bhatta} considered a partial ordering on the set \(\text{CS}(L)\) of all the convex sublattices of a lattice \(L\), namely \(A\leq B\) if and only if \((A]\subseteq (B]\) and \([A)\supseteq [B)\), and begun a corresponding study. The present paper continues this study. We list here some of the results: Theorem 1. This ordering is the smallest ordering on \(\text{CS}(L)\) extending the usual ordering on \(I(L)\) (ideals) and \(D(L)\) (filters) and satisfying an additional necessary condition (named (G)). Theorem 4. \(\text{CS}(L)\) is semimodular if \(I(L)\) and \(D(L)\) are semimodular; if \(L\) is of finite length then the converse also holds. Theorem 11. Let \(L\) be \ complete lattice. Then \(L\) is pseudocomplemented if and only if CS\((L)\) is pseudocomplemented. Reviewer's remark: In the above-mentioned paper, the equivalence in Theorem 11 was proved only for upper continuous lattices.